Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators
In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operators ut + A(t)u = f (t, u), t > τ ; u(τ ) = u0, in a Banach space X. Under suitable conditions, we obtain regularity results for ut(t, x) with respect to its spatial variable x and estimates...
| Autores: | , , , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/130375 |
| Acesso em linha: | https://hdl.handle.net/11441/130375 https://doi.org/10.1016/j.jde.2022.01.030 |
| Access Level: | acceso abierto |
| Palavra-chave: | Nonautonomous parabolic problems Time-dependent linear operators Regularization Smoothing effect Asymptotic dynamics |
| Resumo: | In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operators ut + A(t)u = f (t, u), t > τ ; u(τ ) = u0, in a Banach space X. Under suitable conditions, we obtain regularity results for ut(t, x) with respect to its spatial variable x and estimates for ut in stronger spaces (Xα). We then apply those results to a nonautonomous reaction-diffusion equation ut − div(a(t, x)∇u) + u = f (t, u) with Neumann boundary condition and time-dependent diffusion. From the regularity of ut we derive the existence of classical solutions and from the estimates for ut we prove that the variation of the solution u is bounded in the long-time dynamics. We also prove the existence of pullback attractor, as well as the existence of a compact set that contains the long-time dynamics of the derivatives ut , without requiring any assumption concerning monotonicity or decay in time of a(t, x). |
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