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

Abstract

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). The coefficients D, A are nonnegative functions which may vanish on a set of zero measure in Rn, and A(x)→∞ as |x|→∞, 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}<s<min⁡{p−,q−} and R1>0. In the case p−>2, q(x)=p(x) a.e. in Rn, and f≡f(u), there exists a unique strong solution and the problem has a global attractor in L2(Rn).