Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary

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

“…Furthermore, our results allow a larger class of relax functions which are not necessarily of exponential and polynomial decay. Therefore, this improves earlier results in the literature [22,27].…”

“…Moreover, we note that our result also holds for problem (1.1)-(1.4) with a = 1 and l(t) = 0 and without imposing strong damping term, thus our result improves the one of Bae et al [27]. More precisely, the estimate (3.27) and (3.28) generalizes the exponential and polynomial decay result given in [22,27]. Indeed, we obtain exponential decay for g(t) = c and polynomial decay for g(t) = c(1 + t) -1 , where c is a positive constant.…”

“…Further, our result allows certain kernels which are not necessarily of exponential or polynomial decay. In this way, we improved the results of Santos et al [22], in which they considered problem (1.1)-(1.4) with a = 1 and in the absence of l(t)h (∇u). Moreover, we note that our result also holds for problem (1.1)-(1.4) with a = 1 and l(t) = 0 and without imposing strong damping term, thus our result improves the one of Bae et al [27].…”

“…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. Hence, the main purpose of this article is to answer the above question for system (1.1)-(1.4).…”

“…By denoting k the resolvent kernel of g', he showed that the solution decays exponentially (or polynomially) to zero provided k decays exponentially (or polynomially) to zero. Later on, Santos et al [22] generalized this result to a nonlinear n-dimensional equation of Kirchhoff type of the form…”