Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions

TL;DR

This paper proves the global existence of classical solutions for mass-conserving quadratic and super-quadratic reaction-diffusion systems in three or more dimensions, using advanced regularity and bootstrap techniques.

Contribution

It introduces new methods to establish global classical solutions for complex reaction-diffusion systems with mass conservation in higher dimensions.

Findings

Global existence of classical solutions in 3+ dimensions.

Solutions are uniformly bounded in time under entropy dissipation.

Established exponential convergence to equilibrium.

Abstract

This paper considers quadratic and super-quadratic reaction-diffusion systems for reversible chemistry, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.