Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates
Dieter Bothe, Petra Wittbold · Communications on Pure & Applied Analysis · 2012
We consider reaction-diffusion systems with merely measurable reaction terms to cover thepossibility of discontinuities. Solutions of such problems are defined as solutions toappropriate differential inclusions which, in an abstract form, lead to evolution inclusionsof the form $u' \in - A u + F(t,u)$ on $[0,T], u(0)=u_{0},$ where $A$ is $m$-accretive and $F$ is of upper semicontinuous type.While such problems, in general, can exhibit non-existence of solutions, the present paper showsthat especially for $m$-completely accretive $A$, and under reasonable assumptionson $F$, mild solutions do exist.