Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions

Abstract

<p style="text-indent:20px;">Uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimension are proved. The systems are assumed to dissipate the total mass and to have locally Lipschitz nonlinearities of at most (slightly super-) quadratic growth. This pushes forward the recent advances concerning global existence of reaction-diffusion systems dissipating mass in which a uniform-in-time bound has been known only in space dimension one or two. As an application, skew-symmetric Lotka-Volterra systems are shown to have unique classical solutions which are uniformly bounded in time in all dimensions with relatively compact trajectories in <inline-formula><tex-math id="M1">\begin{document}$ C(\overline{\Omega})^m $\end{document}</tex-math></inline-formula>.