The influence of damping and source terms on solutions of nonlinear wave equations

Mohammad Rammaha

doi:10.5269/bsmp.v25i1-2.7427

Abstract

We discuss in this paper some recent development in the study of nonlinear wave equations. In particular, we focus on those results that deal with wave equations that feature two competing forces.One force is a damping term and the other is a strong source. Our central interest here is to analyze the influence of these forces on the long-time behavior of solutions.

Highlights

In general the proportionality coefficients in the damping term aut + bu3t are nonconstants and they may depend on the longitudinal displacement u, ∇u and other physical quantities
Studying nonlinear wave equations with a damping term which depends on the longitudinal displacement u(x, t) is more subtle
When Gk ≡ 0, k = 0, 1, 2, one can appeal to a variety of methods to show that most solutions of (1.1) and (1.2) blow up in finite time

Summary

For n the proof of Theorem

See [9] for more details. We move the discussion towards recent blow up results. The blow up Theorem that appeared in [9,36,38] are more restrictive and they only deal with weak solutions that satisfy a variational equality with the added requirement that u and |u|kj(ut) ∈ L2(Ω × (0, T )). In order to state Theorem 2.9, we should specify the range of parameters k, p, m for which local generalized solutions do exist, but the condition k + m ≥ p (which results in a global solution) is violated. This leads to the following range of parameters:. There exists a unique local weak solution (u, v) to (3.1) defined on [0, T ] for some T > 0; with u, v ∈ C([0, T ], H01(Ω)), ut, vt ∈ C([0, T ], L2(Ω)), ut ∈ Lm+1(Ω × (0, T )), vt ∈ Lr+1(Ω × (0, T )), and u satisfies the following energy identity:

In addition

The critical points of the functional

There exists θ