Global existence and nonexistence for nonlinear wave equations with damping and source terms

Rammaha, Mohammad; Strei, Theresa

doi:10.1090/S0002-9947-02-03034-9

Global existence and nonexistence for nonlinear wave equations with damping and source terms
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by Mohammad A. Rammaha and Theresa A. Strei PDF

Trans. Amer. Math. Soc. 354 (2002), 3621-3637 Request permission

Abstract:

We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term $\left |u\right |^{m-1}u_t$ and a source term of the form $\left |u\right |^{p-1}u$, with $m, p>1$. We show that whenever $m\geq p$, then local weak solutions are global. On the other hand, we prove that whenever $p>m$ and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.

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