Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

摘要

We study the homogeneous Dirichlet problem for the class of nonlinear parabolic equations with variable nonlinearityu(t) - div(D(x)vertical bar del(u)vertical bar(p(x)-2)del(u)) = f(x, t, u) - A(x)vertical bar u vertical bar(q(x)-2)uin the cylinder Omega x (0, T) with given nonnegative weights D(x), A(x), measurable bounded exponents p(x) is an element of [p(-), p(+)], q(x) is an element of [q(-), q(+)] and a globally Lipschitz function f (x, t, u). Sufficient conditions of existence and uniqueness of weak and strong solutions are derived. We find conditions on the exponents p(x), q(x) which guarantee that the associated semigroup has a compact global attractor in L-2(Omega). It is shown that in case the exponents p(x) and q(x) do not meet the sufficient conditions of existence of a nontrivial global attractor and parallel to u(0)parallel to(L2(Omega)) is sufficiently small, then every solution with bounded parallel to u(t)parallel to(2)(L2(Omega)) either vanishes in a finite time, or decays exponentially as t - infinity.