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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
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