Existence of global solutions to semilinear damped wave equations with nonlinearities of derivative type

In this paper, we would like to consider the semi-linear damped wave equation with the power nonlinearity of derivative type $\mathrm{\mid }{u}_t{\mathrm{\mid }}^p$. The main contribution of this work is to improve the results for global (in time) solution existence in a comparison with the pioneering paper \cite{Matsumura1976} of Matsumura, who first established that the solutions exist globally for $p>1$ ($n=1$) and $p\ge 2$ ($n\ge 2$). More precisely, we have extended such a result for any $p>1$ ($n=1,2$) and $p>3\mathrm{/}2$ ($n=3$). Our approach relies on constructing appropriately weighted solution spaces linked to the delicate application of several tools from Harmonic Analysis and Banach fixed-point theorem.