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Zbl 0648.45006
Bellout, Hamid
Blow-up of solutions of parabolic equations with nonlinear memory.
(English)
[J] J. Differ. Equations 70, 42-68 (1987). ISSN 0022-0396

This paper deals with the initial-boundary value problem $$u\sb t=\Delta u+\int\sp{t}\sb{0}m(t-\tau)f(u(x,\tau))d\tau +g(x)$$ in $Q\sb{\infty}$, $u(x,0)=u\sb 0(x)$ in $\Omega$, $u(x,t)=0$ in $\Gamma\sb{\infty}$; here $\Omega$ is a bounded domain in $R\sp n$ with $C\sp 2$ boundary, $Q\sb t=\Omega \times (0,t)$, and $\Gamma\sb t=\partial \Omega \times (0,t)$, where the functions f, m, g, and $u\sb 0$ satisfy certain smoothness and monotonicity conditions. Under these conditions and specifically $\int\sp{\infty}\sb{0}(F(s))\sp{-}ds<\infty$, where F is an antiderivative of f, the solution u blows up in finite time. This paper aims at characterizing the blow-up set; i.e., the set $\{$ $x\in \Omega:\exists (x\sb n,t\sb n)\to (x,T)\ni u(x\sb n,t\sb n)\to \infty \}$. In much of this paper $f(u)=(u+\lambda)\sp p$, $p>1$, $\lambda >0$. A basic method for dealing with the delay aspect of this equation is to consider the t-derivative of both sides and to use rather strong monotone properties of m'(t) in the hypotheses. \par [A number of misprints was noted by the reviewer, which contributed to difficulties he experienced following details of some of the proofs.]
[G.Seifert]
MSC 2000:
*45K05 Integro-partial differential equations

Keywords: parabolic equations with nonlinear memory; initial-boundary value problem; monotonicity conditions; blow-up set

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