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On the dead core behavior for a semilinear heat equation. (English) Zbl 0934.35075
From the introduction: We consider the problem $$\cases u_t=u_{xx}- u^p,\quad & \text{in }Q_T \equiv(-l,l)\times (0,T),\\ u(x,0)= u_0(x),\quad & x\in(-l,l),\\ u(\pm l,t)= 1,\quad & t\ge 0, \endcases$$ where $0<p<1$, $0<u_0(x)\le 1$ in $(-l,l)$ and $u_0(\pm l)=1$. Throughout this paper we assume that $u$ has a dead core at the finite time $T$ and $u_0$ is smooth and satisfies $\partial^2/ (\partial x^2)u_0(x) -u^p_0(x)\le 0$, i.e., $u_t\le 0$ at $t=0$. In this paper, we show that there are only finitely many dead core points for $u$.

35K60Nonlinear initial value problems for linear parabolic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations