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 $|u_t|^p$. 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 $p > 3/2$ ($n = 3$). 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.