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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.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
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