Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions

TL;DR

This paper studies the blow-up behavior of solutions to nonlinear reaction-diffusion equations with convection terms under dynamical boundary conditions, providing conditions for blow-up, bounds on blow-up time, and blow-up rates.

Contribution

It introduces new criteria for blow-up in reaction-diffusion equations with nonlinear convection and derives bounds on blow-up time and rates for specific nonlinearities.

Findings

Conditions under which solutions blow up in finite time.

Upper bounds for blow-up time for certain nonlinearities.

Determination of blow-up rates for solutions.

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.