Differential and Integral Equations

Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions

Ciprian G. Gal and Mahamadi Warma

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Abstract

We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as generalizations of semilinear reaction-diffusion equations with dynamic boundary conditions and various other phase-field models, such as the isothermal Allen-Cahn equation with dynamic boundary conditions. We thus formulate a class of initial and boundary-value problems whose global existence and uniqueness is proven by means of an appropriate Faedo-Galerkin approximation scheme developed for problems with dynamic boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor.

Article information

Source
Differential Integral Equations, Volume 23, Number 3/4 (2010), 327-358.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019321

Mathematical Reviews number (MathSciNet)
MR2588479

Zentralblatt MATH identifier
1240.35307

Subjects
Primary: 35B41: Attractors 35K55: Nonlinear parabolic equations 37L30: Attractors and their dimensions, Lyapunov exponents 80A22: Stefan problems, phase changes, etc. [See also 74Nxx]

Citation

Gal, Ciprian G.; Warma, Mahamadi. Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions. Differential Integral Equations 23 (2010), no. 3/4, 327--358. https://projecteuclid.org/euclid.die/1356019321


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