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Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $$\mathbb{R}^N$$. (English) Zbl 1336.35207
This paper addresses the asymptotic behavior as $$t\rightarrow \infty$$ of bounded solutions to the Cauchy problem
$u_t-\Delta u=f(u),\,\, x\in\mathbb{R}^n,\,t>0, \quad u(0,x)=u_0(x),\quad x\in\mathbb{R}^n,$
where $$u_0$$ is a non-negative function with compact support and $$f$$ is a function in $$C^1(\mathbb{R})$$ satisfying $$f(0)=0$$. It is also assumed that $$f'$$ is locally Hölder continuous and that $$f$$ satisfies a nondegeneracy condition at a specific subset of its zero points.
It is known that nonconstant equilibria for such equations occur in continua, leading to a complicated set of equilibria with a huge variety of elements. Despite this fact, the main results in this paper show that the compactness assumption on $$u_0$$ is powerful enough to guarantee that the $$\omega$$-limit set of any bounded solution has a rather simple structure. It consists of a unique equilibrium that is either constant or radially symmetric and decreasing around a fixed point, converging to a constant equilibrium as $$|x|\rightarrow \infty$$.

MSC:
 35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian 35K15 Initial value problems for second-order parabolic equations 35K05 Heat equation 35B40 Asymptotic behavior of solutions to PDEs
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