Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866910451267272704 |
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| author | Folino, Raffaele Plaza, Ramón G. Strani, Marta |
| author_facet | Folino, Raffaele Plaza, Ramón G. Strani, Marta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_11155 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion Folino, Raffaele Plaza, Ramón G. Strani, Marta Analysis of PDEs 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. |
| title | Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1907.11155 |