Barbara Gawrońska-Szklarz

Andrzej Siuda

Mateusz Kurzawski

Dariusz Bielicki

Wojciech Marlicz

Marek Droździk
email: drozdzik@sci.pam.szczecin.pl

Effects of CYP2C19, MDR1, and interleukin 1-B gene variants on the eradication rate of Helicobacter pylori infection by triple therapy with pantoprazole, amoxicillin, and metronidazole

Keywords: CYP2C19, MDR1, Interleukin-1β, Genetic polymorphism, Peptic ulcer, H, pylori, Pharmacogenetics B

Introduction

Eradication of H. pylori is an important treatment strategy in a variety of upper gastrointestinal disorders, such as peptic ulcer, chronic gastritis, and mucosa-associated lymphoid tissue lymphoma. Current regimens of eradication consist of proton pump inhibitor (PPI) and two antibiotics (amoxicillin and either clarithromycin or metronidazole). A PPI is a key drug in H. pylori eradication, as it makes antibiotics more stable by raising intragastric pH [START_REF] Grayson | Effect of varying pH on the susceptibility of Campylobacter Eliopoulos pylori to antimicrobial agents[END_REF]. Additionally, a neutral level of intragastric pH allows bacteria to go into the growth phase, and become more sensitive to antibiotics [START_REF] Scott | The life and death of Helicobacter pylori[END_REF]. The PPIs are metabolized in liver by the cytochrome P450 (CYP) system. The principal enzyme involved in the metabolism is CYP2C19, and to a lesser extent, CYP3A4 [START_REF] Andersson | Slow omeprazole metabolizers are also poor Smephenytoin hydroxylators[END_REF][START_REF] Yamazaki | Different contributions of cytochrome P450 2C19 and 3A4 in the oxidation of omeprazole by human liver microsomes: effects of contents of these two forms in individual human samples[END_REF]. Pantoprazole in metabolized via CYP2C19 to demethylated pantoprazole and via CYP3A4 to pantoprazole sulfone [START_REF] Barone | Comparative pharmacology of proton pump inhibitors[END_REF][START_REF] Touw | Clinical implications of genetic polymorphisms and interactions mediated by cytochrome P-450 enzymes[END_REF].

There are interindividual differences in CYP2C19 activity, mainly due to genetic polymorphism: subjects carrying two defective CYP2C19 alleles are poor metabolizers (PMs) of enzyme substrates [START_REF] Kupfer | Pharmacogenetics of mephenytoin: a new drug hydroxylation polymorphism in man[END_REF]. One common variant allele is responsible for the great majority of PM phenotypes in Caucasians (CYP2C19*2), while inactive CYP2C19*3 is additionally found in Asians. It was demonstrated that PPI-based eradication therapy is more effective, not only in PMs, but also in intermediate metabolizers (IM), heterozygotes with one wild-type and one defective CYP2C19 allele, compared with wild-type homozygotes (rapid extensive metabolizers) [START_REF] Gawronska-Szklarz | Effect of CYP2C19 and MDR1 polymorphisms on cure rate in patients with acidrelated disorders with Helicobacter pylori infection[END_REF][START_REF] Sugimoto | Treatment strategy to eradicate Helicobacter pylori infection: impact of pharmacogenomics-based acid inhibition regimen and alternative anitbiotics[END_REF][START_REF] Furuta | Effect of genotypic differences in CYP2C19 on cure rates for Helicobacter pylori infection by triple therapy with a proton pump inhibitor, amoxicillin, and clarithromycin[END_REF]. In PMs and IMs mean plasma levels of PPIs are higher, what makes therapy with standard drugs' doses more effective compared with wild-type homozygotes [START_REF] Shao | Blood concentration of pantoprazole sodium is significantly high in hepatogenic peptic ulcer patients, especially those with a poor CYP2C19 metabolism[END_REF]. Recently, a novel CYP2C19 gene variant (CYP2C19*17), associated with increased gene transcription, and thus faster metabolism of CYP2C19 substrates was described [START_REF] Sim | A common novel CYP2C19 gene variant causes ultrarapid drug metabolism relevant for the drug response to proton pump inhibitors and antidepressants[END_REF]. Contrary to defective CYP2C19*2 and *3 alleles, CYP2C19*17 is likely to cause therapeutic failures in treatment with PPIs. However, limited clinical data are available on the CYP2C19*17 allele's cure rate of H. pylori eradication [START_REF] Sim | A common novel CYP2C19 gene variant causes ultrarapid drug metabolism relevant for the drug response to proton pump inhibitors and antidepressants[END_REF][START_REF] Kurzawski | Effect of CYP2C19*17 gene variant on Helicobacter pylori eradication in peptic ulcer patients[END_REF], contrary to the large number of studies suggesting higher cure rates in CYP2C19*2 and CYP2C19*3 carriers.

The MDR1 gene product P-glycoprotein (P-gp) is probably one of the most important and best defined transporters for drug disposition in humans. P-gp transports a wide range of substrates with diverse chemical structures, among them proton pump inhibitors [START_REF] Hoffmeyer | Functional polymorphisms of the human multidrug-resistance gene: multiple sequence variations and correlation of one allele with P-glycoprotein expression and activity in vivo[END_REF][START_REF] Pauli-Magnus | Interaction of omeprazole, lansoprazole and pantoprazole with Pglycoprotein[END_REF]. P-gp is expressed in the apical membrane of cells with excretory functions such as the liver, kidney, small intestine, stomach, the blood-brain barrier, and functions as an efflux pump for lipophilic and amphipathic substrates. It was documented that MRD1 3435C>T polymorphism is a good predictor of P-gp activity, with 3435TT subjects having low transporter activity [START_REF] Hoffmeyer | Functional polymorphisms of the human multidrug-resistance gene: multiple sequence variations and correlation of one allele with P-glycoprotein expression and activity in vivo[END_REF]. The 3435TT MDR1 genotype has been associated with successful PPI treatment (most cases omeprazole medicated) in patients with acid-related disorders with H. pylori infection [START_REF] Gawronska-Szklarz | Effect of CYP2C19 and MDR1 polymorphisms on cure rate in patients with acidrelated disorders with Helicobacter pylori infection[END_REF].

Interleukin-1β (IL-1β) is not only an amplifier of the immune response, but also a potent inhibitor of stomach acid secretion [START_REF] El-Omar | The importance of interleukin 1beta in Helicobacter pylori associated disease[END_REF]. Studies carried out in Caucasian and Asian populations have demonstrated that the IL-1B polymorphism is associated with an increased risk of hypochlorhydria and gastric carcinoma as well as peptic ulcer disease [START_REF] El-Omar | Interleukin-1 polymorphisms associated with increased risk of gastric cancer[END_REF][START_REF] Chakravorty | Interaction between IL1B gene promoter polymorphisms in determining susceptibility to Helicobacter pylori associated duodenal ulcer[END_REF]. Scarce data were provided for the role of IL-1B polymorphism and treatment outcome of peptic ulcer disease. Furuta et al. and Take et al. demonstrated that IL-1beta-511 polymorphism was one of the determinants of successful eradication of H. pylori using triple therapy with PPI, amoxicillin, and clarithromycin, together with the CYP2C19 genotype [START_REF] Furuta | Polymorphism of interleukin-1beta affects the eradication rates of Helicobacter pylori by triple therapy[END_REF][START_REF] Take | Interleukin-1beta genetic polymorphism influences the effect of cytochrome P 2C19 genotype on the cure rate of 1-week triple therapy for Helicobacter pylori infection[END_REF]. The observations from Asian populations suggest that the IL-1B polymorphism can influence the treatment outcome of peptic ulcer disease. However, any data from Caucasian populations are available. The IL-1B gene displays many SNPs among which polymorphisms at the position +3954 influence the rate of production of the cytokine. Considering the +3954 IL-1B polymorphism, homozygous individuals for the T allele produce a four-fold higher amount of IL-1β than individuals displaying the C⁄C genotype. Heterozygous individuals have intermediary production of IL-1β [START_REF] Pociot | A TaqI polymorphism in the human interleukin-1 beta (IL-1 beta) gene correlates with IL-1 beta secretion in vitro[END_REF].

In the light of the aforementioned data the study was aimed at the evaluation of an association of CYP2C19, MDR1, and IL-1B polymorphisms' with the H. pylori eradication rate in Polish Caucasian patients treated with triple therapy of pantoprazole, amoxicillin, and metronidazole.

Materials and methods

Patients of Caucasian origin with Polish nationality were included in the study after giving informed consent. The local Ethics Committee of the Pomeranian Medical University, Szczecin, Poland approved the protocol of the study. A total of 139 unrelated patients (74 female, 65 male, mean age 51.0±14.4 years, range 20-80 years) suffering from peptic ulcer, positive for H. pylori infection, were included in the study. All subjects underwent physical examination, gastroduodenoscopy, and laboratory tests (rapid urease test, histopathological examination). The patients exhibited normal liver function according to clinical laboratory tests. None of the subjects consumed extensive amounts of alcohol or had a smoking habit. The patients were treated with triple therapy (pantoprazole 40 mg×2 per day, amoxicillin 1,000 mg×2 per day, and metronidazole 500 mg×2 per day) for 1 week, then only 40 mg pantoprazole a day was given for 5 weeks. All the patients confirmed that they fully complied with physicians' instructions. Negative results on all examinations, including gastroduodenoscopy, rapid urease test, histopathological examination, and 13 C-urea breath test at 6 weeks after initiation of the therapy were judged indicative of H. pylori eradication. Patients were subsequently divided into two groups, i.e., group 1, patients with successful eradication after the first cycle of the therapy, and group 2, patients who failed the first cycle of the treatment.

Blood for pantoprazole concentration assay was sampled 3 h post-dose during a control visit at 6 weeks from the beginning of the study, centrifuged for plasma separation and then stored at -20°C till analysis. Plasma pantoprazole concentrations were measured using the HPLC method published by Storms and Stewart [START_REF] Storms | Stability-indicating HPLC assays for the determination of prilocaine and procaine drug combinations[END_REF]. The detection limit of the method (LOQ) was set at 0.5 μg/ml.

Genomic DNA was extracted from peripheral leukocytes using a non-organic and non-enzymatic extraction method [START_REF] Gustincich | A fast method for high-quality genomic DNA extraction from whole human blood[END_REF]. Genotyping of each subject for the presence of common CYP2C19 variants (19154G>A, specific for *2, and 17948G>A, for *3 allele) was performed using previously described a PCR/RFLP assay [START_REF] Goldstein | Genetic tests which identify the principal defects in CYP2C19 responsible for the polymorphism in mephenytoin metabolism[END_REF]. Two single nucleotide polymorphisms (SNPs), specific for CYP2C19*17 alleles, were analyzed by PCR-RFLP (-3402C>T) and allele-specific amplification methods (-806C>T), described by Sim et al. [START_REF] Sim | A common novel CYP2C19 gene variant causes ultrarapid drug metabolism relevant for the drug response to proton pump inhibitors and antidepressants[END_REF].

The MDR1 C3435T polymorphism was identified by the PCR-RFLP method as described by Cascorbi et al. [START_REF] Cascorbi | Frequency of single nucleotide polymorphisms in the Pglycoprotein drug transporter MDR1 gene in white subjects[END_REF]. Digestion with the restriction enzyme Sau3AI resulted in the classification of evaluated genotypes into the following three groups: 3435 CC, CT, and TT.

The IL-1B 3954C>T polymorphism was assessed using the PCR-RFLP method by Hodge at al. [START_REF] Hodge | Failure to detect an association with IL1 genotypes in European Caucasians with generalised early onset periodontitis[END_REF]. Based on the TaqI digestion pattern the patients were classified into the following three genotype groups: 3954 CC, CT, and TT.

Allele and genotype frequencies were determined by the direct counting of the alleles. Concordance of genotype distribution with the Hardy-Weinberg equilibrium and associations between categorical variables were assessed by Fisher's exact test, using Statistica 6.0 (Statsoft). The study with 139 patients has over 80% power to detect differences in allele frequency between the groups stratified according to treatment outcome if it exceeds 14%, under the assumption that the treatment success rate is 75% and the minor allele frequency in the population is 15%. Odds ratios and confidence intervals were calculated using the Newcombe-Wilson method without continuity correction. The effect of the CYP2C19, MDR1, and IL-1B genotypes as well as treatment regimens on eradication of H. pylori infection after the first cycle of triple therapy was evaluated using univariate and multivariate logistic regression analysis. The multivariate analysis was adjusted for age, gender, and smoking status. The Kruskal-Wallis test was applied to evaluate an influence of CYP2C19 genotypes on pantoprazole plasma concentration. A p value less than 0.05 was considered significant.

Results

Based on haplotype estimation analysis, two SNPs specific for the CYP2C19*17 allele (-3402C>T and -806C>T) were found in complete linkage in the studied population. On the other hand, SNPs distinctive for the *17 and *2 alleles occurred separately, none of the patients homozygous for the *17 allele carried the 19154G>A SNP. The genotype frequency distribution was in concordance with the Hardy-Weinberg equilibrium. No significant differences in the frequency or distribution of the CYP2C19 allele (allele *3 was not detected, Table 1) and genotypes (Table 2) were found between the two groups of patients (i. e., with those with successful H. pylori eradication and those with treatment failure). However, no carriers of a defective CYP2C19*2/*2 genotype were found among the patients with treatment failure (Table 2). Mann-Whitney test; unfortunately, we were not able to measure pantoprazole concentrations in *2/*2 patients), and the lowest in the case of *17/*17 (Fig. 1).

Similar to CYP2C19, the MDR1, and IL-1B genotypes were not found to be significantly associated with success or failure H. pylori eradication. Distribution of allele and genotype frequencies were similar in both subgroups of the study (Tables 3,4).

Univariate and multivariate analysis of the evaluated genotypes, i.e., CYP2C19, MDR1, and IL-1B did not reveal any significant association between the genotypes and the H. pylori eradication rate (Table 5).

Discussion

The incidence of peptic ulcer disease in the Polish population is estimated to be 5-10%, with the prevalence of duodenal ulcer (60%) [START_REF] Schabowski | Is there a territorial differentiation in the prevalence of peptic ulcer among rural population?[END_REF] being an important medical pathology. In the present study, a potential association between CYP2C19, MDR1, and IL-1B polymorphisms and H. pylori eradication efficacy in Polish Caucasian patients has been analyzed. It is well documented that the CYP2C19 polymorphism affects cure rates of peptic ulcer. Patients harboring defective alleles CYP2C19*2 and CYP2C19*3 are characterized by higher rates of successful H. pylori eradication, when omeprazole/amoxicillin/clarithromycin or pantoprazole/amoxicillin/metronidazole were administered [START_REF] Gawronska-Szklarz | Effect of CYP2C19 and MDR1 polymorphisms on cure rate in patients with acidrelated disorders with Helicobacter pylori infection[END_REF][START_REF] Furuta | Effect of genotypic differences in CYP2C19 on cure rates for Helicobacter pylori infection by triple therapy with a proton pump inhibitor, amoxicillin, and clarithromycin[END_REF]. Therapy including amoxicillin and clarithromycin are gastric pH-dependent, while the effects of metronidazole are independent [START_REF] Grayson | Effect of varying pH on the susceptibility of Campylobacter Eliopoulos pylori to antimicrobial agents[END_REF][START_REF] Narikawa | Distribution of metronidazole susceptibility factors in obligate anaerobes[END_REF]. Our observations support the published data, as all carriers of CYP2C19*2/*2 genotypes responded to the administered triple therapy of pantoprazole, amoxicillin, and metronidazole. The data for hyperactive allele CYP2C19*17 are very scarce. In our previous study, which included mainly patients medicated with omeprazole, we did not document any association between H. pylori eradication and CYP2C19*17 [START_REF] Kurzawski | Effect of CYP2C19*17 gene variant on Helicobacter pylori eradication in peptic ulcer patients[END_REF]. Our current observations are in keeping with the former report, i.e., no effect of CYP2C19*17 allele on the cure rate in patients medicated with pantoprazole (along with amoxicillin and metronidazole). However, the study revealed that the CYP2C19 polymorphism influences significantly the metabolic rate of pantoprazole, as defective allele carriers showed highest drug concentrations, with the lowest for the hyperactive CYP2C19*17 allele. This finding is in line with data reported by Sim et al. [START_REF] Sim | A common novel CYP2C19 gene variant causes ultrarapid drug metabolism relevant for the drug response to proton pump inhibitors and antidepressants[END_REF], who estimated that CYP2C19*17 homozygotes would attain up to 40% lower omeprazole AUC values than subjects homozygous for the CYP2C19*1 allele after a single dose of the drug. Similar to our observations, higher pantoprazole concentrations in poor CYP2C19 metabolizers have been reported by others [START_REF] Shao | Blood concentration of pantoprazole sodium is significantly high in hepatogenic peptic ulcer patients, especially those with a poor CYP2C19 metabolism[END_REF].

Limited data indicate that the MDR1 3435C>T polymorphism affects PPI (lansoprazol) pharmacokinetics, but no marked PPI pharmacodynamic effects (i.e., intragastric pH) of the polymorphism were determined [START_REF] Kodaira | Effect of MDR1 C3435T polymorphism on lansoprazole in healthy Japanese subjects[END_REF]. There are few data on the association of MDR1 polymorphism with H. pylori eradication. Furuta at al. found that the polymorphism of MDR1 was one of the determinants of successful eradication of H. pylori by the triple therapy with lansoprazole, amoxicillin, and clarithromycin, together with the CYP2C19 genotype and bacterial susceptibility to clarithromycin, with the lowest response rate in 3435TT carriers [START_REF] Furuta | Effect of MDR1 C3435T polymorphism on cure rates of Helicobacter pylori infection by triple therapy with lansoprazole, amoxicillin and clarithromycin in relation to CYP2C19 genotypes and 23 S rRNA genotypes of H. pylori[END_REF]. In our previous study, similar to observations by Furuta et al. [START_REF] Furuta | Effect of MDR1 C3435T polymorphism on cure rates of Helicobacter pylori infection by triple therapy with lansoprazole, amoxicillin and clarithromycin in relation to CYP2C19 genotypes and 23 S rRNA genotypes of H. pylori[END_REF], lower cure rates were noted in MDR1 3435TT H. pylori-positive peptic ulcer patients treated predominantly with omeprazole as the PPI component of triple therapy, with the MDR1 polymorphism being found to be an independent risk factor [START_REF] Gawronska-Szklarz | Effect of CYP2C19 and MDR1 polymorphisms on cure rate in patients with acidrelated disorders with Helicobacter pylori infection[END_REF]. However, the results of the present study did not confirm the significant role of the MDR1 gene polymorphism in H. pylori eradication in patients medicated with pantoprazole, amoxicillin, and metronidazole.

The last genotype evaluated in the present study was functional IL-1B 3954C>T. Take et al. [START_REF] Take | Interleukin-1beta genetic polymorphism influences the effect of cytochrome P 2C19 genotype on the cure rate of 1-week triple therapy for Helicobacter pylori infection[END_REF] reported that IL-1B-511 genetic polymorphism, although not an inde-pendent factor in the treatment outcome of peptic ulcers (PPI applied: omeprazole, lansoprazole, or rabeprazole), influenced the impact of the CYP2C19 genotype on the cure rate of 1-week triple therapy for H. pylori infection. Logistic regression analysis of clinical study by Furuta et al. [START_REF] Furuta | Polymorphism of interleukin-1beta affects the eradication rates of Helicobacter pylori by triple therapy[END_REF] showed that the IL-1B-511C>T polymorphism, as well as the CYP2C19 genotype of patients and the clarithromycin resistance of H. pylori, was associated with successful eradication. Eradication rates for H. pylori, the highest among carriers of the -511 T/T genotypes of IL-1B. Therefore, the IL-1B-511 polymorphism could be one of the determinants of the successful eradication of H. pylori using triple therapy with a PPI (omeprazole or lansoprazole), amoxicillin, and clarithromycin. In the present study we evaluated the effects of different functional SNPs of IL-1B 3954C>T of pantoprazolebased triple therapy on the H. pylori eradication rate in a Caucasian population (contrary to the reports cited above from Asian populations). The IL-1B 3954C>T is functional as it was documented that monocytes from homozygous subjects carrying allele 2 of the IL-1B gene produced significantly more IL-1β than those from heterozygotes [START_REF] Pociot | A TaqI polymorphism in the human interleukin-1 beta (IL-1 beta) gene correlates with IL-1 beta secretion in vitro[END_REF]. Our results did not replicate the observations of Take et al. and Furuta et al., i.e., no significant association of the IL-1B polymorphism and H. pylori eradication in pantoprazole-treated patients was documented. As a summary of our study we performed multivariate analysis, and did not reveal any significant association between the genotypes, i.e., CYP2C19, MDR1, and IL-1B and the H. pylori eradication rate in Polish Caucasian peptic ulcer patients receiving pantoprazole, amoxicillin, and metronidazole. However, other factors may also be responsible for a failure to eradicate H. pylori, i.e., the high incidence of H. pylori resistance to metronidazole (30-40% in Poland) [START_REF] Boyanova | The status of antimicrobial resistance of Helicobacter pylori in eastern Europe[END_REF], an antimicrobial agent included in triple therapy of studied patients, or undetermined drug interactions. It should also be stated that the effects of the CYP2C19 and MDR1 polymorphisms, which influence PPI pharmacokinetics, may be more prominent in patients medicated with PPI/amoxicillin/clarithromycin than in those with PPI/amoxicillin/metronidazole, as the antimicrobial efficacy of amoxicillin and clarithromycin depend on gastric pH, contrary to metronidazole [START_REF] Grayson | Effect of varying pH on the susceptibility of Campylobacter Eliopoulos pylori to antimicrobial agents[END_REF].

On the basis of our results, one can conclude that the CYP2C19 genotype, contrary to the MDR1 and IL-1B genotypes, may have an impact on the efficacy of H. pylori eradication in Polish Caucasian peptic ulcer patients receiving pantoprazole, amoxicillin, and metronidazole.

  Classification of patients into subgroups of extensive, EM, (genotypes *1/ *1, *1/*17, *17/*17); intermediate, IM, (*1/*2, *2/*17); and poor, PM, metabolizers (*2/*2) did not reveal significant differences in cure rates. Eradication was observed in 70.8% (95% CI 61.1-79.2) EM, 83.9% (95% CI 66.3-94.5) IM, and 100% (95% CI 15.8-100) PM patients. Failure of treatment was demonstrated in 29.2% (95% CI 20.8-38.9) EM, 16.1% (95% CI 5.5-33.7) IM, and 0% (95% CI 0-84.2) PM subjects. Pantoprazole plasma concentrations were evaluated in 24 patients: 6 carriers of *1/*1 genotype, 7-*1/*2, 6-*1/*17, 3-*2/*17, and 3-*17/*17. The values of the drug concentrations were as follows: *1/*1-1.18±0.30 μg/ml, *1/*2-2.28±1.15 μg/ml, *1/*17-1.60±0.69 μg/ml, *2/*17-0.97± 0.56 μg/ml, and *17/*17-0.59±0.50 μg/ml. The concentrations of pantoprazole among the studied genotypes differ significantly (p=0.030, Kruskal-Wallis test). The highest concentrations were observed in heterozygous subjects, carriers of CYP2C19*1/*2 genotypes (p=0.073 vs *1/*1,

Fig. 1

 1 Fig. 1 Pantoprazole plasma concentrations (mean, SD) 3 h after administration of 40 mg pantoprazole in patients with different CYP2C19 genotypes

Table 1

 1 CYP2C19 allele frequencies in peptic ulcer patients (n=139)

	CYP2C19 allele	Group 1, successful H. pylori eradication (n=103),	Group 2, failure of H. pylori eradication (n=36),	p value
		n (%; 95% CI)	n (%; 95% CI)
	*1 (n=165)	117 (56.80; 49.73-63.66)	48 (66.67; 54.57-77.34)	0.16
	*2 (n=35)	30 (14.56; 10.05-20.13)	5 (6.94; 2.29-15.74)	0.10
	*17 (n=78)	59 (28.64; 22.57-35.33)	119 (26.39; 16.70-38.10)	0.76

Table 3

 3 MDR1 3435C>T genotype frequencies in peptic ulcer patients (n=139)

	MDR1 3435C>T genotype Group 1, successful H. pylori eradication (n=103),	Group 2, failure of H. pylori eradication (n=36),	p value
		n (%; 95% CI)	n (%; 95% CI)
	3435CC (n=44)	33 (75.0; 59.66-86.81)	11 (25.0; 13.19-40.34)	0.99
	3435CT (n=61)	45 (73.77; 60.93-84.20)	16 (26.23; 15.80-39.07)
	3435TT (n=34)	25 (73.53; 55.64-87.12)	9 (26.47; 12.88-44.36)

Table 4

 4 IL-1B 3954C>T genotype frequencies in peptic ulcer patients (n=139)

	IL-1B 3954C>T	Group 1, successful H. pylori eradication (n=103), n (%;	Group 2, failure of H. pylori eradication (n=36), n	p
	genotype	95% CI)	(%; 95% CI)	value
	3954TT (n=80)	56 (70.00; 58.72-79.74)	24 (30.00; 20.26-41.28)	0.40
	3954CT (n=47)	38 (80.85; 66.74-90.85)	9 (19.15; 9.15-33.26)
	3954CC (n=12)	9 (75.00; 42.81-94.51)	3 (25.0; 5.49-57.19)

Table 5

 5 Univariate and multivariate analysis for factors predicting success of cure of H. pylori infection after the first cycle of triple therapy

	Factor	Univariate analysis OR	p value a	Multivariate analysis b OR	p value a
		(95% CI)		(95% CI)
	CYP2C19*2/*2+*1/*2+*2/*17 vs *1/*1+*1/*17+*17/*17	2.32 (0.85-6.37)	0.10	1.99 (0.68-5.83)	0.20
	MDR1 3435 C carriers vs T/T	0.96 (0.40-2.33)	0.93	0.98 (0.39-2.45)	0.97
	IL-1B 3954 T/T vs C carriers	1.68 (0.75-3.74)	0.20	1.53 (0.66-3.54)	0.32

a No statistically significant differences were documented b Adjusted for age, gender, and smoking status

Acknowledgements The authors highly appreciate the statistical assistance of Dr. Krzysztof Safranow.


H Schmidt

W Gruber

Crystallization Kinetics of Amorphous Si-C-N Ceramics: Dependence on Nitrogen Partial Pressure

Keywords: crystallization, kinetics, precipitation Crystallization, kinetics, precipitation, X-ray diffractometry

The crystallization kinetics of amorphous precursor-derived ceramics of composition Si 26 C 41 N 33 is investigated as a function of temperature and nitrogen partial pressure using X-ray diffractometry. Isothermal annealing at a pressure of 1 bar leads to simultaneous crystallization of Si 3 N 4 and SiC, while for annealing at a reduced pressure of 1 mbar crystalline SiC is formed only. Rate constants of crystallization are determined using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) formalism. For temperatures below 1700 °C crystallization rates are significantly higher for annealing at 1 mbar than for 1 bar, respectively. For an explanation of the results a model is proposed, which is based on diffusion controlled nucleation and growth of crystalline Si 3 N 4 and SiC in an amorphous matrix combined with thermal decomposition of Si 3 N 4 at high temperatures.

Introduction

Non-oxide ceramics of the system Si-C-N, which are derived from pre-ceramic polymers by solid state thermolysis became a field of growing research interest during the last years [START_REF]Precursor-Derived Ceramics[END_REF][START_REF] Bill | [END_REF][3][4][5][6][START_REF] Jansen | High Performance non-oxide ceramics[END_REF].

The resulting materials are free of sinter additives and exhibit a good high-temperature stability and oxidation resistance making them attractive for applications in various branches of technology. The underlying aim of this processing route is to tailor inorganic ceramics on the basis of molecular units in order to control the structure and properties of such a material on an atomic scale.

After thermolysis, Si-C-N ceramics form an amorphous non-equilibrium state, which is stable up to about 1500 °C. Investigations with small and wide angle X-ray and neutron scattering [START_REF] Haug | [END_REF]9] as well as nuclear magnetic resonance experiments [10] proved a phase separation of the material on the scale of 1 nm, where amorphous Si 3+0.25x C x N 4-x (x = 0 -4) domains and graphite-like amorphous carbon domains with sp 2 hybridised structural units coexist [START_REF] Haug | [END_REF]9]. At a temperature of 1500 °C and above the material starts to crystallize, while different types of composites (e.g. nano-/micro-crystalline) are formed [4]. Several publications on polymer-derived ceramics of type Si-C-N can be found in the literature, dealing with synthesis [3,5,6,[START_REF] Jansen | High Performance non-oxide ceramics[END_REF], microstructure [START_REF] Haug | [END_REF][9][10][11], thermochemistry [12,13], oxidation resistance [14][15][16], electronic properties [17,18], and mechanical properties [19,20]. However, quantitative studies on the crystallization kinetics of these materials are very limited. In preceding research work [21,22], we investigated the crystallization kinetics of Si-C-N at a nitrogen partial pressure of 1 bar. We found that microcrystalline α-Si 3 N 4 and nano-crystalline SiC are simultaneously formed. Crystallization was described according to the Johnson-Mehl-Avrami-Kolmogorov (JMAK) formalism, assuming a three-dimensional, diffusion controlled grain growth with a time dependent nucleation rate. The determined rate constants of crystallization are temperature dependent, but equal for both crystallized phases and can be described approximately by an Arrhenius behaviour with an activation enthalpy of about 120.4 kJ/mol (12.5 eV) [21].

In the present study, we investigated the influence of reduced nitrogen partial pressure on crystallization and precipitation behaviour of the model system Si 26 C 41 N 33 in order to get information on an optimized stability of the amorphous state for technological applications and to obtain basic information for the production of tailor-made microstructures. We also present a model explaining the pressure dependence of crystallization kinetics in these materials

Experimental details

Bulk samples under investigation were produced by isostatic pressing and subsequent thermolysis of cross-linked pre-ceramic polymers at 1050 °C in an argon atmosphere. The polyvinylsilazane VT 50 (Hoechst AG, Germany) was used to synthesize amorphous ceramics with a porosity of about 5 %. A detailed description of the preparation procedure can be found in Ref. [3]. The composition was determined by Rutherford backscattering analysis to be Si 26 C 41 N 33 . To obtain defined conditions for X-ray investigations the bulk samples were cut into platelets of about 8 x 8 x 2 mm 3 , polished with diamond paste (particle size: 15 µm, 6 µm, 3 µm, and 1 µm) and cleaned with ethanol in an ultrasonic bath.

To observe the crystallization process, the ceramics were annealed in nitrogen at 1000 mbar and alternatively in a nitrogen/argon mixture with a nitrogen partial pressure of 1 mbar in the temperature range between 1450 and 1700 °C. To avoid decomposition and contamination with oxygen during annealing, the samples were placed in a crucible and embedded in powder of the same material. The investigations with X-ray diffractometry were carried out with a SIEMENS

Results

Annealing of as-thermolyzed amorphous Si-C-N samples in the temperature range between 1450 -1700 °C in nitrogen leads to crystallization. For anneals carried out at a nitrogen partial pressure of 1 bar a mixture of crystalline Si 3 N 4 (grain diameter > 100 nm) and crystalline SiC (grain diameter ~ 30 -40 nm) is precipitated, as shown in Ref. [21]. For temperatures above 1600 °C only SiC and below 1535 only Si 3 N 4 is formed [21]. In contrast, all the XRD measurements of this study revealed, that the samples annealed at a reduced nitrogen partial pressure of 1 mbar between 1450 and 1700 °C show crystallization of SiC only. This behaviour can be explained considering the phase diagram of the system Si-C-N. Fig. 1 shows theoretically calculated isothermal sections at ambient nitrogen partial pressure according to Ref. [23] valid up to 1484 °C (a) and between 1484 and 1841 °C (b). The crystalline phases Si 3 N 4 , SiC, and C form a threephase equilibrium, which is stable up to 1484 °C. The amorphous material is located on the tieline between Si 3 N 4 and C. During annealing a de-mixing of the metastable amorphous material is expected, resulting in crystalline composite material consisting of Si 3 N 4 and C (graphite).

However, TEM and XRD measurements indicate that carbon stays amorphous [START_REF] Bill | [END_REF]21]. In contrast, at temperatures above 1484 °C the phase equilibria change and a three-phase area between SiC, C and gaseous N 2 is now stable wherein the starting material is located. The transition from the phase equilibrium shown in Fig. 1 where Si 3 N 4 reacts with free carbon to SiC and nitrogen gas. From these theoretical considerations annealing of the amorphous ceramics at temperatures higher than 1484 °C should result in the formation of crystalline material composed of SiC by simultaneous loss of gaseous N 2 [23]. However, X-ray analysis shows that up to 1600 °C always a mixture of Si 3 N 4 and SiC is formed. With growing temperature, the fraction of α-Si 3 N 4 decreases and at the same time the fraction of SiC increases, until at 1645 °C only SiC is present. In the present composites, crystalline Si 3 N 4 seems to be stabilized, at least partly, to higher temperatures due to the embedding in an amorphous matrix of Si-C-N (for details see Ref. [21]). Fig. 2 shows the equilibrium temperature of reaction (1) as a function of nitrogen partial pressure as received by thermodynamic calculations according to Ref. [12]. As obvious the equilibrium temperature is shifted to a lower temperature of 1049 °C for a pressure of 1 mbar. This explains why during the experiments with reduced partial pressure SiC crystallizes only.

For a detailed understanding of the crystallization process X-ray diffraction studies were carried out as a function of annealing time. First, the amorphous material was isothermally annealed at a distinct temperature for a given time. Then the material was characterized with XRD and afterwards annealed again. This procedure was repeated until the X-ray diffractograms did not change any more during further annealing, meaning that the degree of crystallization of the material remained constant. A typical example for the time dependent growth of a SiC X-ray peak is shown in Fig. 3. The fraction of crystallized phase, χ i , is calculated from the integrated intensity of the X-ray peak. A detailed description of the procedure can be found in Ref. [21].

An characteristic example for the dependence of the crystallized fraction on annealing time at 1500 °C is illustrated in Fig. 4 for experiments at both partial pressures under study. As obvious annealing at a reduced partial pressure leads to accelerated crystallization of SiC. The crystallization process based on a nucleation and growth process is analysed using Johnson, Mehl, Avrami, and Kolmogorov theory (JMAK-theory) [24]. The crystallized volume fraction of a specimen after time t is given by ( )

n ) t k ( exp 1 ) t ( - - = χ , (2)
where, k is the temperature dependent rate constant of crystallization and n is the JMAK exponent. The experimentally determined volume fractions of crystallized SiC and Si 3 N 4 were analysed as a function of annealing time at different annealing temperatures in terms of Eq. ( 2).

Fitting of the data yields JMAK exponents between n = 2 -2.5, corresponding to a threedimensional diffusion controlled growth with significant nucleation during the growth process (for details see Ref [21]). In Fig. 5 the determined rate constants of crystallization are plotted as a function of reciprocal temperature.

On base of the actual findings and of literature data, the crystallization process can be visualized as follows. At low temperatures and high nitrogen partial pressure silicon nitride directly crystallizes from the amorphous silicon nitride domains by a nucleation and growth process. For higher temperatures and lower nitrogen partial pressure, in the phase-separated amorphous state amorphous silicon nitride domains react with the surrounding carbon phase to amorphous SiC. In a second step, amorphous SiC transforms to crystalline SiC embedded in an amorphous residual matrix by a polymorphous transformation, if the critical nucleus size is reached.

Discussion

Data analysis is carried out on base of the classical nucleation and growth theory, where the rate constant of crystallization is given by [24] (3) where k 0 is the pre-exponential factor, ∆H D is the activation enthalpy of diffusion, ∆Γ is the difference in the interfacial energy between amorphous domains and amorphous/crystalline interfaces and finally ∆G is the difference of the Gibbs energy between the crystalline composite and the amorphous ceramic, the driving force of crystallization. For the crystallization of SiC, we subdivide the last quantity into two contributions

        ∆ ∆ -         ∆ - = T k T G Γ T k H k k B B D 2 ) ( 3
∆G = ∆G c (T) + ∆G r (T, P) (4)
a reaction part, ∆G r , describing the Gibbs energy which is released due to reaction (1) and a crystallization part, ∆G c , describing the Gibbs energy difference between crystalline and amorphous SiC. Further, the components of the Gibbs energy are given by

∆G c = ∆H c -T ∆S c (5)
and ∆G r = ∆H r -T ∆S r + 2k B T ln(P/P 0 ),

respectively. Here, ∆H i (i = r, c) is the enthalpy change and ∆S i the entropy change of the process and P 0 = 1bar. In contrast, for the crystallization of Si 3 N 4 no reaction part is present because the crystallites can be directly formed within the amorphous silicon nitride domains.

For crystallization of SiC, the crystallization part of the precipitation process given by Eq. ( 5) is given according to Ref. [25] in good approximation by ∆G c ≈ ∆H c ≈ const. The entropy contribution is very small and Gibbs energy is approximately independent of temperature at the temperatures under investigation. We take from Ref. [26] a Gibbs energy of ∆G c = -162 kJ for the crystallization of three mole of SiC, which is identical to -54 kJ/(mol SiC). This term simply describes the difference in Gibbs energy between amorphous and crystalline silicon carbide. The first two terms of the reaction part in Eq. ( 6) were calculated at 1 bar by the FACT-Web/Reaction-Web program [27] for reaction (1) to be ∆G r = ∆H r -T ∆S r = 524.9 kJ -306.2 J/K

• T. The last term in Eq. ( 6) can be obtained directly from the experimental conditions. It is zero for annealing at 1 bar and -114.8 J/K • T for annealing at 1 mbar, respectively. Note that the data describing the Gibbs energy refer to equation (1) and are not per mole.

In Eq. ( 3) also the activation enthalpy of self-diffusion for the slowest moving and consequently rate determining constituent in amorphous Si-C-N enters. Experiments on self-diffusion of Si in amorphous VT50-derived Si-C-N ceramics, and of Si, C and N in similar amorphous Si-(B-)C-N ceramics were carried out by our group with stable isotopes by means of secondary ion mass spectrometry [28][29][30]. No large differences between the diffusivities of the three elements were measured (about one order of magnitude) indicating a common diffusion mechanism where all elements contribute nearly equally to atomic re-arrangement processes [30]. The results show a thermally activated behaviour with an activation enthalpy of ∆H D = 5.7 eV (549.0 kJ/mol) for the VT50-derived material [29].

In order to compare our experimental data to our model, the data given above are inserted in Eqs.

(3-6). The only unknown parameters are the difference in interfacial energy, ∆Γ, and the preexponential factor, k 0 , which are obtained by a least-squares fitting procedure of the data obtained at 1 bar. The result is given in Fig. 5. We obtain a quite reasonable value of ∆Γ ≈ 0.5 J/m 2 assuming a density of 1.2 x 10 5 mol/m 3 of Si 0.26 C 0.41 N 0.33 . A considerable lower value of 0.2 J/m 2 is for example found for the crystallization of amorphous silicon [31]. For the description of crystallization at a pressure of 1 mbar the difference of 2k B •T ln(P/1bar) = -114.8

J/K • T is subtracted from the Gibbs energy difference, ∆G, while ∆Γ remains constant. Here, also a good description of the experimentally derived rate constants is obtained (see Fig. 5). The only adjustable parameter is the pre-exponential factor, which changes slightly from 6 x 10 13 s -1 for 1 bar to 3 x 10 12 s -1 for 1 mbar. We see that the fittings perfectly describe the rate constants of SiC precipitation for anneals at 1 bar and 1 mbar, respectively.

As already mentioned, for the crystallization of Si 3 N 4 no reaction part is present because the crystallites can be directly formed within amorphous silicon nitride domains. Further, we assume also a temperature independent crystallization part [25], which is given by ∆G c = -69 kJ [32] for the crystallization of 1 mole of Si 3 N 4 . The corresponding model curve is also shown in Fig. 5.

The rate constants of crystallization of Si 3 N 4 are also described very well for the same difference in interfacial energy as found for the crystallization of silicon carbide.

In Ref. [13] the enthalpy of formation of amorphous Si-C-N ceramics from crystalline Si 3 N 4 , C and a tiny amount of SiC was studied at 25 °C by calorimetric measurements. The determined enthalpies lie between -9.2 and + 12.7 kJ/(mol Si x C y N z ) (x + y +z = 1) within error limits for samples with similar composition (S3 and S4 in Ref [13]). Keeping in mind that in our case about 0.08 mol of crystalline Si 3 N 4 has to be consumed for the formation of 1 mol of amorphous Si 0.26 C 0.41 N 0.33 , the calorimetric measurements correspond to enthalpy values ranging from -115 to +159 kJ/(mol Si 3 N 4 ). This is not contradictive to our assumption of -∆G c = 69 kJ /(mol Si 3 N 4 ).

As also obvious from Fig. 5, silicon nitride and silicon carbide crystallize approximately with the same rate constant in the temperature range investigated. According to our analysis, in case of Si 3 N 4 an Arrhenius straight line is present and in case of SiC a slightly curvature has to be present, which however cannot be resolved by the experimental data. The question whether the similarity of the rate constants of both crystalline phases and their temperature is the product of accidentally coincidence or whether an unknown mechanism is present, e. g. a coupling of the crystallization process of SiC and Si 3 N 4 , has to be clarified in future.

F

  diffractometer in the θ/2θ modus using CoK α radiation (40 kV, 40 mA) on bulk ceramics.

  (a) to that shown in Fig. 1(b) can be expressed by the chemical reaction Si 3 N 4 + 3C 3SiC + 2N 2 ,

Figure 1 :

 1 Figure 1:Isothermal sections of the Si-C-N system at a total pressure of 1 bar (after Ref[23]).

Figure 2 :

 2 Figure 2: Calculated equilibrium temperature of reaction (1) as a function of temperature

Figure 3 :

 3 Figure 3: Exemplary illustration of the isothermal growth of a SiC X-ray peak at 1650 °C. The

Figure 4 :

 4 Figure 4: Normalized volume fraction of crystallized SiC as a function of annealing time at

Figure 5 :

 5 Figure 5: Rate constants of crystallization for SiC and Si 4 N 4 as a function of reciprocal

  Fig. 2 209x297mm (600 x 600 DPI)

  Fig. 3 209x297mm (600 x 600 DPI)

  Fig. 4 209x297mm (600 x 600 DPI)

  Fig.5 209x297mm (600 x 600 DPI)

  In this study we investigated crystallization kinetics of amorphous Si 26 C 41 N 33 ceramics as a function of annealing temperature and pressure. We found that a reduced nitrogen partial pressure suppresses the crystallization of silicon nitride and accelerates the crystallization of SiC below 1700 °C. A model based on diffusion controlled nucleation and growth is presented, where the lower rate constants at a reduced partial pressure can be explained by an increase of the driving force of crystallization. A more negative Gibbs free energy difference between the crystalline and amorphous state is caused by the pressure dependence of the decomposition reaction, Eq. (1).

	5. Conclusion
	o r F o r F o r F o r
	P P P P
	e e e e
	e r e r e r e r
	R R R R
	e e e e
	v i e v i e v i e v i e
	w w w w
	O n l O n l O n l O n l
	y y y y

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

The authors thank P. Gerstel (MPI Stuttgart) for supply of the precursor ceramics and E. Ebeling for ceramographic preparation. Funding by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged.


Hoby Andrianirina

Anne-Laure Foucher

Hoby Andrianirina -Anne

Laure Foucher

Blended learning through the eyes of Malagasy students

Keywords: blended learning, experience of students, didactics French as a foreign language, CMC, mediation, collaboration task

whether they are published or not. The documents may come

The aim of this paper is to report on the impressions and practices of Malagasy learners on their first experience of French at University Level in a blended learning environment. This is part of a wider action research study in the Didactics of Languages and Cultures focusing on learner practices and providing help to language trainers in designing blended learning courses. The premise here is that design should be based on real knowledge of the practices.

A blended setup is characterized by face-to-face teaching linking to distance work and also by collective distance activity with online interaction (Mangenot, 2010). The blending brings two factors into play: the human or mediating factor and the CMC factor (Peraya, 1999).

Furthermore the collaborative work is defined as a collective knowledge building approach where the group plays a supporting and motivational role (Henri et Lundgren-Cayrol, 1998).

The blended course in our study brought into contact, in the first semester of 2008-2009, thirty Malagasy students, non-specialists in French and twelve students, future teachers of French from our university for ten contact teaching and three distance learning sessions.

The two groups were asked to work on two collaborative macro-tasks. The human mediation was done in the contact sessions by the coordinating teachers and the paired Malagasy students and in the distance sessions by the French student pairs. The computer-mediated activity was supported by a learning environment and in particular by the two communication tools: chat and forums. The computer-mediated character of the course was given a special dimension, though, due to the fact that 70% of the Malagasy students had no experience of ICT and that computer provision was rather inadequate.

To know how the Malagasy students perceived the blended nature of the environment and the resulting practices, we will draw on elicited data (Van der Maren, 1999): on the one hand the replies to the final evaluation questionnaire and on the other the computer-mediated exchanges in the chat sessions and forums and data from video recordings made during a session. Generally, the Malagasy students expressed satisfaction: the computer-mediated session conditions were not an obstacle to learning and the "composite" mediation by the coordinators and the French and Malagasy pairs was appreciated. This perception, however, needs to be questioned or nuanced. On examining the interactions and practices, we find many explicit or implicit references to their computer-mediated working conditions in the direction of the French students (who had better facilities) and quite creative ways of adapting to the characteristics of the computer-mediated activities.

The Malagasy students also expressed a feeling of frustration. To conclude, even if the computer-mediated conditions were not perceived as problematic by the Malagasy students, the imbalance between the distant groups did pose a problem and did impact the mediation with the French pairs. This leads us to suggest that in a collaborative approach, the blended nature of a course setup should be thought more in terms of interdependence for the human mediation and technological balance than of performance with the computer-mediated tools.


Jean-Franc ¸ois

Rault Lmpa

Joseph Liouville

A BIFURCATION FOR A GENERALIZED BURGERS' EQUATION IN DIMENSION ONE

Keywords: 2000 Mathematics Subject Classification. Primary: 35A01, 35B32, 35K55 ; Secondary: 35B44 Bifurcation, Existence of solution, Blow-up, Phase plane. 1

We consider the generalized Burgers' equation 8 < :

, Ω a subdomain of R, and where B(u) = 0 denotes some boundary conditions. First, using some phase plane arguments, we study the existence of stationary solutions under the Dirichlet or the Neumann boundary conditions and prove a bifurcation depending on the parameter λ. Then, we compare positive solutions of the parabolic equation with appropriate stationary solutions to prove that global existence can occur when B(u) = 0 stands for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions σ∂tu + ∂ν u = 0. Finally, for many boundary conditions, global existence and blow up phenomena for solutions of the nonlinear parabolic problem in an unbounded domain Ω are investigated by using some standard super-solutions and some weighted L 1 -norms.

Introduction.

Let Ω be a domain of the real line R, not necessarily bounded. Let p be a real number with p > 1, λ ∈ R and ϕ a non-negative continuous function in Ω. Consider the following nonlinear parabolic problem

   ∂ t u = ∂ 2 x u -u∂ x u + u p -λu in Ω for t > 0, B(u) = 0 on ∂Ω for t > 0, u(•, 0) = ϕ in Ω, (1)
where B(u) = 0 stands for the Dirichlet boundary conditions (u = 0), the Neumann boundary conditions (∂ ν u = 0) or the dynamical boundary conditions (σ∂ t u + ∂ ν u = 0 with σ a non-negative smooth function). For the local existence of the positive solutions of this problem, we refer to von Below and Mailly's results [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF] and references therein, [START_REF] Bandle | Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up[END_REF], [START_REF] Below | Dynamical interface transition in ramified media with diffusion[END_REF] and [START_REF] Escher | Quasilinear parabolic systems with dynamical boundary conditions[END_REF] . In the first section, we study the stationary equation u -uu + u|u| p-1 -λu = 0 [START_REF] Bandle | Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up[END_REF] stemming from Problem [START_REF] Amann | Ordinary differential equations[END_REF]. We aim to prove the existence of positive and signchanging solutions using phase plane arguments and dealing with the first order system

u v = v uv -u|u| p-1 + λu . (3)
We prove a bifurcation in the phase plane of this system, depending on the parameters λ and p, which influences the resolution of Equation ( 2) under the Dirichlet, the Neumann and the mixed boundary conditions. Then in a second section, using the comparison method from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF], we deduce from the solutions of the stationary Equation ( 2) some regular super-solutions for the Problem [START_REF] Amann | Ordinary differential equations[END_REF]. Dealing with these super-solutions and with the blow-up results from [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF], we investigate global existence and blow-up phenomena for the Problem [START_REF] Amann | Ordinary differential equations[END_REF] for different values of λ and p, and for the Dirichlet, the Neumann and the dynamical boundary conditions. We also examine both phenomena in unbounded domains: we obtain global existence results with the comparison method and using some well-known super-solutions (we mean explicit functions) for the Dirichlet, the Neumann and the dynamical boundary conditions. The blowing-up concerns the regular solutions of Problem (1) satisfying some growth order at infinity and some boundary conditions such that

• ∂ ν u = 0 (Neumann b.c.), • ∂ ν u = g(u)
with g a polynomial of degree 2 (nonlinear b.c.).

We use some weighted L 1 -norms: our technique is to prove the blowing-up of the solution by proving the blowing-up of appropriate L 1 -norms.

Before starting, let us define the kind of solution we look for:

Definition 1.1. A function u is called a solution (or regular solution) of Equation [START_REF] Bandle | Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up[END_REF] in Ω if u is of class C 2 (Ω) and satisfies the equation in the classical sense. A function u is called a solution (or regular solution) of Problem [START_REF] Amann | Ordinary differential equations[END_REF] in Ω if u is of class C(Ω × [0, T )) ∩ C 2,1 (Ω × (0, T )) and satisfies the equations of Problem [START_REF] Amann | Ordinary differential equations[END_REF] in the classical sense in Ω × [0, T ) where T ∈ (0, ∞] denotes the maximal existence time of the solution u.

2. Stationary equation. In this section, we study the existence of positive and sign-changing solutions of Equation (2) using a phase plane method. Unless otherwise stated, we suppose p ∈ (1, ∞). For the theory of phase planes (nature of equilibrium, regularity, behaviour and uniqueness of trajectories), we refer to H.Amann's book [START_REF] Amann | Ordinary differential equations[END_REF]. Here we consider the system (u , v

) t = F (u, v) with a C 1 (R 2 , R 2 ) function F given by F (u, v) = (v, uv -u|u| p-1 + λu) t
, thus uniqueness and regularity (C 1 )of the solutions (u, v) come from the standard ODE's theorems. With v = u , we deduce that u is of class C 2 . First, we can note that System (3) has three equilibrium points if λ > 0: (0, 0), (λ 1 p-1 , 0) and (-λ 1 p-1 , 0). Using Hartman-Grobman's linearization theorem (see Reference [START_REF] Amann | Ordinary differential equations[END_REF]), we can state that (0, 0) is a saddle point, (λ

1 p-1 , 0) is an unstable and repulsive vortex (if 1 -4(p -1)λ p-3 p-1 < 0), an unstable node (if 1 -4(p -1)λ p-3
p-1 ≥ 0, which degenerates when 1 -4(p -1)λ p-3 p-1 = 0 ). And (-λ 1 p-1 , 0) is a stable and attractive vortex (for 1 -4(p -1)λ p-3 p-1 < 0), a stable node (for 1 -4(p -1)λ p-3 p-1 ≥ 0 with degeneracy when equality occurs). If λ ≤ 0, then (0, 0) is the only equilibrium point of System (3). We will prove later that (0, 0) is a center.

2.1.

Case λ > 0. Let λ be a positive real number and p > 1. We want to study the phase plane of the System (3). First we prove a lemma on the symmetry of the trajectories:

Lemma 2.1. The support of the trajectories of the System (3) are symmetric with respect to the ordinates axis.

Proof. Let (u, v) denote a solution of the System (3) in (-a, a) for some a ∈ (0, ∞], and define

w(x) = -u(-x) z(x) = v(-x)
for all x ∈ (-a, a).

A simple calculus of the derivatives implies

w (x) = u (-x) = v(-x) = z(x), and
z (x) = -u(-x) = -v (-x) = -u(-x)v(-x) -u(-x)|u(-x)| p-1 + λu(-x) = w(x)z(x) -w(x)|w(x)| p-1 + λw(x).
Then (w, z) is also a trajectory of the System (3), and it is symmetric to (u, v) with respect to the ordinates axis.

Thus, we can reduce our phase plane analysis to the half plane R + × R. In order to draw the phase plane of the System (3), we write the ordinate v as a function depending on the abscissa u: v = f (u). We do not know the function f , but we can deduce its variations and convexity using the equations (3). For the variations, when v = 0, we have

dv du = uv -u|u| p-1 + λu v = u v v -|u| p-1 + λ , (4)
in particular, it vanishes along the axis {u = 0} and along the curve {v = |u| p-1 -λ}.

For u < λ

1 p-1 , we have dv du v=0 = ∞ whereas for u > λ 1 p-1 dv du v=0 = -∞.
Then we have dv du > 0 in the sets {u > 0, v > 0, v > |u| p-1 -λ} and {u > 0, v < 0, v < |u| p-1 -λ}. On the other hand, dv du < 0 in the sets {u > 0, v < 0, v > |u| p-1 -λ} and {u > 0, v > 0, v < |u| p-1 -λ}. Next, we compute the convexity of the function f and we obtain

d 2 v du 2 = 1 + 1 v 2 (λ -p|u| p-1 )v -u(λ -|u| p-1 ) dv du . (5)
We have

d 2 v du 2 < 0 in {u > 0, v > 0, v < |u| p-1 -λ} and d 2 v du 2 > 0 in {u > 0, v < 0, v < |u| p-1 -λ}. Since d 2 v du 2 u=0 = 1 + λ v and d 2 v du 2 v=|u| p-1 -λ = (1 -p) |u| p-1 v ,
the convexity is sign-changing in {u > 0, v > |u| p-1 -λ}. These arguments are sufficient to know the profile of the trajectories in the half plane {v < |u| p-1 -λ}.

We do not need to know how the trajectories behave in {u > 0, v < 0, v > |u| p-1 -λ} to solve Equation [START_REF] Bandle | Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up[END_REF]. In {u > 0, v > 0, v > |u| p-1 -λ}, things are different: unbounded trajectories can appear (see §2.3). To ensure the occurrence of bounded trajectories, we need an additional hypothesis:

p ≥ 3. ( 6
)
Lemma 2.2. Under hypothesis [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF], all the trajectories of the System (3) are bounded in

A = {u > 0, v > 0, v > |u| p-1 -λ}.
Proof. Let v 0 > 0 and consider (u, v) the solution of the System (3) with initial data (u(0), v(0)) = (0, v 0 ). The calculus of the variations (see Equation ( 4)) ensures that (u(t), v(t)) ∈ A for small t > 0. We prove that there exist 0

< τ < ∞ such that v(τ ) = |u(τ )| p-1 -λ. It means that (u, v) is bounded in A. Since (u, v) belongs to A, we have dv du = u + λu v - u|u| p-1 v ≤ u + λu v .
Then dv du ≥ 0 in A implies v > v 0 as long as (u, v) ∈ A, and we obtain

dv du ≤ u 1 + λ v 0 . Integration gives v ≤ 1 2 1 + λ v 0 u 2 + v 0 . If p > 3, the intersection {v = |u| p-1 -λ} ∩ {v = 1 2 1 + λ v0 u 2 + v 0 } is non-empty for all v 0 > 0. If p = 3, we need to choose v 0 sufficiently big such that 1 2 1 + λ v 0 < 1.
Then, the trajectory (u, v) belongs to the compact

{u ≥ 0, v ≥ |u| p-1 -λ, v ≤ 1 2 1 + λ v 0 u 2 + v 0 },
and, using dv du ≥ 0, we know that there exist 0 < τ < ∞ such that v(τ ) = |u(τ )| p-1λ. This argument proves that each solution of the System (3) with initial data (u(0), v(0)) = (0, v 0 ) is bounded in A if v 0 is big enough. Thanks to uniqueness of solution, it also proves the result for all the solutions initiated in A.

Then, we complete this phase plane analysis by proving the existence of periodic trajectories.

Lemma 2.3. Assume that hypothesis (6) is fulfilled. Then, there exists periodic trajectories of the System (3). Proof. Thanks to the symmetry (see Lemma 2.1), we just need to prove that for some initial data belonging to {0} × (0, ∞), there exists a trajectory which attains a point belonging to {0} × (-∞, 0). First, consider a trajectory (u, v) initiated at (0, v 1 ) with v 1 > 0. According to hypothesis (6), we know that (u, v) is bounded, and using its variations and its convexity (Equations ( 4) and ( 5)), we can deduce that (u, v) attains the x-axis at a point (u 1 , 0) with u 1 > λ 1 p-1 (see Figure 1). Then, using the reverse system

u v = -v -uv + u|u| p-1 -λu ,
and one of its trajectories initiated at (0, v 2 ) with v 2 < -λ) (trajectories of reverse system and of System (3) have same support), one can note that for u 0 > λ 1 p-1 , there exists a trajectory (w, z) of (3) with w(0) = u 0 and z(0) = 0 (see Figure 1). Finally, let us consider the trajectory (a, b) of System (3) containing the point (u 2 , 0), where u 2 > max{u 0 , u 1 }. Thanks to the uniqueness of the solutions, and using the information on the variations and the convexity, we deduce that there exist two real numbers s < t such that a(s) = a(t) = 0, b(s) = v 0 and b(t) = v 3 (see Figure 1). Thus, the trajectory (a, b) is the periodic trajectory we look for. Now, analysing the phase plane of the System (3), we deduce the following results concerning the Equation (2).

Theorem 2.4. Assume hypothesis (6) and λ > 0. For each boundary conditions

• u(-α) = u(α) = 0 (Dirichlet b.c.) , • u (-α) = u (α) = 0 (Neumann b.c.) , • u(-α) = u (α) = 0 (mixed-1 b.c.), • u (-α) = u(α) = 0 (mixed-2 b.c.
), there exists a positive solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (-α, α) for some α > 0.
Proof. We use the phase plane of System (3), see Figure 1. Consider the trajectory (a, b) between the points • (0, v 0 ) and (0, v 3 ): we obtain the Dirichlet solution,

• (0, v 0 ) and (u 2 , 0): we obtain the mixed-1 solution,

• (u 2 , 0) and (0, v 3 ): we obtain the mixed-2 solution.

For the Neumann solution, consider 0 < µ 0 < λ 1 p-1 and the trajectory (µ, ν) of System (3) initiated at (µ 0 , 0). Since (µ, ν) can not cross the trajectory (u, v) ( see Figure 1), it must cross the x-axis at (µ 1 , 0) with λ 1 p-1 < µ 1 < u 1 . Thus, the abscissa of this trajectory is the Neumann solution we look for. Finally, the length (2α) of the existence interval is governed by the time needed by the trajectory to go from its initial data to its "final data".

Theorem 2.5. Assume hypothesis [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF] and λ > 0. For some α > 0, there exists a periodic sign-changing solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in R.
Proof. We just need to choose one of the periodic trajectories of the System (3) built in Lemma 2.3.

Remark 1. Using the periodic solutions in the previous theorem, and restricting them to some suitable subintervals (non-trivial), we can build four sign-changing solutions satisfying the four boundary conditions: Dirichlet, Neumann, mixed-1 and mixed-2 (see Theorem 2.4). Now, suppose that hypothesis ( 6) is not achieved. Then, we do not know if the solutions are bounded in {v > |u| p-1 -λ}: we will see in §2.3 that unbounded solutions appear. But in {v < |u| p-1 -λ}, the behaviour of the trajectories do not change.

Theorem 2.6. Let λ > 0. For some α > 0, there exists a positive solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (-α, α)
with the mixed boundary conditions u (-α) = u(α) = 0. In addition, if

1 -4(p -1)λ p-3 p-1 < 0, (7)
then there exists a positive solution of the Equation ( 2) under the Neumann boundary conditions.

Proof. The first part of the statement comes from Theorem 2.4, the solution with mixed-2 boundary conditions is located in {v < |u| p-1 -λ}. The other part stems from Equation ( 7): in this case, the equilibrium (λ

1 p-1 , 0) is an unstable vortex. If we consider u 0 > 0 such that |λ 1 p-1 -u 0 | is sufficiently small, the trajectory (u, v) of the System (3), with u(0) = u 0 and v(0) = 0, whirls around (λ 1 p-1 , 0). Thus, there exists τ > 0 such that v(τ ) = 0 and u(t) > 0 for all t ∈ [0, τ ].
Without hypothesis (6), we can not construct positive solutions anymore for the Dirichlet, Neumann or mixed-1 boundary conditions. If we do not impose the positivity, we obtain this result: Theorem 2.7. Let λ > 0. For each boundary conditions

• u (-α) = u (α) = 0 (Neumann b.c.) , • u(-α) = u (α) = 0 (mixed-1 b.c.),
there exists a solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (-α, α) for some α > 0.
Proof. As we mentioned before, we consider the part {v < |u| p-1 -λ} of the phase plane of the System (3) (see Figure 1). For the Neumann solution, we consider the trajectory (a, b) between (u 2 , 0) and (-u 2 , 0). For the mixed-1 solution, we can also consider the trajectory (a, b), but only between (0, v 3 ) and (-u 2 , 0). Remark 2. The Neumann solution built above is sign changing, whereas the mixed-1 solution is negative.

Remark 3. In the general case, we can not build any solution with the Dirichlet boundary conditions using our phase plane method. Indeed, we will give a criterion in Theorem 2.18 concerning nonexistence of the Dirichlet solution.

Concerning the solutions in infinite interval, we can state: Theorem 2.8. Let λ > 0. Then the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 admits • a positive solution u in (-∞, 0] satisfying u (-∞) = u (0) = 0 (Neumann). • a positive solution v in (-∞, 0] satisfying v (-∞) = v(0) = 0 (mixed-2). • a sign-changing solution w in R satisfying w (-∞) = w (∞) = 0 (Neumann). • a negative solution u in [0, ∞) satisfying z(0) = z (∞) = 0 (mixed-1). Proof. Consider µ 0 > 0 with µ 0 > λ 1 p-1 and with |λ 1 p-1 -µ 0 | small enough such that there exists a trajectory (µ, ν) of the System (3) satisfying µ(-∞) = λ 1 p-1 , ν(-∞) = 0 and µ(0) = µ 0 , ν(0) = 0.
Since (λ 1 p-1 , 0) is repulsive, the existence of (µ, ν) is clear. Hence, u = µ in (-∞, 0] is suitable for the first statement. Then, the trajectory (µ, ν) can be continued in the part {u > 0, v < 0} using the information on its behaviour (see Equations ( 4) and ( 5)) until (µ, ν) attains the ordinate axis. Denote t 1 > 0 the time such that µ(t 1 ) = 0 and ν(t 1 ) < 0. We obtain the second statement setting v(t) = µ(t + t 1 ) for all t ∈ (-∞, 0]. Finally, these results and the symmetry of the trajectories (see Lemma 2.1) imply the third and the fourth statements with the following definitions:

w(t) = v(t) ∀ t ≤ 0 -v(-t) ∀ t > 0 and z(t) = -v(-t) for all t ≥ 0.
2.2. Case λ ≤ 0. First note that the System (3) has only one equilibrium point (0, 0). As in the previous case, we can reduce our phase plane analysis to the halfplane R + ×R since Lemma 2.1. Again, we obtain some information on the variations of the trajectories of the System (3) using Equation ( 4). We have dv du = 0 along the curves {u = 0} and {v = |u| p-1 -λ}. For u > 0

dv du v=0 = -∞ whereas for u < 0 dv du v=0 = +∞.
Then, we have dv du ≥ 0 in {u > 0, v < 0} ∪ {v ≥ |u| p-1 -λ} and dv du ≤ 0 in {u > 0, v > 0, v ≤ |u| p-1 -λ}. In addition, thanks to Equation ( 5), we know that

d 2 v du 2 ≤ 0 in {u > 0, v > 0, v ≤ |u| p-1 -λ}, d 2 v du 2 ≥ 0 in {u > 0, v < 0} while it is sign-changing in {u > 0, v ≥ |u| p-1 -λ}.
In this last part of the plane, we use the following lemma, similar to Lemma 2.2: Lemma 2.9. Let λ ≤ 0 and (u, v) be a trajectory of the System (3) with initial data (0, v 0 ).

If v 0 > -λ satisfies v 0 > -λ if p ≥ 3 , v 0 ≤ -λ + (p -1) p-1 3-p -1 2 (p -1) 2 3-p if p < 3 , ( 8
)
then the trajectory (u, v) is bounded in A = {u > 0, v ≥ |u| p-1 -λ}.
Proof. The calculus of the variations (see Equation ( 4)) ensures that (u(t), v(t)) ∈ A for small t > 0. We prove that there exists 0

< τ < ∞ such that v(τ ) = |u(τ )| p-1 -λ.
It means that (u, v) is bounded in A. Since (u, v) belongs to A and thanks to λ ≤ 0, we have

0 ≤ dv du = u + λu v - u|u| p-1 v ≤ u.
Then, integration between 0 and u gives

v ≤ 1 2 u 2 + v 0 . Hypothesis (8) implies that {u > 0, v = |u| p-1 -λ} ∩ {u > 0, v = 1 2 u 2 + v 0 } is not empty. Thus, the trajectory (u, v) belongs to the compact {u ≥ 0, v ≥ |u| p-1 -λ, v ≤ 1 2 u 2 + v 0 }.
Using dv du ≥ 0, we know that there exist τ > 0 such that v(τ ) = |u(τ )| p-1 -λ. Now, the phase plane of the System (3) can be drawn, see Figure 2. Theorem 2.11. Let λ ≤ 0. For all α > 0, the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (-α, α)
admits no positive solution under the Neumann boundary conditions.

Proof. Ab absurbo, suppose that there exists u a positive solution of (2) under the Neumann boundary conditions, and denote v = u . Then the curve (u, v) is a trajectory of the System (3) located in R + × R with initial data on the axis {v = 0}. Then Equations ( 4) and ( 5) prove that (u, v) can not cross the axis {v = 0} once again without going into R -× R. A contradiction with the positivity of u.

Theorem 2.12. Let λ ≤ 0. For some α > 0, the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (-α, α)
admits a sign-changing solution under the Neumann boundary conditions.

Proof. Using the phase plane of System (3) (see Figure 2), consider the trajectory (a, b) between the points (a 0 , 0) and (-a 0 , 0).

To conclude this section, let us give this result concerning the periodic solutions:

Theorem 2.13. Let λ ≤ 0. For some α > 0, there exists a sign-changing periodic solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in R.
Proof. As in Lemma 2.3, we can build periodic trajectories of (3) using the symmetry (Lemma 2.1).

2.3. Unbounded solutions. In the above paragraphs, we proved that all the trajectories of the System (3) are bounded for p ≥ 3, but if 1 < p < 3 we do not have a general answer: for example, we obtain some bounded trajectories when λ ≤ 0 (see Lemma 2.9), but with our method, we do not have (yet) any result when λ > 0. In this paragraph, we show that there exists unbounded trajectories for every λ ∈ R and for all p ∈ (1, 3). We start with a trajectory (u, v) with an initial data (0, v 0 ).

Lemma 2.14. Let p ∈ (1, 3) and λ ∈ R. Suppose that

v 0 > 2 max{-λ, 0} + 2 • 8 p-1 3-p . ( 9
)
Then the trajectory (u, v) is not bounded.

Proof. We will show that under hypothesis (9), the trajectory (u, v) always lies above the curve v = 2u p-1 + 2 max{-λ, 0} . Thus, using dv du ≥ 0 (Equation ( 4)), we obtain that (u, v) is not bounded. Ab absurdo, suppose that there exists x * > 0 such that u(x * ) = u 1 > 0 and v(x * ) = v 1 > 0 satisfy

v 1 = 2u p-1 1 + 2 max{-λ, 0}, (10)
and

v(x) > 2u(x) p-1 + 2 max{-λ, 0} ∀ x ∈ [0, x * ).
Thus in [0, x * ), we have

λ -u p-1 v > - 1 2 . ( 11
)
On the other hand, Equation (4) gives

dv du = u + u λ -u p-1 v ,
and thanks to condition (11), we obtain

dv du ≥ 1 2 u ≥ 0. (12)
Then v(u) ≥ u 2 4 + v 0 . Hence, for u = u 1 , we have:

v 1 = v(u 1 ) ≥ u 2 1 4 + v 0 ,
and by definition (10) of u 1 , we have

2u p-1 1 + 2 max{-λ, 0} ≥ u 2 1 4 + v 0 .
Hypothesis (9) implies

-2 • 8 p-1 3-p > u 2 1 4 -2u p-1 1 . (13)
Meanwhile, if we study both cases u 1 < 8 1 3-p and u 1 > 8 1 3-p , we remark that

u 2 1 4 -2u p-1 1 = u p-1 1 4 u 3-p 1 -8 ≥ -2 • 8 p-1 3-p . (14)
Equations ( 13) and ( 14) are not compatible. Thus, the trajectory (u, v) can not attain the curve v = 2u p-1 + 2 max{-λ, 0} .

Concerning Equation (2), we obtain the following results:

Theorem 2.15. Let p ∈ (2, 3) and λ ∈ R. For some α > 0, there exists a positive and unbounded solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (-α, α). satisfying u(-α) = 0 and lim x→α u(x) = ∞.
Proof. The existence comes from the previous lemma. We just need to prove that the length of the existence interval is finite. Ab absurdo, suppose that there exists a positive and unbounded solution u of the Equation ( 2) in [0, ∞). Let b > 0 such that u > |2λ| 1 p-1 in [b, ∞), and define w(x, t) = u(x + t) for all x ∈ [b, b + 1] and for all t ∈ [0, ∞). Thanks to the choice of b, we have

∂ 2 x u -u∂ x u + u p -λu ≥ ∂ 2 x u -u∂ x u + u p 2 in [b, b + 1] × [0, ∞).
Because the solution u corresponds to a trajectory of the System (3) located in R × R + , we have ∂ t w = ∂ x u > 0. Thus, w is super-solution of the following problem

   ∂ t v = ∂ 2 x v -v∂ x v + 1 2 v p in [b, b + 1] × (0, ∞), ∂ t v + ∂ ν v = 0 on {±b} × (0, ∞), v(•, 0) = |2λ| 1 p-1 in [b, b + 1].
By the comparison principle from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF], w ≥ v where v is the solution of the previous problem. But, according to [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF], the solution v blows up in finite time. Since w ≥ v, this contradicts the global existence of w. Thus, w can not exist on [b, b+1]×(0, ∞), and the solution u exists only in a finite interval. For 1 < p ≤ 2, we do not have the blowing-up argument and we are not sure that the existence interval of the solution is finite. With some assumption on the parameter λ, we can also build an unbounded trajectory (u, v) with an initial data (u 0 , 0) belonging to the abscissa axis. Lemma 2.17. Let p ∈ (1, 3) and λ ∈ R + . Suppose that there exists β > 1 such that

λ > max β -1 2β 2β 2 β -1 2 3-p , β 2β 2 β -1 p-1 3-p (15)
If

u 0 = 2β 2 β -1 1 3-p , ( 16
)
then the trajectory (u, v) is not bounded.

Proof. We use the same method as in Lemma 2.14: we prove that, under hypotheses (15) and ( 16), the trajectory (u, v) always lies above the curve {v = βu p-1 -λ}.

Ab absurdo, suppose that there exist x * > 0 such that u(x * ) = u 1 and

v 1 = v(x * ) verify v 1 = βu p-1 1 -λ, ( 17
) and v(x * ) > βu(x * ) p-1 -λ ∀ 0 < x < x * .
Thus, in [0, x * ), we have

λ -u p-1 v ≥ -1 β . ( 18
)
Equation ( 4) gives

dv du = u + u λ -u p-1 v ,
and condition (18

) implies dv du ≥ β -1 β u ≥ 0.
Integration between u 0 and u 1 leads to

v(u 1 ) ≥ β -1 2β (u 2 1 -u 2 0 ), definition (17) gives βu p-1 1 -λ ≥ β -1 2β (u 2 1 -u 2 0 ),
and we obtain

u p-1 1 1 - β -1 2β 2 u 3-p 1 ≥ 1 β λ -u 2 0 β -1 2β . ( 19
)
Since u 0 < u 1 , Equations ( 15) and ( 16) imply

λ -u 2 0 β -1 2β > 0 and 1 - β -1 2β 2 u 3-p 1 < 0.
Hence, Equation ( 19) is a contradiction.

Concerning Equation ( 2), and reasoning as in Theorem 2.15, we obtain the following result.

Theorem 2.18. Let p ∈ (1, 3) and λ ∈ R verifying Equation (15). For some α ∈ (0, ∞], there exists a positive and unbounded solution of the Equation ( 2)

u -uu + u|u| p-1 -λu = 0 in (0, α). satisfying u (0) = 0 and lim x→α u(x) = ∞.
In addition, if p ∈ (2, 3), then α is finite.

2.4.

Limiting case p = 1. In this paragraph, we study the case where the exponent p attains the limit 1. Then, Equation (2) becomes

u -uu + (1 -λ)u = 0 in R,
and the System (3) is written

u v = v u(v + λ -1) . ( 20
)
For λ = 1, (0, 0) is the only equilibrium point, while for λ = 1 the axis {v = 0} is a continuum of equilibria. We begin with the case λ = 1. Here, we have dv du = u, then

v(u) = 1 2 u 2 + c,
where c depends on the initial data. Thus, the phase plane is easily drawn, see Figure 3. Now, suppose λ = 1. One can compute the explicit trajectory

u e (x) = (1 -λ)x v e (x) = (1 -λ) ∀ x ∈ R
Then, using the following equations

dv du = u + u v (λ -1) and d 2 v du 2 = 1 + + λ -1 v 2 v -u dv du
we can draw the phase plane of the System (20), see Figure 3. 2.5. Bifurcation. According to the previous paragraphs, we can state that there exists a bifurcation of the phase plane of the System (3). First, we note that, for a fixed exponent p, the value of λ influences the phase plane of the System (3): for λ > 0, the System (3) admits three equilibrium points (a saddle point, an attractive equilibrium and a repulsive equilibrium). The distance between these equilibria goes to 0 when λ → 0, and for λ = 0, they collapse and generate a unique center, which persists for all negative λ (see Figure 4). Now, for a fixed λ, the value of the exponent p has an important role. With λ, the value of p governs the type of the equilibrium points (node, improper node, vortex). The exponent p also establishes if all the trajectories of the System (3) are bounded (p ≥ 3) or if there exists unbounded trajectories (1 ≤ p < 3). Moreover, when p attains the limit 1, the critical value of λ changes from 0 (if p > 1) to 1 (for p = 1). The case λ = 1 is special because when p → 1, the three equilibria of the System (3) (a saddle point, an attractive vortex and a repulsive vortex) generate a continuum of equilibria when p attains the limit 1 (see Figure 5). 3. Parabolic problem. In this section, we study the positive solutions of the parabolic Problem (1) for many boundary conditions. First, we use the results concerning the stationary Equation (2) when the domain Ω is bounded. Then, we consider the case of unbounded domains: we investigate global existence using the comparison method, and blow-up phenomenon thanks to a L 1 -norm technique.

3.1. Comparison. We begin with the Dirichlet problem

   ∂ t u = ∂ 2 x u -u∂ x u + u p -λu in [-α, α] × (0, ∞), u = 0 on {±α} × (0, ∞), u(•, 0) = ϕ in [-α, α], (21)
where

α > 0, p > 1, λ ∈ R and ϕ ∈ C 0 ([-α, α]
) is non-negative. Thanks to the comparison principle [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF] and with the results of the previous sections, we have:

Theorem 3.1. Let p > 1 and λ ∈ R. For some α > 0, there exists a global positive solution u ∈ C([-α, α] × [0, ∞)) ∩ C 2,1 ([-α, α] × (0, ∞)) of Problem (21) if the initial data ϕ ∈ C 0 ([-α, α]) is sufficiently small. Proof. If p ≥ 3 and λ > 0, consider β ∈ C 2 ([-α, α]
) a solution of (2) with the Dirichlet boundary conditions (see Theorem 2.4). Suppose that ϕ is small enough:

ϕ ≤ β in [-α, α]. Then, we obtain    ∂ t β = 0 = ∂ 2 x β -β∂ x β + β p -λβ in [-α, α] × (0, ∞), β = 0 on {±α} × (0, ∞), β(•, 0) ≥ ϕ in [-α, α].
Thus, β is a non-negative upper solution of (21), and the constant 0 is a lower solution of (21). Using the comparison method from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF], we prove that there exists a solution u of (21) satisfying 0 ≤ u ≤ β for all (x, t) ∈ [-α, α] × (0, ∞). Thus, u is a global positive solution. If 1 < p < 3 and λ > 0, then we just need to choose a positive solution β given in Theorem 2.6 (even if β(±α) > 0). For λ ≤ 0, we consider the Dirichlet solution given in Theorem 2.10. Now, we replace the Dirichlet boundary conditions by the dynamical boundary conditions. Consider the following problem

   ∂ t u = ∂ 2 x u -u∂ x u + u p -λu in [-α, α] × (0, ∞), σ∂ t u + ∂ ν u = 0 on {±α} × (0, ∞), u(•, 0) = ϕ in [-α, α], (22)
with α > 0, p > 1, λ ∈ R and where ϕ ∈ C([-α, α]) and σ(±α, •) ∈ C 1 ([0, ∞)) are non-negative. We obtain two results, depending on the sign of λ.

Theorem 3.2. Let p > 1 and λ > 0. There exists a global positive solution

u ∈ C([-α, α] × [0, ∞)) ∩ C 2,1 ([-α, α] × (0, ∞))
of Problem ( 22)

• for all α > 0 if ϕ ≤ λ 1 p-1 . • for some α > 0 if ϕ-λ 1 p-1 is sign-changing and max{ϕ-λ 1 p-1 , 0} is sufficiently close to 0. • for no α > 0 if ϕ > λ 1 p-1 and p > 2.
Proof. For the first statement, we just need to note that the constant function λ Then, using these super-solutions and the comparison principle from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF], we prove first and second assertions. For the third statement, consider c > 0 such that

ϕ > c > λ 1 p-1 .
The comparison principle from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF] implies that u > c, where u denote the solution of (22) with the initial data ϕ. Hence, there exists d > 0 such that u p -λu ≥ du p for all x ∈ [-α, α] and for all t > 0.

Thus, u verifies

   ∂ t u ≥ ∂ 2 x u -u∂ x u + du p in [-α, α] for t > 0, σ∂ t u + ∂ ν u = 0 on {-α, α} for t > 0, u(•, 0) > c > 0 in [-α, α].
Then, blow-up results from [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF] imply the blowing-up in finite time of u.

Theorem 3.3. Let p > 2 and λ ≤ 0. For all α > 0, the positive solution u of Problem (22) blows up in finite time if the initial data ϕ satisfies

ϕ ≥ 0, ϕ ≡ 0, ϕ ∈ C([-α, α]) Proof. Since λ ≤ 0, the function u verifies    ∂ t u ≥ ∂ 2 x u -u∂ x u + u p in [-α, α] for t > 0, σ∂ t u + ∂ ν u = 0 on {-α, α} for t > 0, u(•, 0) > 0 in [-α, α].
Thanks to the blow-up results from [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF] and [START_REF] Rault | Phénomène d'explosion et existence globale pour quelques problèmes paraboliques sous les conditions au bord dynamiques, Thèse doctorale à l'Université du Littoral Côte d'Opale[END_REF], we know that u blows up in finite time.

Remark 4. The Neumann boundary conditions are included here, with the special case σ ≡ 0.

3.2.

Global existence in unbounded domains. We study the Problem (1) under the Dirichlet, the Neumann and the dynamical boundary conditions when Ω is an unbounded domain. Using some explicit super-solutions, we look for global existence in the three types of unbounded domains: (-∞, 0), (0, ∞) and R. We begin with the case λ > 0: Proof. As in the proof of Theorem 3.2, we consider the constant function v(x, t) = λ 1 p-1 for all (x, t) ∈ Ω × (0, ∞). Then, v satisfies Burgers' Equation. On the boundary, we have:

v ≥ 0 (Dirichlet). ∂ ν v = 0 (Neumann). ∂ ν v + av ≥ 0 (Robin). σ∂ t v + ∂ ν v = 0 (Dynamical).
The choice of ϕ implies ϕ ≤ v(•, 0) in Ω. Thus, v is super-solution of (1) for the four boundary conditions above, and we conclude with the comparison method from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF].

If λ ≤ 0, we must add some restrictions, and we obtain the following results. Proof. We deal with the comparison principle [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF] and the explicit function v(x, t) = Ae αx+(t+t0) 2 defined in R + × R + . Computing the partial derivatives, we have

∂ t v(x, t) = 2(t + t 0 )v. ∂ x v(x, t) = αv. ∂ 2 x v(x, t) = α 2 v. Choosing t 0 ≥ 1 2 α 2 -λ , we obtain ∂ t v -∂ 2 x v + v∂ x v -v p + λv ≥ v 2 α -v p-2 .
Thanks to p ≤ 2 and with αx

+ (t + t 0 ) 2 ≥ 0 in R + × R + , we have v p-2 ≤ A p-2 . Choosing A p-2 ≤ α, we obtain ∂ t v -∂ 2 x v + v∂ x v -v p + λv ≥ 0.
Since v ≥ 0, the case of the Dirichlet boundary conditions is trivial. Choosing t 0 ≥ α 2σ , the case of the dynamical boundary conditions is verified thanks to

σ∂ t v + ∂ ν v = v 2σ(t + t 0 ) -α ≥ 0.
Finally, choosing A ≥ sup Ω ϕ, v is a super-solution of Problem (1) under the above boundary conditions. Thus, using the comparison method from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF], we prove that there exist a positive solution of Problem (1) bounded by v, and then, this solution must be global. Remark 5. In the previous proof, one can see that the dynamical boundary conditions are satisfied for a more general coefficient σ verifying σ(x, t) ≥ α 2(t + t 0 ) .

And replacing the function v by w(x, t) = Ae αx+(t+t0) n , we can consider smaller coefficients σ > 0 with σ(x, t) ∼ Proof. As in the previous theorem, we consider v(x, t) = Ae αx+(t+t0) 2 . Thanks to p = 2 and with some appropriate constants A and α, we have

∂ t v -∂ 2 x v + v∂ x v -v p + λv ≥ 0 in Ω × [0, ∞). v(•, 0) ≥ ϕ in Ω.
The case Ω = R (no boundary) and the case of Dirichlet boundary conditions are trivial. For Ω = (-∞, 0) (the boundary is {0}), we have defined in R -× R + with A > 0, γ = 1 p-1 and y = -2σγ. A simple calculation leads to

∂ ν v = ∂ x v = αv > 0 for x = 0.
∂ t v -∂ 2 x v + v∂ x v -v p = v 2(t + 1) -2γ + 1 -(x + y)v -v p-1 .
By definition of γ and p > 3, we have -2γ + 1 > 0. Since v p-1 ≤ A p-1 , and because -(x + y) > 0 for all x ∈ Ω, we obtain ∂ t v -∂ 2 x v + v∂ x v -v p ≥ 0 by choosing A small enough. The case of the Dirichlet boundary conditions is clear because v ≥ 0. For the dynamical boundary conditions and the Neumann boundary conditions (σ ≡ 0), we use the definition of y and we have σ∂ t v(0, t) + ∂ ν v(0, t) ≥ v(0, t) 2(t + 1) -2σγ -y ≥ 0.

Thus, v is a super-solution of the Problem (1) as soon as we choose 0 ≤ ϕ ≤ v(•, 0) in Ω.

3.3. Blow up in unbounded domains. Here, using some weighted L 1 -norms, we examine blow-up phenomena for some solutions of Problem (1) in unbounded domains satisfying the Neumann, the Robin, and some nonlinear boundary conditions. We only consider regular solutions satisfying this standard growth order condition at infinity: for all a > 0 and for all t > 0 lim Unless otherwise stated, we always suppose Ω = (0, ∞). We begin with a lemma which gives a criterion for the blowing-up of the solution. Then, for all τ > 0 there exists c > 0 such that u(0, t) ≥ c for all t ≥ τ .

Proof. Let u be the positive solution of Problem (1) with one of the above boundary conditions (denoted by B(u) = 0), and with the initial data ϕ. Let v be the positive solution of the following problem

       ∂ t v = ∂ 2 x v -v∂ x v + v p -λv in [0, 1] × [0, ∞), B(v) = 0 on {0} × [0, ∞), v = 0 on {1} × [0, ∞), v(•, 0) = ϕ 1 in [0, 1],
where B(v) = 0 denote the same boundary conditions as in B(u) = 0, where ϕ 1 ∈ C 2 ([0, 1]) satisfies ϕ 1 (1) = 0, ∂ 2 x ϕ 1 -ϕ 1 ∂ x ϕ 1 + ϕ p 1 -λϕ 1 ≥ 0 and 0 ≤ ϕ 1 ≤ ϕ in [0, 1]. We refer to [START_REF] Below | Blow Up for some non linear parabolic problems with convection under dynamical boundary conditions[END_REF] for the existence of v. Thanks to u(•, 0) ≥ v(•, 0) in [0, 1] and u(1, t) ≥ 0 = v(1, t) for all t > 0, the comparison principle from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF] implies u(x, t) ≥ v(x, t) for all x ∈ [0, 1] and t > 0.

Then, the comparison principle and the maximum principle from [START_REF] Below | A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions[END_REF] imply ∂ t v(x, t) ≥ 0 and v(x, t) > 0.

for all x ∈ [0, 1] and t > 0, see Lemma 2.1 in [START_REF] Below | Blow Up for Reaction Diffusion Equations Under Dynamical Boundary Conditions[END_REF]. Thus, for all τ > 0, we obtain u(0, t) ≥ v(0, t) ≥ v(0, τ ) > 0 for all t ≥ τ.

Remark that, we have v(0, τ ) ≥ ϕ(0), and if ϕ(0) > 0, we can choose c = ϕ(0).

Figure 1 .

 1 Figure 1. Phase plane for p ≥ 3 and λ > 0.

Figure 2 .

 2 Figure 2. Phase plane for λ ≤ 0.

Theorem 2 .

 2 10. Let λ ≤ 0. For each boundary conditions • u(-α) = u(α) = 0 (Dirichlet b.c.) , • u(-α) = u (α) = 0 (mixed-1 b.c.), • u (-α) = u(α) = 0 (mixed-2 b.c.), there exists a positive solution of the Equation (2) u -uu + u|u| p-1 -λu = 0 in (-α, α) for some α > 0. Proof. We use the phase plane of System (3), see Figure 2. Consider the trajectory (a, b) between the points • (0, b 0 ) and (0, b 1 ): we obtain the Dirichlet solution, • (0, b 0 ) and (a 0 , 0): we obtain the mixed-1 solution, • (a 0 , 0) and (0, b 1 ): we obtain the mixed-2 solution.

Theorem 2 .

 2 16. Let p ∈ (1, 2] and λ ∈ R. For some α ∈ (0, ∞], there exists a positive and unbounded solution of the Equation (2) u -uu + u|u| p-1 -λu = 0 in (0, α). satisfying u(-α) = 0 and lim x→α u(x) = ∞.

Figure 3 .

 3 Figure 3. Phase planes for p = 1.

Figure 4 .

 4 Figure 4. Abscissa of the equilibrium points of the System (3) depending on the parameter λ.

Figure 5 .

 5 Figure 5. Phase planes of the System (3) with different parameters.

1 p- 1 1 p- 1 .

 111 is a super-solution of (22) when 0 ≤ ϕ ≤ λ For the second statement, we consider two cases: when p ≥ 3, we consider a positive solution w of Equation (2) under the Neumann boundary conditions, see Theorem 2.4. Choosing ϕ such that 0 ≤ ϕ ≤ w, w becomes a non-negative super-solution of (22). If 1 < p < 3, we consider a trajectory (µ, γ) of System (3) with 0 < µ(0) < λ 1 p-1 and γ(0) = 0. According to Equation (4), for a small x * > 0, we have γ(-x) < 0 and γ(x) > 0 for all x ∈ (0, x * ). Thus, µ satisfies ∂ ν µ(-x * ) = -γ(-x * ) > 0 and ∂ ν µ(x * ) = γ(x * ) > 0, and it is a super-solution of (22) when 0 ≤ ϕ ≤ µ in [-x * , x * ].

Figure 6 .

 6 Figure 6. Trajectory (µ, γ).

Theorem 3 . 4 . 1 p- 1 ,

 3411 Let p > 1, λ > 0 , ϕ ∈ C(Ω) a non-negative function, and let Ω be any unbounded domain. Then, the Problem (1) admits a global positive solution if the initial data satisfies 0 ≤ ϕ ≤ λ and when B(u) = 0 stands for the Dirichlet, the Neumann, the Robin (∂ ν u + au = 0 with a ≥ 0) or the dynamical boundary conditions.

Theorem 3 . 5 .

 35 Assume Ω = (0, ∞) and let p ∈ (1, 2], λ ≤ 0 and ϕ ∈ C(Ω) a non-negative function. Then, the Problem (1) admits a global positive solution if the initial data is bounded and when B(u) = 0 stands for the Dirichlet boundary conditions or the dynamical boundary conditions with σ > 0 constant.

t→∞ t -n+1 . Corollary 2 .

 2 Suppose Ω = (-∞, 0) or Ω = R. Let p = 2, λ ≤ 0 and ϕ ∈ C(Ω). Then the Problem (1) admits a global positive solution if there exists C > 0 and a > 0 such that 0 ≤ ϕ(x) ≤ Ce ax in Ω and when B(u) = 0 stands for the Dirichlet, the Neumann or the dynamical boundary conditions with σ ≥ 0.

  Thus, the Neumann boundary conditions and the dynamical boundary conditions with σ ≥ 0 are verified. When λ = 0, Ω = (-∞, 0) and p > 3, the Green function of the heat equation is a suitable super-solution for the Problem (1). Theorem 3.6. Assume Ω = (-∞, 0), p > 3, λ = 0 and ϕ ∈ C(Ω). Then the Problem (1) admits a global positive solution if the initial data ϕ is sufficiently small and when B(u) = 0 stands for the Dirichlet, the Neumann or the dynamical boundary conditions with σ ≥ 0 constant. Proof. Consider the function v(x, t) = A(t + 1) -γ e -(x+y) 2 4t+4

  |x|→∞ u(x, t)e -a|x| = 0 and lim |x|→∞ |∂ x u(x, t)|e -a|x| = 0. (23)

Lemma 3 . 7 . 0 eLemma 3 . 8 .

 37038 Let u be a solution of Problem[START_REF] Amann | Ordinary differential equations[END_REF] which satisfies the condition (23). If there exists α > 0 such thatN α (t) := ∞ 0 u(x, t)e -αx dxblows-up in finite time, then u also blows-up in finite time.Proof. Consider α > 0 such that N α blows-up in finite time. Using the following inequalityN α (t) ≤ ∞ -αx/2 dx • sup Ω u(x, t)e -α 2 x = 2 α sup Ω u(x, t)e -α 2 x ,and because N α blows up, we can deduce the blowing up in finite time of the function u(x, t)e -α 2 x . Then, thanks to the growth order condition (23), the solution u must blow up too.We also need this technical lemma. Let u be a solution of Problem[START_REF] Amann | Ordinary differential equations[END_REF] where the boundary conditions are the Neumann, the Robin, or some nonlinear boundary conditions ∂ ν u = g(u).

E-mail address: jfrault@lmpa.univ-littoral.fr

Acknowledgments. The author would like to thank Professor Joachim von Below for his valuable discussions and advices.

Theorem 3.9. Let λ < 0 and p ≥ 2. Then the Problem (1) admits no global positive solution when B(u) = 0 stands for the Neumann boundary conditions.

Proof. We aim to prove the existence of α > 0 and β > 0 such that N α ≥ βN p Using the growth order condition (23) and integrating by parts, we obtain

Thus, we have

(24)

Thanks to Lemma 3.8, and considering u from a time τ > 0, we can assume that

Then, if α is small enough (α ≤ c/2), we have -αu(0, t) + u 2 (0,t) 2 ≥ 0. Then, the Neumann boundary conditions imply

Shrinking α, we can suppose α ≤ -2λ and α ≤ 1. When u(x, t) ≤ 1, we have -λ -αu(x, t)/2 > 0. On the other hand, if u(x, t) ≥ 1, we have u p-1 (x, t)αu(x, t)/2 ≥ u p-1 (x, t)/2. Hence, we obtain:

Hölder inequality

Finally, we prove the blowing-up of N α in finite time. Integrating the differential inequality N α (t) ≥ βN p α (t) between 0 and t > 0, we obtain

and

Since of -1 p-1 < 0, the right hand side term blows up at t =

(p-1)β > 0. We conclude with Lemma 3.7.

Corollary 3. Let λ < 0 and p ≥ 2. Then the Problem (1) admits no global positive solution when B(u) = 0 stands for the nonlinear boundary conditions ∂ ν u = g(u), where g is a function such that there exists δ > 0 and ε ≤ 1/2 satisfying

Proof. We follow the proof of Theorem 3.9. We just change the choice of α: let α > 0 such that α ≤ δ, and use the following minoration in Equation ( 24):

Then, we return to Equation (25) and we can prove that there exists a β > 0 such that N α (t) ≥ βN p α (t) for t ∈ (0, T ) . When λ = 0, the choice of α is too strict. Meanwhile, we obtain some blow-up results imposing some restrictions on the exponent p and on the initial data.

Let β ∈ (0, 1) and put it into the previous equation:

If u ≤ 2α, we have α 2 -αu/2 ≥ 0, whereas u > 2α implies

(26) Thanks to 1 < p ≤ 3, Equation ( 26) is achieved by choosing α > 0 sufficiently small and β ∈ (0, 1) depending on p. Thus, we obtain

Then, we can suppose that u(0, t) > c > 0 for all t > 0 (see Lemma 3.8), and with α < c/2 we have -αu(0, t)

As in the proof of Theorem 3.9, we use Hölder inequality and we are led to N α ≥ δN p α with δ > 0 depending on α, β and p. Hence, N α blows-up in finite time, so does the solution u, see Lemma 3.7. Proof. The proof is similar to the previous one. Go back to Equation (26): since p > 3, we must choose α such that

Under this condition, N α satisfies the differential inequality

Because α can not be too small, we must use the assumption ϕ(0) > 2 p-3 , we obtain -αu(0, t)+ u 2 (0,t) 2 ≥ 0. Hence, we have

We conclude with Hölder inequality and the blowing up of N α .

Proof. Return to the proof of Theorem 3.9. Under the Neumann boundary conditions and introducing β and δ ∈ (0, 1) in Equation ( 24), we obtain

Studying both cases u ≥ 2αδ and u ≤ 2αδ, we obtain δα 2 -αu/2

Since of u 2 /2 -αu ≥ -α 2 /2 and using Hölder inequality we have

where γ = (1 -β)

∞ 0 e -αx dx 1-p > 0. First, consider this minoration

Thus, N α satisfies

In particular, N α (0) ≥ (2 -2δ) -1 + A. Choosing δ > 0 close to 0 and with β ∈ (0, 1) close to 1, Hypothesis (27) implies N α (0) > (2 -2δ) -1 . Thus, A is positive and we obtain

From Equation (28), we deduce

Hence N α blows-up, and the solution u blows up too, see Lemma 3.7.

Finally, if Ω = (-∞, 0), we must change the weight in N α and we obtain this results concerning the nonlinear boundary conditions. Theorem 3.12. Let λ ≤ 0 and p ≥ 2. Then the Problem (1) admits no global positive solution when B(u) = 0 stands for the nonlinear boundary conditions ∂ ν u = g(u), where g is a function such that there exists c > 0 and d > 0 satisfying g(η) ≥ cη 2 + dη.

Proof. As in the case of Ω = (0, ∞), we use a weighted L 1 -norm:

u(x, t)e αx dx , with α > 0.

We compute N α (t) = 0 -∞ ∂ t u(x, t)e αx dx, and using the equations of Problem (1), integration by parts leads to N α (t) = 0 -∞ (α 2 u + αu 2 + u p )e αx dx + ∂ x u(0, t) -αu(0, t) -α 2 u 2 (0, t).

Thanks to ∂ ν u(0, t) = ∂ x u(0, t) in (-∞, 0), choosing α = min{2c, d}, we obtain

u p e αx dx.

Hölder inequality leads to the differential equation N α (t) ≥ γN p α (t) with γ > 0. Hence N α and the solution u blow up in finite time.