Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions. (English) Zbl 1240.35307

Gal, Ciprian G.; Warma, Mahamadi

Differ. Integral Equ. 23, No. 3-4, 327-358 (2010).

Summary: 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 proved 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.

MSC:

35K91	Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35K61	Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37L30	Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems

Keywords:

semilinear parabolic equation; global attractor; well posedness