# Well-posedness of the weak formulation for the phase-field model with memory

@article{Colli1997WellposednessOT, title={Well-posedness of the weak formulation for the phase-field model with memory}, author={Pierluigi Colli and Gianni Gilardi and Maurizio Grasselli}, journal={Advances in Differential Equations}, year={1997}, volume={2}, pages={487-508} }

A phase–field model based on the Gurtin–Pipkin heat flux law is considered. This model consists in a Volterra integrodifferential equation of hyperbolic type coupled with a nonlinear parabolic equation. The system is then associated with a set of initial and Neumann boundary conditions. The resulting problem was already studied by the authors who proved existence and uniqueness of a smooth solution. A careful and detailed investigation on weak solutions is the goal of this paper, going from the…

## 54 Citations

Uniqueness of Weak Solutions to the Phase-Field Model with Memory

- Mathematics
- 1998

The paper deals with a phase-field model based on the Gurtin-Pipkin heat flux law. A Volterra integrodifferential equation is coupled with a nonlinear parabolic equation in the resulting sys- tem,…

Existence and Stabilization of Solutions to the Phase-Field Model with Memory

- Mathematics
- 1998

A phase field model is considered when the classical Fourier law is replaced by the linearized GurtinPipkin constitutive assumption for the heat flux. The resulting system of partial differential…

Regularity and convergence results for a phase–field model with memory

- Mathematics
- 1998

A phase field model based on the Coleman-Gurtin heat flux law is considered. The resulting system of non-linear parabolic equations, associated with a set of initial and Neumann boundary conditions,…

Weak solutions for the fully hyperbolic phase-field system of conserved type

- Mathematics
- 2007

Abstract.We consider a conserved phase-field system coupling two nonlinear hyperbolic integro-differential equations. The model results from the assumption that the material undergoing phase…

Hyperbolic Phase-Field Dynamics with Memory

- Mathematics
- 2001

Abstract We consider a non-conserved phase-field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations. This model is derived from two basic assumptions: the heat…

Asymptotic Analysis of a Phase-Field Model with Memory

- MathematicsFree boundary problems:
- 2019

A phase-field model accounting for memory effects is considered. This model consists of a hyperbolic integrodifferential equation coupled with a parabolic differential inclusion. The latter relation…

Well-Posedness and Regularity for a Parabolic-Hyperbolic Penrose-Fife Phase Field System

- Mathematics
- 2005

This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The…

Convergence of phase field to phase relaxation models with memory

- MathematicsANNALI DELL UNIVERSITA DI FERRARA
- 1996

This paper is concerned with phase field models accounting for memory effects and based on the linearized Gurtin-Pipkin constitutive assumption for the heat flux. After recalling and generalizing…

Singular limit of a transmission problem for the parabolic phase-field model

- Mathematics
- 2000

A transmission problem describing the thermal interchange between two regions occupied by possibly different fluids, which may present phase transitions, is studied in the framework of the…

Asymptotic Justification of the Conserved Phase-Field Model with Memory

- Mathematics
- 2000

We consider a conserved phase-field model with memory in which the Fourier heat conduction law is replaced by a constitutive assumption of Curtin-Pipkin type; the system is conserved in the sense…

## References

SHOWING 1-10 OF 16 REFERENCES

Weak solution to hyperbolic Stefan problems with memory

- Mathematics
- 1997

Abstract. A model for Stefan problems in materials with memory is considered. This model is mainly characterized by a nonlinear Volterra integrodifferential equation of hyperbolic type. Colli and…

Hyperbolic phase change problems in heat conduction with memory

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 1993

Synopsis The aim of this paper is to formulate and study phase transition problems in materials with memory, based on the Gurtin–Pipkin constitutive assumption on the heat flux. As different phases…

Existence and asymptotic results for a system of integro-partial differential equations

- Mathematics
- 1996

A non-Fourier phase field model is considered. A global existence result for a Dirichlet, or generalized Neumann, initial-boundary value problem is obtained, followed by a discussion of the…

Linear and Quasilinear Equations of Parabolic Type

- Mathematics
- 1968

linear and quasi linear equations of parabolic type by o a ladyzhenskaia 1968 american mathematical society edition in english, note citations are based on reference standards however formatting…

Reviews of Modern Physics: Addendum to the Paper "Heat Waves"

- Physics
- 1989

papers which should have been cited have come to our attention. It appears that our effort to write a relatively complete chronology of thought about heat waves fell somewhat short of the mark. We…

Non-homogeneous boundary value problems and applications

- Mathematics
- 1972

7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{{M_k}}}\left( H \right)$$.- 1.3 The Spaces $${D_{{M_k}}}\left( H…

Heat Waves

- Environmental SciencePlastics Engineering
- 2021

In 1982, the Belgian Pilots' Guild raised the question of what effect exposure to radar radiation--for example, that encountered in passing a pilot launch's radar--might have on the human body.…

Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert

- Mathematics
- 1967

SuntoSi studia il problema di Cauchy per equazioni differenziali lineari astratte in spazi di Hilbert, in una impostazione che permette di trattare contemporaneamente sia le equazioni differenziali…