Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term

Abstract

We consider the nonlinear model of the wave equation ytt − ∆y + f0 (∇y) = 0 subject to the following nonlinear boundary conditionsWe show existence of solutions by means of Faedo-Galerkin method and the uniform decay is obtained by using the multiplier technique.

General decay for a wave equation of Kirchhoff type with a boundary control of memory type

“…Although the subject is important, there are few mathematical results in the presence of the nonlinearity given by h(∇u), see [24][25][26]. In light of this and previous articles [17,22], it is interesting to investigate whether we still have the similar general decay result as in [17] for nondissipative distributed system (1.1) with the memory-type damping acting on a part of the boundary.…”

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

“…The wellposedness of problem (2.1) can be treated by the standard semigroup theory, as in [7] or Faedo-Galerkin procedure as in [1]. In fact, we have the following preliminary result: (i) For any initial data u 0 , u 1 ∈ V × L 2 ( ) there exists a unique weak solution of (2.1) in the class…”

“…where is a bounded domain of R n (n ≥ 2) having a smooth boundary := ∂ , such that = 0 ∪ 1 with 0 , 1 ]. Let f : → R be a smooth function; define ω 1 to be a neighbourhood of 1 , and subdivide the boundary 0 into two parts: * 0 = {x ∈ 0 ; ∂ ν f > 0} and 0 \ * 0 .…”

Global solutions for a nonlinear hyperbolic equation with boundary memory source term

“…We refer to [8,9] to see the details. In many works concerned with equations of type (1.1), we cite Aassila et al [1], where the following wave equation was considered:…”