## Convergence to a steady state for asymptotically autonomous semilinear heat equations on $$\mathbb R^N$$.(English)Zbl 1263.35035

The authors treat semilinear parabolic equations of the form $u_t=\Delta u+f(u)+h(x,t),\qquad(x,t)\in\mathbb{R}^N\times(0,\infty), \tag{1}$ that are asymptotically autonomous in the sense that $$\|h(\cdot,t)\|_\infty\to0$$ as $$t\to\infty$$. Here $$f$$ is assumed to be continuously differentiable and to satisfy $$f(0)=0$$ and $$f'(0)<0$$. Consider a global, bounded and nonnegative solution $$u$$ of (1) that decays in $$x$$, uniformly in $$t$$: $\lim_{|x|\to\infty}\sup_{t>0}u(x,t)=0.$ Define its $$\omega$$-limit set in a suitable Banach space of states by $\omega(u):=\{v: u(\cdot,t_k)\to v\text{ for some sequence } t_k\to\infty\}.$ A positive solution of $$\Delta u+f(u)=0$$ on $$\mathbb{R}^N$$ that decays to $$0$$ as $$|x|\to\infty$$ is called a ground state. A ground state is always radially symmetric about some point and radially decaying.
Under these assumptions it is shown that either $$\omega(u)=\{0\}$$ or that $$\omega(u)$$ consists entirely of ground states. If in addition $$h(\cdot,t)$$ decays exponentially in a suitable Hölder class, then either $$\omega(u)=\{0\}$$ or $$\omega(u)$$ consists of exactly one ground state. Since $\lim_{t\to\infty}\text{dist}_\infty(u(\cdot,t),\omega(u))=0,$ these results yield, respectively, quasiconvergence and convergence of $$u$$ in the $$L^\infty$$-sense.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K58 Semilinear parabolic equations 35B09 Positive solutions to PDEs 35B07 Axially symmetric solutions to PDEs

### Keywords:

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