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Remarks on blow up for a nonlinear parabolic equation with a gradient term. (English) Zbl 0768.35047
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 gives an easy characterization of initial data implying finite time blow-up in one dimension: they’re greater than or equal to the unique positive equilibrium solution. Key steps in the proof are that global, increasing solutions (in any number of dimensions) are uniformly bounded in $$L^ \infty$$ and that they converge to an equilibrium solution. The proof is completed in one dimension by showing that this positive equilibrium solution is unstable.

##### 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:
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