Sharp behavior of semilinear damped wave equations driven by mixed local-nonlocal operators

This paper investigates the Cauchy problem for the semilinear damped wave equation $u_{tt}+\mathcal{L}_{a,b}u+u_t=|u|^p$ with the mixed local-nonlocal operator $\mathcal{L}_{a,b}:=-a\Delta+b(-\Delta)^{\sigma}$, where $a,b\in\mathbb{R}_+$ and $\sigma\in(0,1)\cup (1,+\infty)$. We determine the critical exponent for this problem being $p_{\mathrm{crit}}=1+\frac{2\min\{1,\sigma\}}{n}$, which sharply separates global in-time existence and finite-time blow-up of solutions. Furthermore, for the super-critical case $p>p_{\mathrm{crit}}$, we establish the asymptotic profiles of global in-time solutions, showing the anomalous diffusion when $\sigma\in(0,1)$ and the classical diffusion when $\sigma\in(1,+\infty)$, together with the sharp decay estimates. For solutions blowing up in finite time when $1<p\leqslant p_{\mathrm{crit}}$, we derive the sharp estimates for upper and lower bounds of lifespan. Our results reveal the crucial influence of mixed operators on the qualitative properties of solutions, fundamentally governing their critical phenomena, large-time behavior and blow-up dynamics, via $\max\{1,\sigma\}$ or $\min\{1,\sigma\}$.