Existence and Blow up Time Estimate for a Negative Initial Energy Solution of a Nonlinear Cauchy Problem

Abstract

In this paper, we consider nonlinear wave equations with dissipation having the form

$$ u_{tt} - \text{div}\bigl(|\nabla u|^{\gamma -2} \nabla u \bigr) + b(t , x)|u_{t}|^{m-2} u_{t} = g(x , u) $$

for $(t, x)\in [0 , \infty )\times \mathbb{R}^{n}$. We obtain existence and blow up results under suitable assumptions on the positive function $b(t , x)$ and the nonlinear function $g(x , u)$. The existence result was obtained using the Galerkin approach while the blow up result was obtained via the perturbed energy method. Our result improves on the perturbed energy technique for unbounded domains.