Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

Abstract

This paper is devoted to the study of a system of nonlinear equations with nonlinear boundary conditions. First, on the basis of the Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, the exponential decay property of the global solution via the construction of a suitable Lyapunov functional is presented. Copyright © 2013 John Wiley & Sons, Ltd.

Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms

“…By employing the nonlinear semigroups and the theory of monotone operators, the well-posedness of (1.2) is moderately investigated. An important result obtained in [7] and further in [14] is that every weak solution blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping involved in the system. In [3,4], Cavalcanti et al studied the existence of global solutions, and showed the relation between the asymptotic behavior of the energy and the degenerate system of wave equations with boundary conditions of memory type.…”

“…Accounting for the Kirchhoff-Love plate theory in shear deformations, the system (1.1) is closely related to the Reissner-Mindlin plate equations (see [9]), structured by three coupled wave and wave-like equations involving the influence of nonlinear damping and source terms. Mathematically, systems of wave equations have been extensively studied by many authors, see [1,4,7,14,15,16] and references therein where the existence, regularity and the asymptotic behavior of solutions are investigated.…”

“…In recent years, various types of wave equations with linear or nonlinear damping and sources have been solved by using the Galerkin approximation (see, for instance, [2,12,14,15,17]). Based on a priori estimates, weak convergence, and compactness techniques, and via the construction of a suitable Lyapunov functional, the existence, regularity, blow-up, and exponential decay estimates of solutions for such typical wave equations have been proved in [2,12,17].…”

“…Nevertheless, the circumstance for the coupled system (1.1)- (1.4) is still open and its rigorous treatment is technically demanding in this direction. In [14], the authors have solved the similar coupled system where the results are controlled not only by the damping orders r 1 , r 2 , but also by the involved parameters on the boundary. Note that compared again to [14] with the homogeneous Dirichlet boundary condition for u, our paper needs a careful adaptation to handle several different parameters at the same time.…”

Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

“…Below we present an example such that the functions f 1 , f 2 , G satisfy the assumptions (H 3 ) , (H 4 ), respectively. In this example, f 1 , f 2 are more general than the functions given in [19,21].…”