Blow-Up Phenomena for Nonlinear Reaction-Diffusion Equations under Nonlinear Boundary Conditions

Abstract

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinear boundary conditions:g(u)t=∇·au∇u+fu in Ω×0,T, ∂u/∂n=bx,u,t on ∂Ω×(0,T), u(x,0)=u0(x)>0, in Ω¯,whereΩ⊂RN (N≥2)is a bounded domain with smooth boundary∂Ω. We obtain the conditions under which the solutions either exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles. Moreover, the upper estimates of the “blow-up time,” the “blow-up rate,” and the global solutions are also given.