Explosive solutions of parabolic stochastic partial differential equations with L$\acute{e}$vy noise
In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a L$\acute{\mbox{e}}$vy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that the positive solutions will blow up in finite time in mean $L^{p}$-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrated the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by L$\acute{\mbox{e}}$vy noise has a global solution.
Many physical phenomena can be described by biexponential curve. To study the physical phenomena, there is a need to obtain their mathematical expressions. Generally, the expressions are acquired based on the fitting of experimental data, and the general method to fit exponential curve is nonlinear least squares (NLLS). In the NLLS method, it is crucial to choose good enough starting values for the parameters, otherwise, the fitting might fail to converge. In this paper, one new method is proposed to solve the problem, which utilizes the Fourier transform to transform the non-linear exponential fitting model into a linear one. It makes the estimation of the parameters independent of the initial values and easy to converge. The fitting result shows that the proposed method can be used to fit the curve expressed by difference of double exponentials.