Reaction diffusion equations with non-linear boundary conditions, blowup and steady states

摘要

In this paper we study the following problem: u(t) - Delta u = -f(u) in Omega x (0, T) = Q(T), partial derivative u/partial derivative n = g(u) on partial derivative Omega x (0, T) = S-T, u(x, 0) = u(0) (x) in Omega, where Omega subset of R-N is a smooth bounded domain, f and g are smooth Functions which are positive when the argument is positive, and u(0)(x) > 0 satisfies some smooth and compatibility conditions to guarantee the classical solution Ic(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper-lower solutions using related equations. (C) 1998 B. G. Teubner Stuttgart-John Wiley & Sons, Ltd. [References: 27]