Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$

Fašangová, Eva

Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 525-544.

Résumé

We show that nonnegative solutions of $$ \begin{aligned} u_{t}-u_{xx}+f(u)=0,\quad x\in \Bbb R,\quad t>0, \\ u=\alpha \bar u,\quad x\in \Bbb R,\quad t=0, \quad \operatorname{supp}\bar u \hbox{ compact } \end{aligned} $$ either converge to zero, blow up in $\operatorname{L}^{2}$-norm, or converge to the ground state when $t\to\infty$, where the latter case is a threshold phenomenon when $\alpha>0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha>0$, provided $\operatorname{supp}\bar u$ is sufficiently small.

Détail

Classification : 35B05, 35B40, 35K55
Keywords: parabolic equation; stationary solution; convergence