Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{l}\partial _t {\sf u}(t,{\bf x}) +\sum\nolimits_{k=1}^d {\bf A}^k(t,{\bf x})\partial _{k} {\sf u}(t,{\bf x})={\sf f}(t,{\bf x},{\sf u}(t,{\bf x}))\\ {\sf u}(0,\cdot)={\sf v}\end{array}\right\}$$\end{document}is considered, with each matrix function Ak being diagonal, bounded and locally Lipschitz in x. Discrete models for the Boltzmann equation furnish examples of such systems. For bounded initial data, and right-hand side that is locally Lipschitz and locally bounded in u, local existence and uniqueness results in L∞ are well known, together with some estimates on weak solutions. More precise estimates for weak solutions of the above Cauchy problem will be given, supplemented by estimates on the maximal time of existence for the solution, as well as the local existence and uniqueness in Lp setting (1 < p < ∞).