Viability of a Moving Set with Respect to a Semilinear Reaction-Diffusion System with delay

Monica-Dana Burlica^* and Daniela Rosu

Let $A:D(A)\subseteq X\to X$ and $B:D(B)\subseteq Y\to Y$ be the infinitesimal generators of two $C_0$-semigroups, let $\sigma > 0$, let $C_{\sigma}=C([\,-\sigma,0\,];X)\times C([\,-\sigma,0\,];Y)$, let $K:[\,a,b)\leadsto X\times Y$ and $F:\mathcal K\leadsto X$ be multi-functions with nonempty values and let $G:\mathcal K\to Y$, where $\mathcal{K}=\{(t,\varphi,\psi)\in [\,a,b)\times C_{\sigma};\ (\varphi(0),\psi(0))\in K(t)\}.$ Some necessary and sufficient conditions for viability of $\mathcal{K}$ with respect to the reaction-diffusion system $$\left\{\begin{array}{ll} u'(t)\in Au(t)+F(t,u_t,v_t), \\ v'(t)=Bv(t)+G(t,u_t,v_t) \end{array}\right.$$ are obtained and a concrete application is discussed.

Mathematics Subject Classification: 35K57 47J35

Keywords: Differential delay evolution systems; nonlocal delay initial condition; reaction-diffusion systems; locally closed graph; viability; tangency condition

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