Nonexistence of global solutions of a quasilinear hyperbolic equation

Abstract

Abstract. In this work, the nonexistence of the global solutions to a class of initial boundary value problems with dissipative terms in the boundary conditions is considered for a quasilinear hyperbolic equation. The nonexistence proof is achieved by the use of a lemma due to O. Ladyzhenskaya and V. K. Kalantarov and by the usage of the so called concavity method. In this method one writes down a functional which reflects the properties of dissipative boundary conditions and represents the norm of the solution in some sense, then proves that this functional satisfies the hypotheses of Ladyzhenskaya-Kalantarov lemma. Hence from the conclusion of the lemma one concludes that in finite time t 2 , this functional and hence the norm of the solution blows up.

Global nonexistence and stability of solutions of inverse problems for a class of Petrovsky systems

“…There are numerous papers devoted to the study of stability and global nonexistence results for direct problems and the existence, uniqueness of solutions of inverse problems for various evolutionary partial differential equations (see [2,6,7,11,[13][14][15]). But less is known about the global nonexistence for solutions of hyperbolic and parabolic inverse problems.…”

Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

“…In this paper, we consider the following initial-boundary value problem [1,2,3] as well as in geophysics and ocean acoustics, where for example, the coefficient () ax represents the "effective tension" [6].…”

“…In [1], Bayrak and Can considered the following a nonlinear wave equation with initial-boundary conditions…”

Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term

“…Amroun and Benaissa [1] investigated (1.2) with f(u, u t ) = b|u| p-2 u-h(u t ) and proved the global existence of solutions by means of the stable set method in H 2 0 (Ω) combined with the Faedo-Galerkin procedure. In [3], Messaoudi studied problem (1.2) with f(u, u t ) = b|u| p-2 u-a|u t | m-2 u t .…”

“…Research of global existence, blow-up and energy decay of solutions for the initial boundary value problem (1.2) has attracted a lot of articles (see [1][2][3][4] and references there in).…”