Existence of pullback \(\mathcal{D}\)-attractors for Kirchhoff wave equations with strong damping and delay

Abstract

This paper studies the existence of pullback $\mathcal{D}$-attractors for non-autonomous Kirchhoff wave equations with strong damping and delay effects in the phase space $\mathcal{E} = C_{H^1_0(\Omega)} \times C_{L^2(\Omega)}$. The model contains a strong damping term $-\Delta \partial_t u$, a nonlocal Kirchhoff term $\Psi(\|\nabla u\|^2)$, a state-dependent delay term $\phi(t, u_t)$, and a time-dependent external force $h(x, t)$. There appear to be no results on the pullback attractors of Kirchhoff wave equations involving both strong damping and delay effects in the literature. To this end, we establish delicate uniform estimates and employ the contraction function method to prove the pullback asymptotic compactness of the associated process, which yields the existence of pullback $\mathcal{D}$-attractors.