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 equationut−div(D(x)|∇u|p(x)−2∇u)+A(x)|u|q(x)−2u=f(x,t,u).ut−div(D(x)|∇u|p(x)−2∇u)+A(x)|u|q(x)−2u=f(x,t,u). The coefficients D, A   are nonnegative functions which may vanish on a set of zero measure in RnRn, and A(x)→∞A(x)→∞ as |x|→∞|x|→∞, f(x,t,u)f(x,t,u) is globally Lipschitz with respect to u  . The exponents p,q:Rn↦(1,∞) 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−=essinfRn⁡p(x)>max⁡{2nn+2,1}, q−=essinfRn⁡q(x)>2, A−2q(x)−2∈L1(Rn) and D−sp(x)−s∈L1(BR1(0)) with constants max⁡{1,2nn+2}0R1>0. In the case p−>2p−>2, q(x)=p(x)q(x)=p(x) a.e. in RnRn, and f≡f(u)f≡f(u), there exists a unique strong solution and the problem has a global attractor in L2(Rn)L2(Rn).