Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

Abstract

In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi-valued hyperbolic differential inclusionswith memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: h(∇u) = −∇φ · (a∇u) and prove the general decay estimates of equivalent energy.

Blow‐up of solution of wave equation with internal and boundary source term and non‐porous viscoelastic acoustic boundary conditions

“…After [25] many authors showed decay results of the energy to problems involving memory and acoustic boundary conditions, see [3,4,[15][16][17][18]23,26]. We highlight the work of Boukhatem and Benabderrahmane [4] where was showed blow up of solutions.…”

Exponential stabilization of wave equation with acoustic boundary conditions and its application to memory‐type boundary

“…The asymptotic behavior was obtained in Beale 3 (Theorem 2.6). Recently, wave equations with acoustic boundary conditions have been studied by many authors 5–19 …”

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

“…For more papers on acoustic boundary conditions studying the interaction with memory or delay terms, we can refer to previous studies. [8][9][10] Generally, the case of variable coefficients (A(x) ≠ I) reflects the inhomogeneous nature of materials which has a wide application in natural sciences. Many researchers investigated wave equations with variable coefficients.…”

“…In addition, we know from Gal et al 7 that the dynamic boundary conditions has a connection with acoustic boundary conditions. For more papers on acoustic boundary conditions studying the interaction with memory or delay terms, we can refer to previous studies 8‐10 …”