Regular attractor of the β-evolution equation with fractional damping on Rn

Abstract

We study the well-posedness and longtime dynamics of the β-evolution equation with fractional damping: ∂t2u+(−Δ)βu+γ(1−Δ)α∂tu+f(u)=g(x) on the whole space Rn, with β > 2α > 0. First, we find a critical exponent p*=n+4αn−2β for the well-posedness of energy solutions. In fact, if the nonlinear term grows with the order p ∈ [1, p*) and satisfies some dissipative conditions, then the equation is globally well-posed in the energy space. Moreover, both u and ∂tu have a smoothing effect as a parabolic equation. Finally, we show that the solution semigroup has a global attractor A in the energy space. The main difficulties come from the non-compactness of the Sobolev embedding on Rn and the nonlocal characteristic of the equation. We overcome them by establishing some new commutator estimates.