Effect of different additional $L^{m}$ regularity on semi-linear damped $\sigma$-evolution models

TL;DR

This paper investigates how additional $L^{m}$ regularity influences the global existence of solutions to semi-linear damped $\sigma$-evolution models, revealing that the critical exponent varies with these regularities without affecting decay rates.

Contribution

It demonstrates that different $L^{m}$ regularities alter the critical exponent for global existence in semi-linear $\sigma$-models, providing new insights into regularity effects.

Findings

Critical exponent depends on additional regularity parameters.

Two different critical values are identified under certain conditions.

Decay estimates remain unaffected by the additional regularity.

Abstract

The motivation of the present study is to discuss the global (in time) existence of small data solutions to the following semi-linear structurally damped $\sigma$-evolution models: \begin{equation*} \partial_{tt}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma/2}\partial_{t}u=\left|u\right| ^{p}, \ \sigma\geq 1, \ \ p>1, \end{equation*} where the Cauchy data $(u(0,x), \partial_{t}u(0,x))$ will be chosen from energy space on the base of $L^{q}$ with different additional $L^{m}$ regularity, namely \begin{equation*} u(0,x)\in H^{\sigma,q}(\mathbb{R}^{n})\cap L^{m_{1}}(\mathbb{R}^{n}) , \ \ \partial_{t}u(0,x)\in L^{q}(\mathbb{R}^{n})\cap L^{m_{2}}(\mathbb{R}^{n}), \ \ q\in(1,\infty),\ \ m_{1}, m_{2}\in [1,q). \end{equation*} Our new results will show that the critical exponent which guarantees the global (in time) existence is really affected by these different additional regularities and will take \textit{two different values} under some restrictions on $m_{1}, m_{2}$, $q$, $\sigma$ and the space dimension $n\geq1$. Moreover, in each case, we have no loss of decay estimates of the unique solution with respect to the corresponding linear models.