Attractors for degenerate parabolic type equations by Messoud Efendiev

By Messoud Efendiev

This booklet offers with the long-time habit of options of degenerate parabolic dissipative equations coming up within the research of organic, ecological, and actual difficulties. Examples contain porous media equations, -Laplacian and doubly nonlinear equations, in addition to degenerate diffusion equations with chemotaxis and ODE-PDE coupling platforms. For the 1st time, the long-time dynamics of assorted sessions of degenerate parabolic equations, either semilinear and quasilinear, are systematically studied when it comes to their worldwide and exponential attractors. The long-time habit of many dissipative structures generated through evolution equations of mathematical physics could be defined when it comes to international attractors. relating to dissipative PDEs in bounded domain names, this attractor often has finite Hausdorff and fractal measurement. accordingly, if the worldwide attractor exists, its defining estate promises that the dynamical method decreased to the attractor comprises the entire nontrivial dynamics of the unique procedure. furthermore, the decreased section area is de facto "thinner" than the preliminary part house. in spite of the fact that, not like nondegenerate parabolic sort equations, for a really huge classification of degenerate parabolic style equations, their international attractors may have limitless fractal measurement. the most aim of the current booklet is to provide an in depth and systematic research of the well-posedness and the dynamics of the semigroup linked to special degenerate parabolic equations when it comes to their worldwide and exponential attractors. primary issues contain life of attractors, convergence of the dynamics and the speed of convergence, in addition to the selection of the fractal size and the Kolmogorov entropy of corresponding attractors. The research and ends up in this publication express that there are new results regarding the attractor of such degenerate equations that can't be saw with regards to nondegenerate equations in bounded domain names. This booklet is released in cooperation with genuine Sociedad Matemática Española (RSME)

Show description

Read Online or Download Attractors for degenerate parabolic type equations PDF

Best differential equations books

The Asymptotic Solution of Linear Differential Systems: Applications of the Levinson Theorem

The trendy idea of linear differential platforms dates from the Levinson Theorem of 1948. it is just in additional contemporary years, although, following the paintings of Harris and Lutz in 1974-7, that the importance and diversity of purposes of the concept became liked. This e-book provides the 1st coherent account of the wide advancements of the final 15 years.

Extra resources for Attractors for degenerate parabolic type equations

Sample text

1. As was shown in [9] the assumption that S(t) : Φ → Φ be continuous for every fixed t can be replaced by the closedness of the graph {(u0 , S(t)u0 ), u0 ∈ Φ}. 2. Although the global attractor usually has a very complicated geometric structure, there exists one exceptional class of DS for which the global attractor has a relatively simple structure which is completely understood, namely the DS having a global Lyapunov function. , L(S(t)u0 ) = L(u0 ) for some t > 0 and u0 implies that u0 is an equilibrium of S(t).

3. 17) dimF (MS , H1 ) ≤ C1 , where dimF is the fractal dimension in H1 . , SMS ⊂ MS . 18) distH1 (S(n)B, MS ) ≤ C2 e−αn , n ∈ N, for some α > 0. 19) distsymm (MS1 , MS2 ) ≤ C3 S1 − S2 H1 κ S, where the positive constants Ci , i = 1, 2, 3, α, and κ only depend on B, H, H1 , δ, and K but are independent of the concrete choice of S ∈ Sδ,K (B). Proof. We first fix a finite covering of the set B by δ/K-balls in the space H (such a covering exists since the embedding H1 ⊂ H is compact). Let V0 := {h1 , .

In particular, this means that the dynamics restricted to the regular attractor A are also preserved (up to homeomorphisms) under perturbations. 1 (whenever necessary) to a class of PDEs arising in mathematical physics. We especially emphasize that one of the challenging questions in the theory of attractors is, in which sense are the dynamics on the global attractor finite dimensional? As already mentioned, the global attractor is usually not a manifold but has a rather complicated geometric structure.

Download PDF sample

Rated 4.28 of 5 – based on 7 votes