Advances in Differential Equations

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

Pierluigi Colli, Gianni Gilardi, and Maurizio Grasselli

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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 aspects of the approximation to the proof of continuous dependence estimates. In addition, a sufficient condition for the boundedness of the phase variable is given.

Article information

Adv. Differential Equations, Volume 2, Number 3 (1997), 487-508.

First available in Project Euclid: 23 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
Secondary: 35K55: Nonlinear parabolic equations 73B30 80A20: Heat and mass transfer, heat flow


Colli, Pierluigi; Gilardi, Gianni; Grasselli, Maurizio. Well-posedness of the weak formulation for the phase-field model with memory. Adv. Differential Equations 2 (1997), no. 3, 487--508.

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