Attractors and asymptotic regularity for nonclassical diffusion equations in locally uniform spaces with critical exponent

Article type: Research Article

Authors: Zhang, Fang-hong^{a; b; *} | Wang, Li-hong^{a; b} | Gao, Jin-ling^{a; b}

Affiliations: [a] Regional Circular Economy Key Laboratory of Gansu Higher Institutions, Lanzhou, China | [b] Department of Mathematics, Longqiao College of Lanzhou University of Finance and Economics, Lanzhou, China

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations in locally uniform spaces ut−Δut−Δu+f(u)=g(x),x∈RN. First, we prove the well-posedness of solution for the nonclassical diffusion equations with critical nonlinearity in locally uniform spaces, and then the existence of (Hlu1(RN),Hρ1(RN))-global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appears to be optimal and the existence of a bounded (in (Hlu2(RN))) subset which attracts exponentially every initial Hlu1(RN)-bounded set with respect to the Hlu1(RN)-norm.

Keywords: nonclassical diffusion equations, global attractor, asymptotic regularity, critical exponent, locally uniform spaces

DOI: 10.3233/ASY-161382

Journal: Asymptotic Analysis, vol. 99, no. 3-4, pp. 241-262, 2016