Electron. J. Differential Equations, Vol. 2020 (2020), No. 123, pp. 1-13.

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

Gurusamy Arumugam, Andre H. Erhardt

We establish the existence and uniqueness of weak solutions to the parabolic system with nonstandard growth condition and cross diffusion,

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.

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Gurusamy Arumugam
Discipline of Mathematics
Indian Institute of Technology Gandhinagar
Gandhinagar, 382355 Gujarat, India
email: gurusamy.a@iitgn.ac.in, guru.poy@gmail.com
André H. Erhardt
Department of Mathematics
University of Oslo, P.O. Box 1053 Blindern
N-0316 Oslo, Norway
email: andreerh@math.uio.no

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