Nonlinear pseudoparabolic equations as singular limit of reaction–diffusion equations

Abstract

In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross diffusion, modelling the reaction and diffusion of two biological, chemical, or physical substances, is reduced then such an hyperbolic equation is obtained. For regular solutions even uniqueness can be shown, although the needed regularity can only be proved in two dimensions.

Keywords:

• Pseudoparabolic equation
• Reaction-diffusion equations
• Galerkin's method

Keywords:

• AMS Subject Classifications:
• 35K50
• 35K57
• 35K55
• 35K60
• 35K70

Acknowledgement

This work was supported by SFB 359 Reaktive Strömungen, Diffusion und Transport, University of Heidelberg.