On Critical Exponents for a Semilinear Parabolic System Coupled in an Equation and a Boundary Condition

Abstract

<p>In this paper, we consider the system \arraycolsep0.14em\begin{array}{rcl {\hskip2em}rcl {\hskip2em}c}u_t&=&\Delta u+v^p,&v_t&=&\Delta v&x\in{\Bbb R}_{+}^N,t>0,\\ \displaystyle-{\partial u\over\partial x_t}&=&0,&\displaystyle-{\partial v\over\partial x_t}&=&u^q&x_1=0,t>0,\\ u(x,0)&=&u_0(x),&v(x,0)&=&v_0(x)&x\in{\Bbb R}_{+}^N, whereR" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">R<em>N</em>+={(<em>x</em>1, <em>x</em>′)|<em>x</em>′∈R" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">R<em>N</em>−1, <em>x</em>1>0}, <em>p</em>, <em>q</em>>0, and<em>u</em>0, <em>v</em>0are nonnegative and bounded. We prove that if<em>pq</em>≤1 every nonnegative solution is global. When<em>pq</em>>1 we let α=(<em>p</em>+2)/2(<em>pq</em>−1), β=(2<em>q</em>+1)/2(<em>pq</em>−1). We show that if max(α, β)><em>N</em>/2 or max(α, β)=<em>N</em>/2 and<em>p</em>, <em>q</em>≥1, then all nontrivial nonnegative solutions are nonglobal; whereas if max(α, β)<<em>N</em>/2 there exist both global and nonglobal nonnegative solutions.</p>