Let $X$ be a real Banach space, $J = [0,a]\subset\mathbf{R}$ , $A: D(A)\subset X\rightarrow 2^X\backslash\emptyset$ an $m$ -accretive operator and $f:J \times X\rightarrow X$ 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
$$u' + Au\ni f(t,u)\mathrm{on}J = [0,a]$$ .
More generally, we provide conditions under which this evolution system has
mild solutions satisfying time-dependent constraints $u(t)\in K(t)$ on $J$ . This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of type $$u_t =\Delta\Phi(u) + g(u)\mathrm{in}(0,\infty)\times\Omega,\Phi(u(t,\cdot))|_{\partial\Omega} = 0,u(0,\cdot) = u_0$$ under certain assumptions on the set $\Omega\subset\mathbf{R}^n$ the function $\Phi(u_1,\ldots,u_m)=(\varphi_1(u_1),\ldots,\varphi_m (u_m))$ and $g: \mathbf{R}^{m}_{+}\to\mathbf{R}^m$ .