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Blow-up for semilinear parabolic equations with a gradient term. (English) Zbl 0768.35049
Consider the parabolic PDE $$u_ t=\Delta u-|\nabla u|^ q+\lambda u^ p$$ in $$D(0,T)$$ with zero Dirichlet boundary data and given initial data for some smooth bounded domain $$D$$ in $$\mathbb{R}^ n$$, with $$\lambda>0$$, and $$p,q>1$$. Assuming that $$p<(n+2)/(n-2)$$ if $$n>2$$ and $$q<2p/(p+1)$$ (or $$q=2p/(p+1)$$, $$n=1$$, and $$p$$ sufficiently large), M. Chipot and F. B. Weissler [(*) SIAM J. Math. Anal. 20, No. 4, 886-907 (1989; Zbl 0682.35010)] and B. Kawohl and L. A. Peletier [(**) Math. Z. 202, No. 2, 207-217 (1989; Zbl 0661.35053)] have shown that solutions of this initial-boundary value problem must blow up in finite time for suitable initial data while $$p\leq q=2$$ implies that all solutions exist globally in time. The author shows that there are nonnegative initial data with finite time blow-up in one dimension in case $$2p/(p+1)<q<p$$ and $$\lambda$$ is sufficiently large. His method is quite different from those of Chipot and Weissler (*), Kawohl and Peletier (**), or M. Fila [Proc. Am. Math. Soc. 111, No. 3, 795-801 (1991; Zbl 0768.35047)], who gives an alternative proof of the results previously mentioned; the key step is to show that $$u_{xx}$$ behaves like $$-\lambda u^ p$$ except on a small set.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [1] and , ’Blow up in Rn for a parabolic equation with a damping nonlinear gradient term’, preprint. [2] Chipot, SIAM J. Math. Anal. 20 pp 886– (1989) [3] Fila, Proc. Amer. Math. Soc. [4] Kawohl, Math. Z. 202 pp 207– (1989)
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