Existência de atratores globais para equações de evolução não lineares

Abstract

In this work we develop the theory of global attractor for differential partial equation, denominate evolution equation :

ut +2 (ux)2 + uxxx + η(Hux + Huxxx) + μu = f(x), u(., 0) = φ(.) (1)

e

zt + zzx + zxxx + η(Hzx + Hzxxx) + μz = g(x), z(·, 0) = ψ(·) (2)

where H denote the Hilbert transform Hf(x) = 1 πP Z ∞ −∞ f(y) y − x dy, f ∈ S(R),

We show that above initial value problem are globally well-possed in Sobolev space. Moreover, we prove existence of global attractor, i.e. invariant set, compact set, which attracts every bounded set of initial conditions.