## Differential and Integral Equations

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

#### 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

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