The heat semigroup on sectorial domains,highly singular initial values and applications |
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Authors: | Hattab Mouajria Slim Tayachi Fred B Weissler |
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Institution: | 1.Laboratoire équations aux Dérivées Partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis,Université de Tunis El Manar,Tunis,Tunisie;2.Université Paris 13-Sorbonne Paris Cité, CNRS UMR 7539 LAGA,Villetaneuse,France |
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Abstract: | We show that the heat semigroup is well defined on the Banach space \({\mathcal{X}_{m,\gamma} = \{ \psi:\Omega_m \to \mathbb{R} ; |x|^{\gamma +2m}(\prod_{i=1}^m x_i)^{-1}\psi(x) \in L^\infty(\Omega_m)\},}\) \({0 < \gamma < N}\), where \({\Omega_m=\{(x_1,\, x_2,\, \ldots,\, x_N) \in \mathbb{R}^{N}; x_i > 0,\, 1\leq i\leq m\},}\) \({1\leq m\leq N}\). We then investigate the large time behavior of solutions of the heat equation \({u_{t}-\Delta u=0}\) for t > 0 and \({x \in \Omega_m.}\) Using certain notions from dynamical systems, we show that the large time behavior is related to the spatial asymptotic behavior of its initial value. Since the space \({\mathcal{X}_{m, \gamma}}\) contains highly singular initial data, which can be extended to all of \({\mathbb{R}^{N}}\) by antisymmetry, we also obtain new results on the complexity in the asymptotic behavior of solutions for the heat equation on the whole space. |
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