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

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