# American Institute of Mathematical Sciences

September  2014, 13(5): 1907-1933. doi: 10.3934/cpaa.2014.13.1907

## Reaction-diffusion equations with a switched--off reaction zone

 1 Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main 2 Institute of Mathematics, Johann Wolfgang Goethe University, 60054 Frankfurt (Main) 3 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074

Received  March 2013 Revised  May 2013 Published  June 2014

Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solution first exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay. This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead core in a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is established firstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.
Citation: Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907
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