Optimal \(L^p\)-\(L^q\)-type decay rates of solutions to the three-dimensional nonisentropic compressible Euler equations with relaxation (Q2064733)

Summary: In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of \textit{Y. Wu} et al. [ibid. 2021, Article ID 5512285, 13 p. (2021; Zbl 1490.35290)], we show the existence and uniqueness of the global small \(H^k\) (\(k \geqslant 3\)) solution only under the condition of smallness of the \(H^3\) norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal \(L^p\)-\(L^q\) (\(1 \leqslant p \leqslant 2\), \(2 \leqslant q \leqslant \infty\))-type decay rates of the solution and its higher-order derivatives.