# Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions

@article{Calatroni2012GlobalST, title={Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions}, author={Luca Calatroni and Pierluigi Colli}, journal={Nonlinear Analysis-theory Methods \& Applications}, year={2012}, volume={79}, pages={12-27} }

## 60 Citations

### The Allen–Cahn equation with dynamic boundary conditions and mass constraints

- Mathematics
- 2014

The Allen–Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint,…

### Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type

- MathematicsAdvances in Nonlinear Analysis
- 2018

Abstract The well-posedness of a system of partial differential equations with dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk…

### Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn–Hilliard type with singular potential

- MathematicsAsymptotic Analysis
- 2021

We consider a class of Cahn–Hilliard equation that models phase separation process of binary mixtures involving nontrivial boundary interactions in a bounded domain with non-permeable wall. The…

### On a Cahn–Hilliard system with convection and dynamic boundary conditions

- Mathematics
- 2017

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn–Hilliard type; an additional convective term with a forced velocity…

### Global Existence for a Nonstandard Viscous Cahn-Hilliard System with Dynamic Boundary Condition

- MathematicsSIAM J. Math. Anal.
- 2017

A model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli is studied, and a dynamic boundary condition involving the Laplace--Beltrami operator for the order parameter is considered.

### Limiting Problems for a Nonstandard Viscous Cahn–Hilliard System with Dynamic Boundary Conditions

- Mathematics
- 2017

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions.…

### Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials

- Mathematics
- 2015

### A Cahn-Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

- Mathematics
- 2022

. A system with equation and dynamic boundary condition of Cahn–Hilliard type is considered. This system comes from a derivation performed in Liu–Wu (Arch. Ration. Mech. Anal. 233 (2019), 167–247)…

### Lagrange multiplier and singular limit of double obstacle problems for the Allen–Cahn equation with constraint

- Mathematics
- 2015

We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the…

## References

SHOWING 1-10 OF 28 REFERENCES

### The non-isothermal Allen-Cahn equation with dynamic boundary conditions

- Mathematics
- 2008

We consider a model of nonisothermal phase transitions taking
place in a bounded spatial region. The order parameter $\psi$ is governed by
an Allen-Cahn type equation which is coupled with the…

### Convergence to steady states of solutions of the Cahn–Hilliard and Caginalp equations with dynamic boundary conditions

- Mathematics, Chemistry
- 2006

We consider a solution of the Cahn–Hilliard equation or an associated Caginalp problem with dynamic boundary condition in the case of a general potential and prove that under some conditions on the…

### Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

- MathematicsNumerische Mathematik
- 2003

Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u0.

### ON THE ASYMPTOTIC BEHAVIOR OF THE CAGINALP SYSTEM WITH DYNAMIC BOUNDARY CONDITIONS

- Mathematics
- 2008

We consider a phase-field system of Caginalp type on a
three-dimensional bounded domain. The order parameter $\psi $
fulfills a dynamic boundary condition, while the (relative)
temperature $\theta…

### Abstract approach to evolution equations of phase-field type and applications

- Mathematics
- 1999

A general system of abstract nonlinear parabolic equations deriving from phase-field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite…

### Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation

- Mathematics
- 2000

We consider an initial and boundary value problem for a nonlinear Volterra integrodifferential equation. This equation governs the evolution of a pair of state variables, u and θ, which are mutually…

### Asymptotic behavior of a parabolic-hyperbolic system

- Mathematics
- 2004

We consider a parabolic
equation nonlinearly coupled with a damped semilinear wave
equation. This system describes the evolution of the relative
temperature $\vartheta$ and of the order parameter…

### ROBUST EXPONENTIAL ATTRACTORS FOR SINGULARLY PERTURBED PHASE-FIELD TYPE EQUATIONS

- Mathematics
- 2002

In this article, we construct robust (i.e. lower and upper semi- continuous) exponential attractors for singularly perturbed phase-eld type equations. Moreover, we obtain estimates for the symmetric…