Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion

Abstract

Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In the one-dimensional model with a balanced bistable reaction function, it is well-known that there is persistence of metastable patterns for an exponentially long time, i.e. a time proportional to $\exp(C/\e)$ where $C,\e$ are strictly positive constants and $\e^2$ is the diffusion coefficient. In this paper, we extend such results to the case when the linear diffusion flux is substituted by the mean curvature operator both in Euclidean and Lorentz--Minkowski spaces. More precisely, for both models, we prove existence of metastable states which maintain a transition layer structure for an exponentially long time and we show that the speed of the layers is exponentially small. Numerical simulations, which confirm the analytical results, are also provided.

27 pages, 5 figures

References in corpus (1)

• Heteroclinic traveling fronts for a generalized Fisher-Burgers equation with saturating diffusion

Cited by in corpus (4)

• Layered patterns in reaction-diffusion models with Perona-Malik diffusions
• Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms
• Long-time behavior of solutions to the generalized Allen-Cahn model with degenerate diffusivity
• Minimization of a Ginzburg-Landau functional with mean curvature operator in 1-D