Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

Jacson Simsen

doi:10.14232/ejqtde.2019.1.68

Abstract

Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

Highlights

In this work, we will study the following problem ∂us ∂t (t) −div( Ds |∇us (t)| ps (x)−2 ∇us (t)) +C(t)|us (t)|ps(x)−2us =B(t, us (t)), t > τ, us(τ) = uτs, (1.1)under homogeneous Neumann boundary conditions, uτs ∈ H := L2(Ω), Ω ⊂ Rn (n ≥ 1) is a smooth bounded domain, Ds ∈ [1, ∞), ps(·) ∈ C(Ω ), p−s := minx∈Ωps(x) > 2, and there exists a constant a > 2 such that p+s := maxx∈Ωps(x) ≤ a, for all s ∈ N
We prove continuity of the flow and weak upper semicontinuity of the family of pullback attractors as s goes to infinity for the problem (1.1) with respect to the couple of parameters (Ds, ps), where ps is the variable exponent and Ds is the diffusion coefficient
In the result we prove the continuity of the solutions of (2.1) with respect to the initial data and exponent parameter

Summary

Introduction

Under homogeneous Neumann boundary conditions, uτs ∈ H := L2(Ω), Ω ⊂ Rn (n ≥ 1) is a smooth bounded domain, Ds ∈ [1, ∞), ps(·) ∈ C(Ω ), p−s := minx∈Ωps(x) > 2, and there exists a constant a > 2 such that p+s := maxx∈Ωps(x) ≤ a, for all s ∈ N. Under homogeneous Neumann boundary conditions, uτs ∈ H := L2(Ω), Ω ⊂ Rn (n ≥ 1) is a smooth bounded domain, B : H → H is a globally Lipschitz map with Lipschitz constant L ≥ 0, Ds ∈ [1, ∞), C(·) ∈ L∞([τ, T]; R+) is bounded from above and below and is monotonically nonincreasing in time, ps(·) ∈ C(Ω ), p−s := minx∈Ωps(x) ≥ p, p+s := maxx∈Ωps(x) ≤ a, for all s ∈ N, when ps(·) → p in L∞(Ω) and Ds → ∞ as s → ∞, with a, p > 2 positive constants They proved continuity of the flows and upper semicontinuity of the family of pullback attractors.

The family of nonautonomous ODEs and its limit problem

Continuity of the flow and weak upper semicontinuity of attractors

Final remarks