Article
Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators
Author/s | Boldrin Belluzi, Maykel
Caraballo Garrido, Tomás Nascimento, Marcelo J.D. Schiabel, Karina |
Department | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Publication Date | 2022-01-24 |
Deposit Date | 2022-03-03 |
Published in |
|
Abstract | 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 ... 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). |
Citation | Boldrin Belluzi, M., Caraballo Garrido, T., Nascimento, M.J.D. y Schiabel, K. (2022). Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators. Journal of Differential Equations, 314, 808-849. |
Files | Size | Format | View | Description |
---|---|---|---|---|
Smoothing effect and asymptotic ... | 481.0Kb | [PDF] | View/ | |