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1997 Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations
Eduard Feireisl, Hana Petzeltová
Differential Integral Equations 10(1): 181-196 (1997).

Abstract

A complete characterization of the long-time behaviour of nonnegative solutions to the problem $$ u_t -\Delta u + u + \sum b_j u^{r_j} - \sum a_i u^{p_i} = 0 , \ u=u(x,t) , \ x \in \rn , \ t>0 $$ $$ 1 <r_j < p_i \leq {N\over N-2}, \ a_i, \ b_j > 0 $$ is given. For any nonnegative compactly supported datum $ \bar u \not\equiv 0 $, there is a constant $ \alpha_c >0 $ such that the solution converges to a ground state for $ u(0)= \alpha_c \bar u $, blows up at a finite time if $ u(0) = \alpha \bar u ,$ $ \alpha > \alpha_c $ and tends to zero provided $ u(0) = \alpha \bar u $ with $ 0 \leq \alpha < \alpha_c $. The proof is based on a combination of the concentrated compactness, comparison theorems and some recent results on uniqueness for the corresponding stationary equation.

Citation

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Eduard Feireisl. Hana Petzeltová. "Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations." Differential Integral Equations 10 (1) 181 - 196, 1997.

Information

Published: 1997
First available in Project Euclid: 6 May 2013

zbMATH: 0879.35023
MathSciNet: MR1424805

Subjects:
Primary: 35K55
Secondary: 35B40

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.10 • No. 1 • 1997
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