Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

Abstract

Abstract. We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, α > 0, or only of Gevrey class, α = 0. We establish the existence of a global attractor for each α ∈ [0, 1], and we show that the family of global attractors is upper-semicontinuous as α → 0. Furthermore, for each α ∈ [0, 1], we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensional compact set in the weak topology of the phase space. This result insures the corresponding global attractor also possess finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter α. In both settings, attractors are found under minimal assumptions on the nonlinear terms.

Finite element error analysis of wave equations with dynamic boundary conditions:L2 estimates

“…Strong damping is of great relevance in engineering due as it increases the robustness of systems against perturbations. Boundary conditions involving strong damping are particularly interesting for applications for physical phenomena that exhibit both elasticity and viscosity when undergoing deformation and for wave-structure interactions, see Graber & Shomberg (2016), Graber & Lasiecka (2014) and Nicaise (2017). We seek u :…”

Longtime behavior of the semilinear wave equation with gentle dissipation

“…(1) has parabolic-like characteristics. Therefore, one should expect the corresponding semilinear problem to exhibit parabolic-like characteristics (See for example [19]).…”

Identification of source terms in wave equation with dynamic boundary conditions

“…The study of wave equations with kinetic boundary conditions of dynamic type requires careful consideration of both the PDE describing the evolution of the system inside the bulk and the evolution on its surface. Taking into account the coupled nature of the wave system, recent research has been done for the physical derivation [17], well-posedness and regularity of such equations, see [3,13,15,[18][19][20] and [31,37,[39][40][41][42][43]. However, in terms of inverse problems and controllability, the existing literature is quite limited compared to the static case (Dirichlet, Neumann, and Robin), despite their importance in applications.…”

“…|f (x, 0)| 2 e 2sϕ(x,0) dx + Γ 1 |g(x, 0)| 2 e 2sϕ(x,0) dS = Ω |f (x, 0)| 2 e 2se λ(ψ 0 (x)+C 1 ) dx +Γ 1 |g(x, 0)| 2 e 2se λ(ψ 0 (x)+C 1 ) dS e 2se λ(d 2 0 +C 1 ) f (•, 0) 2 L 2 (Ω) + g(•, 0) 2 L 2 (Γ 1 ) ,(4.20)where d 0 := min x∈Ω µ(x). By (4 19…”