Attractors For A Weakly Dissipative Hyperbolic Equation On Unbounded Domain

We consider an integro-partial differential equation of hyperbolic type with a cubic nonlinearity on R³ u_tt-k(0)Δu-∫₀^∞k′(s)Δu(t-s)ds+g(u)=f(x) In which no dissipation mechanism is present, except for the convolution term accounting for the past memory of the variable. In the autonomous case, the existence of a attractor is achieved. In order to prove that we apply the so-called gradient systems establish a strongly continuous semigroupS(t) z = (u(t), u_t(t),η^t) of the equation. The first step is then to show the existence of a bounded absorbing set for the semigroup. Here, assumpution on the decy of the memory kernel plays a fundamental role. The second step is to prove that the image of the absorbing ste under the semigroup is asymptotically compact. Since the Poicaréembedding is no longer compact in the unbounded domain case, we split the semigroup S(t) z = (u(t), u_t(t),η^t)into the sum S(t) = L(t) + N(t),. Where L(t)z =(v(t), v_t(t),ξ^t) is exponentially decays arbitrarily small in the long time, and N(t)=(ω(t),ω_t(t),ζ^t) has the finite propagation speed property, whose support set is finite set in the finite time, so we can prove the compactness property of N(t) by virtue of the compactness property of the finite time, so we can prove the compactness property of N(t) by virtue of the Poicarécompactness property .