Nonlinear delay reaction-diffusion systems with nonlocal initial conditions having affine growth

Abstract

We consider a class of abstract evolution reaction-diffusion systems with delay and nonlocal initial data of the form $$\begin{cases} \displaystyle u'(t)\in Au(t)+F(t,u_t,v_t)&\text{for } t\in \mathbb{R}_+,\\ v'(t)\in Bv(t)+G(t,u_t,v_t) & \text{for } t\in \mathbb{R}_+,\\ u(t)=p(u,v)(t)& \text{for } t\in [-\tau_1,0],\\ v(t)=q(u,v)(t)& \text{for } t\in [-\tau_2,0], \end{cases}$$ where $\tau_i\geq 0$, $i=1,2$, $A$ and $B$ are two $m$-dissipative operators acting in two Banach spaces, the perturbations $F$ and $G$ are continuous, while the history functions $p$ and $q$ are nonexpansive functions with affine growth. We prove an existence result of $C^0$-solutions for the above problem and we give an example to illustrate the effectiveness of our abstract theory