Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions

Abstract

We investigate blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term ${\partial }_{t}u=\Delta u-g(u)\cdot \nabla u+f(u)$ in a bounded domain of ${\mathbb{R}}^N$ under the dissipative dynamical boundary conditions $\sigma {\partial }_{t}u+{\partial }_{\nu }u=0$. 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 determinated.

arXiv comment20 pages