Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion

Abstract

We establish the existence and uniqueness of weak solutions to the parabolic system with nonstandard growth condition and cross diffusion, ∂tu - div α(x, t, ∇u)) = div |F|p(x,t)-2 F), ∂tv - div α(x, t, ∇v)) = δ∆u, where δ ≥ 0 and ∂tu, ∂tv denote the partial derivative of u and v with respect to the time variable t, while ∇u and ∇v denote the one with respect to the spatial variable x. Moreover, the vector field α(x, t, ‧) satisfies certain nonstandard p(x, t) growth, monotonicity and coercivity conditions.