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

Abstract

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided.

Corrigendum to “Existence, blow‐up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions”

“…By giving corrections to the energy functionals E(t) and I(t) as above, with adding the functional (g * u)(t), we also have suitable corrections to our papers. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] Note more that, in case of the problem considered containing the term ∫ t 0 g(t − s)Δu(x, s)ds, we onlygive corrections to the functional I(t) with adding the term ∫ t 0 g(t − s) ‖∇u(t) − ∇u(s)‖ 2 ds in the definition of I(t), where, for example, u ∈ C 0 ( R + ; H 1 0…”