The Life Span of Blow-up Solutions for a Weakly Coupled System of Reaction-Diffusion Equations

Abstract

We consider the weakly coupled system of reaction-diffusion equations \[ u_t=\Delta u+|x|^{\sigma_1}v^p, \quad v_t=\Delta v+|x|^{\sigma_2}u^q , \] \[ u(x,0)=\lambda^\mu\varphi(x), \quad v(x,0)=\lambda^\nu\psi(x) \] where $x\in\mathbf{R}^N$, $t>0$, $p$, $q>1$ and $0\leq\sigma_1<N(p-1)$, $0\leq\sigma_2<N(q-1)$. The existence of solutions, blow-up conditions, and global solutions of the above equations are studied by Mochizuki-Huang. In this paper, we consider the estimate of maximal existence time of blow-up solutions in $I^{\delta_1}\times I^{\delta_2}$ as $\lambda$ goes to 0 or $\infty$.