article

Title: Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $R^N$

Authors: Peter Polacik and Yihong Du

Issue: Volume 64 (2015), Issue 3, 787-824

Abstract:

$We consider bounded solutions of the Cauchy problem \[ \begin{cases} u_t-\Delta u=f(u),&x\in\mathbb{R}^N,\ t>0,\\ u(0,x)=u_0(x),&x\in\mathbb{R}^N, \end{cases} \] where $u_0$ is a non-negative function with compact support and $f$ is a $C^1$ function on $\mathbb{R}$ with $f(0)=0$. Assuming that $f'$ is locally H\"older continuous, and that $f$ satisfies a minor nondegeneracy condition, we prove that, as $t\to\infty$, the solution $u(\cdot,t)$ converges to an equilibrium $\varphi$ locally uniformly in $\mathbb{R}^N$. Moreover, either the limit function $\varphi$ is a constant equilibrium, or there is a point $x_0\in\mathbb{R}^N$ such that $\varphi$ is radially symmetric and radially decreasing about $x_0$, and it approaches a constant equilibrium as $|x-x_0|\to\infty$. The nondegeneracy condition only concerns a specific set of zeros of $f$, and we make no assumption whatsoever on the nonconstant equilibria. The set of such equilibria can be very complicated, and indeed a complete understanding of this set is usually beyond reach in dimension $N\geq2$. Moreover, because of the symmetries of the equation, there are always continua of such equilibria. Our result shows that the assumption "$u_0$ has compact support" is powerful enough to guarantee that, first, the equilibria that can possibly be observed in the $\omega$-limit set of $u$ have a rather simple structure; and, second, exactly one of them is selected. Our convergence result remains valid if $\Delta u$ is replaced by a general elliptic operator of the form $\sum_{i,j} a_{ij}u_{x_ix_j}$ with constant coefficients $a_{ij}$.$