Existence of a moving attractor for semi-linear parabolic equations (original) (raw)

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...

Attractors of Parabolic Equations Without Uniqueness

Journal of Dynamics and Differential Equations, 2001

In this paper we study the existence of global compact attractors for nonlinear parabolic equations of the reaction-diffusion type and variational inequalities. The studied equations are generated by a difference of subdifferential maps and are not assumed to have a unique solution for each initial state. Applications are given to inclusions modeling combustion in porous media and processes of transmission of electrical impulses in nerve axons.

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.

Traveling wave solutions of parabolic systems

1994

The theory of travelling waves described by parabolic equations and systems is a rapidly developing branch of modern mathematics. This book presents a general picture of current results about wave solutions of parabolic systems, their existence, stability, and ...

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.