Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density

Abstract

In this paper, we consider the initial boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force termUnder suitable assumptions, we prove the existence of a global solution by means of the Galerkin method, establish the exponential stability result by using only one simple auxiliary functional, and give the polynomial stability result. MSC: 35L05; 35L15; 35L70

Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems

“…We will give some classical examples to illustrate possible applications of the developed abstract theory. However, the approach is quite flexible and can be easily applied to other types of wave equations with memory and variable density [15], outside a star-shaped domain [28] and orbital stability [30].…”

Uniqueness theorem for partially observed elliptic systems and application to asymptotic synchronization

“…However, the approach is quite flexible and can easily be applied to other types of wave equations with variable density, or outside a star-shaped domain (cf. [5,12,13]).…”

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

“…Aassila, Cavalcanti and Soriano [7] established exponentially decaying and polynomially decaying estimates for the energy of a constant-coefficient wave equation governing the vibration of materials occupying a domain whose boundary is of viscoelasticity under different conditions, respectively; in the meanwhile, they justified that the assertion that the energy approaches zero as time goes to infinity holds for all linear viscoelastic wave equations on bounded domains subject to homogeneous Dirichlet boundary condition. The idea in References [5,7] is strikingly illuminating for later study of problems for viscoelastic wave equations; see References [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and the vast references cited therein. For example, Cavalcanti, Domingos Cavalcanti and Ferreira [8] considered the following initial boundary value problem…”