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Blow-up at the boundary for degenerate semilinear parabolic equations. (English) Zbl 0733.35009
Consider the semilinear problem: $xu\sb t=u\sb{xx}+u\sp p$ in $\Omega\times (0,\infty)$, $u(0,t)=u(1,t)=0$ for $t>0$, $u(x,0)=u\sb 0(x)\ge 0$ for $x\in \Omega$ where $\Omega$ is the unit interval (0,1), $p>1$ and the initial function $u\sb 0\in C\sp 1({\bar \Omega})$ with $u\sb 0(0)=u\sb 0(1)=0$. The author proves the theorems: (i) Suppose that $1<p\le 2$ and the solution u blows up at $t=T$. If $(\partial /\partial x)(u\sb 0(x)/x)\le 0$ for $x\in (0,1)$, then $S=\{0\}$, where S is the set of blow-up points. (ii) Suppose that $1<p\le 2$, $(\partial /\partial x)(u\sb 0(x)/x)\le 0$ and u blows up at $t=T.$ Then given $\delta >0$, there exists a $C\sb 2>0$ such that $s\sb 2(t)\le C\sb 2(T-t)\sp{1/3+\delta}$. Under the additional assumption $u\sb 0''+u\sb 0\sp p\ge 0$, there exists a $C\sb 1>0$ such that $s\sb 1(t)\ge C\sb 1(T-t)\sp{1/3}$ for all $t\in (0,T)$, where s(t) be any point such that $u(s(t),t)=\sup\sb{x\in \Omega}u(x,t)$, and $s\sb 1(t)$ is the infimum and $s\sb 2(t)$ the supremum over all such s.

35B40Asymptotic behavior of solutions of PDE
35K65Parabolic equations of degenerate type
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI
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