Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems
where L = — — ~ dx2 + x dx dt' and b is a constant less than 1. If Ta = oo, then z (< 1) exists globally. If Ta < oo, then lim max{z(x,/): 0 < x < a} = 1". '-77 In this latter case, z is said to quench at Ta (cf. Acker and Walter [2]), and Ta is called the quenching time. Recently, we [3] studied the critical length a* for the above problem such that Ta = oo for a < a*, and Ta < oo for a > a*\ an upper bound for the quenching time is also determined. For further references, we refer to the above paper, and those written recently by Acker and Kawohl [ 1 ] and Chan and Kaper [4]. So far, only problems involving one scalar differential equation have been studied. The main purpose here is to extend our study [3] on critical lengths for global existence of solutions and upper bounds for the quenching time for a scalar differential equation to the following coupled parabolic problem: