Nonlinear convection in reaction-diffusion equations under dynamical boundary conditions

Abstract

We study the blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term $partial_t u = Delta u - g(u) cdot abla u + f(u)$ in a bounded domain of $mathbb{R}^N$ under the dissipative dynamical boundary conditions $sigma partial_t u + partial_u u =0$. Some conditions on g and f are discussed to state if the positive solutions blow up in finite time or not. Moreover, for certain classes of nonlinearities, an upper-bound for the blow-up time can be derived and the blow-up rate can be determined.

Keywords

• Nonlinear parabolic problem
• dynamical boundary conditions
• lower and upper-solution
• blow-up
• global solution