On the long-time limit of positive solutions to the degeneratelogistic equation
Yihong Du, Yoshio Yamada · Discrete and Continuous Dynamical Systems · 2009
We study the long-time behavior of positive solutionsto the problem$u_t-\Delta u=a u-b(x)u^p \mbox{ in } (0,\infty)\times \Omega, Bu=0\mbox{ on } (0,\infty)\times \partial \Omega, $ where $a$ is areal parameter, $b\geq 0$ is in $C^\mu(\bar{\Omega})$ and $p>1$ is aconstant, $\Omega$ is a $C^{2+\mu}$ bounded domain in $R^N$ ($N\geq2$), the boundary operator $B$ is of the standard Dirichlet,Neumann or Robyn type. Under the assumption that$\overline\Omega_0$:=$\{x\in\Omega: b(x)=0\}$ has non-empty interior,is connected, has smooth boundary and is contained in $\Omega$, itis shown in [8] that when $a\geq \lambda_1^D(\Omega_0)$, forany fixed $x\in \overline{\Omega}_0$, $\overline{\lim}_{t\to\infty}u(t,x)$=$\infty$, and for any fixed $x\in \overline{\Omega}\setminus\overline{\Omega}_0$,$\overline{\lim}_{t\to\infty}u(t,x)\leq \overline{U}_a(x),\underline{\lim}_{t\to\infty}u(t,x)\geq \underline{U}_a(x), where $\underline{U}_a$ and $\overline{U}_a$ denote respectively the minimal and maximal positive solutions of the boundary blow-up problem$-\Delta u=au-b(x)u^p \mbox{ in} \\Omega\setminus\overline{\Omega}_0,\ Bu=0 \mbox{ on}\ \partial\Omega,\ \ u=\infty \mbox{ on}\ \partial \Omega_0.$The main purpose of this paper is to show that, under the aboveassumptions,$\lim_{t\to\infty} u(t,x)=\underline U_a(x),\forall x\in\overline\Omega\setminus \overline\Omega_0.$This proves aconjecture stated in [8]. Some extensions of this result arealso discussed.