# American Institute of Mathematical Sciences

January  2021, 14(1): 321-330. doi: 10.3934/dcdss.2020326

## Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  April 2019 Revised  November 2019 Published  April 2020

We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper $L^{\infty}$-bounds that exploits the entropic gradient structure of the system.

Citation: Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
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