Flow invariance for perturbed nonlinear evolution equations

Abstract

Let $X$ be a real Banach space, $J=\left[0,a\right]\subset \mathbf{R}$, $A:D\left(A\right)\subset X\to 2^X\backslash \phi$ an $m$-accretive operator and $f:J\times X\to X$ continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets $K\subset X$ for the evolution system $$u^{\prime }+Au∍f\left(t,u\right)\text{on}J=\left[0,a\right]$$. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraints $u\left(t\right)\in K\left(t\right)$ on $J$. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of type $$u_t=\Delta \Phi \left(u\right)+g\left(u\right)\text{in}\left(0,\infty \right)\times \Omega ,\Phi \left(u\left(t,\cdot \right)\right){|}_{\partial \Omega }=0,u\left(0,\cdot \right)=u_0$$ under certain assumptions on the set$\Omega \subset {\mathbf{R}}^n$ the function $\Phi \left(u_1,\dots ,u_m\right)=\left({\varphi }_1\left(u_1\right),\dots ,{\varphi }_m\left(u_m\right)\right)$ and $g:{\mathbf{R}}_{+}^m\to {\mathbf{R}}^m$.