Long-time behaviour for a non-autonomous Klein-Gordon-Zakharov system
The aim of this paper is to study the long-time dynamics of solutions of the evolution system \[ \begin{cases} u_{tt} - Δu + u + η(-Δ)^{\frac{1}{2}}u_t + a_ε(t)(-Δ)^{\frac{1}{2}}v_t = f(u), & \; (x, t) \in Ω\times (τ, \infty), \\ v_{tt} - Δv + η(-Δ)^{\frac{1}{2}}v_t - a_ε(t)(-Δ)^{\frac{1}{2}}u_t = 0, & \; (x, t) \in Ω\times (τ, \infty), \end{cases} \] subject to boundary conditions \[ u = v = 0, \;\; (x, t)\in \partialΩ\times (τ, \infty), \] where $Ω$ is a bounded smooth domain in $\mathbb{R}^n$, $n \geq 3$, with the boundary $\partialΩ$ assumed to be regular enough, $η> 0$ is constant, $a_ε$ is a Hölder continuous function and $f$ is a dissipative nonlinearity. This problem is a non-autonomous version of the well known Klein-Gordon-Zakharov system. Using the uniform sectorial operators theory, we will show the local and global well-posedness of this problem in $H_0^1(Ω) \times L^2(Ω) \times H_0^1(Ω) \times L^2(Ω)$. Additionally, we prove existence, regularity and upper semicontinuity of pullback attractors.