Differential and Integral Equations

Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations

Eduard Feireisl and Hana Petzeltová

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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.

Article information

Source
Differential Integral Equations, Volume 10, Number 1 (1997), 181-196.

Dates
First available in Project Euclid: 6 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367846890

Mathematical Reviews number (MathSciNet)
MR1424805

Zentralblatt MATH identifier
0879.35023

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Feireisl, Eduard; Petzeltová, Hana. Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Differential Integral Equations 10 (1997), no. 1, 181--196. https://projecteuclid.org/euclid.die/1367846890


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