Upper Semicontinuity for Attractors of Parabolic Problems with Localized Large Diffusion and Nonlinear Boundary Conditions

Patterns in Parabolic Problems with Nonlinear Boundary Conditions

In this paper we show the existence of stable nonconstant equilibrium (patterns) for reaction-diffusion equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator is such domains. This information is used to show that the asymptotic dynamic of the heat equations in this domain is eq...

Exponential Attractors for Parabolic Equations with Dynamic Boundary Conditions

We study exponential attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains. First, we give the existence of the exponential attractor in L2(Ω) × L2(Γ) by proving that the corresponding semigroup satisfies the enhanced flattering property. Second, we apply asymptotic a priori estimate and obtain the exponential attractor in Lp(Ω) × L(Γ). Finally, we sh...

Attractors for Parabolic Equations with Nonlinear Boundary Conditions, Critical Exponents, and Singular Initial Data

Blow-up in the Parabolic Problems under Nonlinear Boundary Conditions

In this paper, I consider nonlinear parabolic problems under nonlinear boundary conditions. I establish respectively the conditions on nonlinearities to guarantee that ( , ) u x t exists globally or blows up at some finite time. If blow-up occurs, an upper bound for the blow-up time is derived, under somewhat more restrictive conditions, lower bounds for the blow-up time are also derived.

Upper Semicontinuity of Pullback Attractors for Non-autonomous Generalized 2d Parabolic Equations

This paper is concerned with a generalized 2D parabolic equation with a nonautonomous perturbation −∆ut + α ∆ut + μ∆ u+∇ · −→ F (u) +B(u, u) = ǫg(x, t). Under some proper assumptions on the external force term g, the upper semicontinuity of pullback attractors is proved. More precisely, it is shown that the pullback attractor {Aǫ(t)}t∈R of the equation with ǫ > 0 converges to the global attract...