Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$ - Karachalios, Nikos; Stavrakakis, Nikos M. (Topological Methods in Nonlinear Analysis, 2001)
We discuss the asymptotic behavior of solutions of the nonlinearlydamped wave equation$$u_{tt} +δ \vert u_t\vert ^{m-1}u_t -φ (x)Δ u = λ u\vert u\vert ^{β -1}, \quad x \in \mathbb R^n, \ t ≥ 0,$$with theinitial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1(x)$, in the case where $N ≥ 3$, $ δ > 0$ and $(φ(x))^{-1} =g (x)$ is a positive function lying in $L^{p}(\mathbb R^n)\cap L^{∞}(\mathbb R^n)$, for some $p$. We proveblow-up of solutions when the source term dominates over thedamping, and the initial energy is assumed to be positive. We alsodiscuss global existence energy decay of solutions.