Global attractor for a class of inhomogeneous reaction–diffusion equation in unbounded domain

Abstract

We consider the existence of the global attractor for the following inhomogeneous reaction–diffusion equation ut−Δu−V(x)u+|u|p−2u=g,inRn×R+,u(0)=u0∈L2(Rn)∩Lq(Rn),where n≥3, p>2 and the function V(x) satisfies some suitable assumptions. Since the primary operator −Δ−V is not positive definite in H1(Rn), so we cannot obtain that the corresponding semigroup has a bounded absorbing set in L2(Rn) by the Gronwall inequality. Thus, through the cutting function technique and the method of iteration, we prove that for any q>p, the equation has a bi-spaces global attractor A, which attracts any bounded subset B∈L2(Rn)∩Lq(Rn) in the topology of Lq(Rn).