Well-posedness and long term behavior of supercritical wave equations driven by nonlinear colored noise on Rn

Abstract

This paper is concerned with the well-posedness and long term behavior of the non-autonomous random wave equations driven by nonlinear colored noise on Rn with n≤5. The drift nonlinearity has a supercritical growth exponent (n+2)/(n−2). We first prove the existence and uniqueness of solutions in the energy space by showing the non-concentration of energy via the Morawetz identity and the uniform Strichartz estimates. We then prove the existence and uniqueness of tempered pullback random attractors of the non-autonomous random dynamical system associated with the equation. The asymptotic compactness of solutions is obtained by the idea of energy equation due to Ball and the uniform tail-ends estimates in order to circumvent the difficulty caused by the lack of compactness of Sobolev embeddings on unbounded domains.