Blow-up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions

Abstract

In this paper, we consider the following reaction diffusion systems with gradient nonlinearity under nonlinear boundary condition

$$\{\begin{array}{cc}{u}_t=△u+u^p{v}^q-\mid \nabla u{\mid }^{\alpha },(x,t)\in \Omega \times (0,t^{\ast })\text{;}\hfill & \hfill \\ v_t=△v+v^r{u}^s-\mid \nabla v{\mid }^{\alpha },(x,t)\in \Omega \times (0,t^{\ast })\text{;}\hfill & \hfill \\ \frac{\partial u}{\partial \nu }=g\left(u\right),\frac{\partial v}{\partial \nu }=h\left(v\right),(x,t)\in \partial \Omega \times (0,t^{\ast })\text{;}\hfill & \hfill \\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),x\in \Omega \hfill & \hfill \end{array}$$

where $\Omega \subset R^N(N\ge 1)$ is a bounded region with smooth boundary $\partial \Omega$, $p,q,r,s\ge 0$, $\alpha >1$, $t^{\ast }$ is a possible blow-up time when blow-up occurs. By constructing an appropriate auxiliary functions, and by means of Payne–Weinberger or Scott’s method, a lower bound on blow-up time when blow-up occurs is derived.