Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation (original) (raw)
Existence and regularity of a global attractor for doubly nonlinear parabolic equations
2002
In this paper we consider a doubly nonlinear parabolic partial differential equation ∂β(u) ∂t −∆pu+ f(x, t, u) = 0 in Ω× R, with Dirichlet boundary condition and initial data given. We prove the existence of a global compact attractor by using a dynamical system approach. Under additional conditions on the nonlinearities β, f , and on p, we prove more regularity for the global attractor and obtain stabilization results for the solutions.
Existence and Attractors of Solutions for Nonlinear Parabolic Systems
2001
We prove existence and asymptotic behaviour results for weak solutions of a mixed problem (S). We also obtain the existence of the global at- tractor and the regularity for this attractor in H2() 2 and we derive estimates in @ (0;T ) (b1(u1(x; 0);b2(u2(x; 0)) = (b1('0(x));b2( 0(x))) in where is a bounded open subset in RN , N 1, with a smooth boundary @ : (S) is an example of nonlinear parabolic systems modelling a reaction dif- fusion process for which many results on existence, uniqueness and regularity have been obtained in the case where bi(s) = s ( see, for instance (6; 7; 18)). The case of a single equation of the type (S) is studied in (1; 2; 3; 4; 5; 8; 9; 19): The purpose of this paper is the natural extension to system (S) of the results by (8), which concerns the single equation @ (u) @t u +f(x;t;u) = 0: Actually, our work generalizes the question of existence and regularity of the global attractor obtained therein. In the rst section of this paper, we give some assu...
2010
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.
Global attractors for multivalued semiflows with weak continuity properties
Nonlinear Analysis: Theory, Methods & Applications, 2014
A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.
Attractors for Nonautonomous Parabolic Equations without Uniqueness
International Journal of Differential Equations, 2010
Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.
Pullback Attractors for a Non-Autonomous Semi-Linear Degenerate Parabolic Equation
Glasgow Mathematical Journal, 2010
In this paper, using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semi-linear degenerate parabolic equation in an arbitrary domain, without restriction on the growth order of the polynomial type non-linearity and with a suitable exponential growth of the external force. The obtained results improve some recent ones for the non-autonomous reaction–diffusion equations.
On a Semilinear Strongly Degenerate Parabolic Equation in an Unbounded Domain
2013
We study the existence and long-time behavior of solutions to a semilinear strongly degenerate parabolic equation on R under an arbitrary polynomial growth order of the nonlinearity. To overcome some significant difficulty caused by the lack of compactness of the embeddings, the existence of global attractors is proved by combining the tail estimates method and the asymptotic a priori estimate method.
Continuity of attractors for a nonlinear parabolic problem with terms concentrating in the boundary
2012
We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter goes to zero. Also, we suppose that the "inner boundary" of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to. Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, we also show the lower semicontinuity of the attractors at = 0.