Non-existence of global solutions to nonlinear wave equations with positive initial energy

Abstract

We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained. 2000 Mathematics Subject Classification. Primary: 35B44, 35L71; Secondary: 35Q74.

Blow-up in damped abstract nonlinear equations

“…I thank Professor Howard Levine for calling our attention to his work with Professor Todorova, [18]. We also thank to Professor Varga Kalantarov for sharing his articles, especially [1]. Finally, we thank to the anonymous referees for their valuable comments that improved the final form of this article.…”

“…Ψ(u 0 ). Moreover, by (2), there exist exactly two different roots of J (s) = 1 2 Φ(u 0 , u 1 ), denoted by α δ and β δ , such that…”

“…We shall use the functions σ ν and µ λ,δ to find the values of α δ and β δ , respectively. Remember that these are the roots of J (s) = 1 2 Φ(u 0 , u 1 ). To find α δ , we consider the equation (10) J…”

“…In [1], the abstract problem (P) is studied with a more general linear damping term. In that paper, the nonexistence of global solutions is analyzed for any positive value of the initial energy.…”

A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term

“…Using the concavity approach, the blow-up of solutions was investigated. We also refer to [9][10][11][12][13][14][15], where the large-time behavior of solutions to nonlinear wave equations has been studied by the energy and concavity methods. The applications of fractional calculus are broad.…”