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

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+(-Δ)^σu+(-Δ)^{σ/2}\partial_{t}u=\left|u\right| ^{p}, \ σ\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^{σ,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 $n\ge 1$. Moreover, in each case, we have no loss of decay estimates of the unique solution with respect to the corresponding linear models.

This paper deals with a generalized class of Cauchy data and its influence on the so-called critical exponent. All comments are welcome!