A semilinear parabolic system with a free boundary

Abstract

This paper deals with a semilinear parabolic system with free boundary in one space dimension. We suppose that unknown functions $u$ and $v$ undergo nonlinear reactions $u^q$ and $v^p$, and exist initially in a interval $\{0\leq x\leq s(0)\}$, but expand to the right with spreading front $\{x=s(t)\}$, with $s(t)$ evolving according to the free boundary condition $s'(t)=-\mu (u_x+\rho v_x)$, where $p,\, q,\, \mu, \,\rho$ are given positive constants. The main purpose of this paper is to understand the existence, uniqueness, regularity and long time behavior of positive solution or maximal positive solution. Firstly, we prove that this problem has a unique positive solution $(u,v,s)$ defined in the maximal existence interval $[0,T_{\max})$ when $p,\,q\geq 1$, while it has a unique maximal positive solution $(u,v,s)$ defined in the maximal existence interval $[0,T_{\max})$ when $p<1$ or $q<1$. Moreover, $(u,v,s)$ and $T_{\max}$ have property that either (i) $T_{\max}=+\infty$, or (ii) $T_{\max}<+\infty$ and $$ \limsup_{T\nearrow T_{\max}}\|u,\,v\|_{L^{\infty}([0,T]\times[0,s(t)])}=+\infty.$$ Then we study the regularity of $(u,v)$ and $s$. At last, we discuss the global existence ($T_{\max}=+\infty$), finite time blow-up ($T_{\max}<+\infty$), and long time behavior of bounded global solution.