Existence and Nonexistence of Global Solutions for a Wave System with Different Structural Damping Terms

Abstract

In this article, we study the Cauchy problem for a weakly coupled system of semi-linear wave equations with different structural damping terms. The main goal is to find the threshold, which classifies the existence of small data global (in time) solutions or the nonexistence of global solutions under the growth condition of the nonlinearities.

Appendix

Proposition 2 (Fractional Gagliardo-Nirenberg inequality)

Let $1<r_{0}, r_{1}, r_{2}<\infty$, σ > 0 and s ∈ [0,σ). Then, it holds:

$$ \|u\|_{\dot{H}^{s}_{r_{0}}}\lesssim \|u\|_{L^{r_{1}}}^{1-\theta} \|u\|_{\dot{H}^{\sigma}_{r_{2}}}^{\theta} $$

for all $u\in L^{r_{1}} \cap \dot {H}^{\sigma }_{r_{2}}$, where $\theta = \theta _{s,\sigma ,n}(r_{0},r_{1},r_{2})= \frac {\frac {1}{r_{1}}-\frac {1}{r_{0}}+\frac {s}{n}}{\frac {1}{r_{1}}-\frac {1}{r_{2}}+\frac {\sigma }{n}}$ and $\frac {s}{\sigma }\leq \theta \leq 1$.

For the proof one can see [16].