Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 123, pp. 1-13.

Title: Existence and uniqueness of weak solutions to parabolic problems with
       nonstandard growth and cross diffusion
       
Authors: Gurusamy Arumugam (Indian Inst. of Technology Gandhinagar, Gujarat, India)
         Andre H. Erhardt (Univ. of Oslo,  Oslo, Norway)

Abstract:
 We establish the existence and uniqueness of weak solutions to the  parabolic system with
 nonstandard growth condition and cross diffusion,
 $$\displaylines{
 \partial_tu-\text{div}a(x,t,\nabla u))
 =\text{div}|F|^{p(x,t)-2}F),\cr
 \partial_tv-\text{div}a(x,t,\nabla v))=\delta\Delta u,
 }$$
 where $\delta\ge0$ and $\partial_tu,~\partial_tv$ denote the partial derivative of u
 and v with respect to the time variable t, while $\nabla u$ and $\nabla v$ denote
 the one with respect to the spatial variable x. Moreover, the vector field $a(x,t,\cdot)$
 satisfies certain nonstandard p(x,t) growth, monotonicity and coercivity conditions.

Submitted September 18, 2019. Published December 17, 2020.
Math Subject Classifications: 35A01, 35D30, 35K65.
Key Words: Nonlinear parabolic problem; nonstandard growth; cross diffusion.