The Allen-Cahn equation is a (parabolic) reaction-diffusion equation with a balanced bistable reaction term, which describes phase transition processes. It is well-known that when the diffusion coefficient is very small, the solutions exhibit very interesting phenomena. In the one-dimensional case, we have an example of metastable dynamics, while in the multi-dimensional case the Allen-Cahn equation is strictly related to the mean curvature flow. In this paper we discuss such phenomena in the case of some hyperbolic variations of the Allen-Cahn equation. In particular, in the one-dimensional case we focus the attention on the assumptions needed to have metastability and we show some numerical solutions in the case such assumptions are not satisfied.

Motion of interfaces for hyperbolic variations of the Allen-Cahn equation

C. Lattanzio;
2020

Abstract

The Allen-Cahn equation is a (parabolic) reaction-diffusion equation with a balanced bistable reaction term, which describes phase transition processes. It is well-known that when the diffusion coefficient is very small, the solutions exhibit very interesting phenomena. In the one-dimensional case, we have an example of metastable dynamics, while in the multi-dimensional case the Allen-Cahn equation is strictly related to the mean curvature flow. In this paper we discuss such phenomena in the case of some hyperbolic variations of the Allen-Cahn equation. In particular, in the one-dimensional case we focus the attention on the assumptions needed to have metastability and we show some numerical solutions in the case such assumptions are not satisfied.
1-60133-023-5
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11697/186512
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact