Existence, blow-up and exponential decay estimates for the nonlinear Kirchhoff-Carrier wave equation in an annular with nonhomogeneous Dirichlet conditions

Abstract

This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.

Existence and general decay estimates of the Dirichlet problem for a nonlinear Kirchhoff wave equation in an annular with viscoelastic term

“…In the proofs, we use the multiplier technique combined with a suitable Lyapunov functional. Our results can be regarded as an extension and improvement of the corresponding results of [7], [10] - [12], [18] - [22], [28], [29].…”

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…”

Blow up of Coupled Nonlinear Klein-Gordon System with Distributed Delay, Strong Damping, and Source Terms

“…ð38Þ By ( 35)-(37), we get (32). And by using Young's inequality, ( 14), (16), and ( 17) in (38), we obtain (33). Now, we define the functional…”