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

Bilgesu A Bilgin,Varga K Kalantarov

doi:10.3934/cpaa.2018048

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.

Highlights

The paper is devoted to the question of nonexistence of global solutions of second order abstract wave equations of the formP utt + Au + Qut = F (u), (1)under the initial conditions u(0) = u0, ut(0) = u1 (2)in a real Hilbert space H with the inner product (·, ·) and the corresponding norm ·
As far as we know first result about nonexistence of a global solution of an evolution equation of the form (1) in a Hilbert space H is a global non-existence theorem obtained by using the energy method in [34] for the equation (1) with P = I, Q = 0 and a nonlinear gradient operator F (·) with potential G(u) that satisfy the conditions (F (u), u) ≥ νG(u), (F (u), u) ≥ G( u 2)
Our main goal is to show that non-existence of global solutions with arbitrary positive initial energy of the problem (1) can be established for wider class of nonlinear wave equations than equations considered in the preceeding papers by using the Lemma 1.1 and a modification of the following theorem on growth of solutions of the problem (1), (2) with Q = 0

Summary

Introduction

In a real Hilbert space H with the inner product (·, ·) and the corresponding norm ·. As far as we know first result about nonexistence of a global solution of an evolution equation of the form (1) in a Hilbert space H is a global non-existence theorem obtained by using the energy method in [34] for the equation (1) with P = I, Q = 0 and a nonlinear gradient operator F (·) with potential G(u) that satisfy the conditions (F (u), u) ≥ νG(u),. Our main goal is to show that non-existence of global solutions with arbitrary positive initial energy of the problem (1) can be established for wider class of nonlinear wave equations than equations considered in the preceeding papers by using the Lemma 1.1 and a modification of the following theorem on growth of solutions of the problem (1), (2) with Q = 0.

Set t

Integrating this inequality and rearranging we get

Translating back this inequality equivalently reads

Since p m

The energy equality in this case has the form