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Convergence to Steady States of Solutions of Non-autonomous Heat Equations in $$\mathbb{R}^{N}$$

Authors:

Institution:

(1) Laboratoire de Mathématiques et Applications de Metz et CNRS, Université Paul Verlaine – Metz, UMR 7122, Bat. A, Ile du Saulcy, 57045 Metz Cedex 1, France;(2) Département de Mathématiques, Faculté des Sciences de Bizerte, 7021 Jarzouna Bizerte, Tunisie

Abstract:

Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation

$u_t - \Delta u + f(u) = g(t,x)$

in $\mathbb{R}^{N}$ (N ≥ 3) converge to a steady state. Dedicated to Prof. Pavol Brunovsky on the occasion of his 70th birthday.

Keywords:

Heat equation Lojasiewicz inequality convergence

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