Localized Solutions of a Semilinear Parabolic Equation with a Recurrent Nonstationary Asymptotics

We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations $u_t=\Delta u+f(u)$ on $\mathbb{R}^N$. Here $f\in C^1$, $f(0)=0$, and a localized solution refers to a solution $u(x,t)$ which decays to 0 as $x\to\infty$ uniformly with respect to $t>0$. In all previously known examples, bounded, localized solutions are convergent or at least quasi-convergent in the sense that all their limit profiles as $t\to\infty$ are steady states. If $N=1$, then all positive bounded, localized solutions are quasi-convergent. We show that such a general conclusion is not valid if $N\ge 3$, even if the solutions in question are radially symmetric. Specifically, we give examples of positive bounded, localized solutions whose $\omega$-limit set is infinite and contains only one equilibrium.

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