The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior

The paper addresses the questions of existence and asymptotic behavior of solutions to the Cauchy problem for the equation u(t) - div =(D(x)vertical bar del u vertical bar(p(x)-2)del u) + A(x)vertical bar u vertical bar(q(x)-2)u = f(x, t, u). The coefficients D, A are nonnegative functions which may vanish on a set of zero measure in R-n, and A(x) -> infinity as vertical bar x vertical bar -> infinity, f (x,t,u) is globally Lipschitz with respect to u. The exponents p, q : R-n bar right arrow (1, infinity) are given measurable functions. We prove that the problem admits at least one weak solution in a weighted Sobolev space with variable exponents, provided that p(-) = ess inf(Rn) p(x) > max {2n/n+2, 1}, q(-) = ess inf(Rn) q(x) > 2, A(-) 2/q(x)-2 is an element of L-1(R-n) and D- s/p(x)-s is an element of L-1 (B-R1 (0)) with constants max {1, 2n/n+2} < s < min {p(-), q(-)} and R-1 > 0. In the case p(-) > 2, q(x) = p(x) a.e. in R-n, and f = f (u), there exists a unique strong solution and the problem has a global attractor in L-2(R-n). (c) 2016 Elsevier Inc. All rights reserved.