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

In this paper we study the following problem: u_t?Δu=?f(u) in Ω×(0, T)≡Q_T, ?u ?n=g(u) on ?Ω×(0, T)≡S_T, u(x, 0)=u₀(x) in Ω , where Ω??^N is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u₀(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(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. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.