| Abstract: |
| We consider the nonlinear delay differential evolution equation in a Banach space $ X $
$\begin{equation*}
\begin{cases}
x`(t)=Ax(t)+f(t,x_t)\qquad & t\in[0,T]\
x(t) \in g(x)(t)\qquad & t\in[-r,0].
\end{cases}
\end{equation*}$
where the linear operator $ A $ generates a $ C_0 $-semigroup of contractions and the function $ x_t $, defined by $ x_t(s)=x(t+s) $ for $ s\in[-r,0] $, is the delay term. We prove the existence of mild solutions satisfying the nonlocal, multivalued, Cauchy condition defined by the multimap $ g:C([-r,T],X)\multimap C([-r,0],X) $, whether the semigroup generated by $ A $ is compact or not. Our approach involves a suitable degree argument. We then apply our results to a transport equation in the form
$\begin{equation*}
\begin{cases}
\displaystyle u_t(t,y)+a\cdot\nabla u(t,y)= \Phi\left(\int_{\mathbb{R}^n}|u(t,\xi)|^pd\xi\right)\cdot h\left(\int_{t-r}^t\int_{\mathbb{R}^n}|u(s,\xi)|^p\,\text{d}\xi\,{\mathrm d}s\right)\cdot \ell(t,u(t,y))\quad&[0,T]\times\mathbb{R}^n\
\displaystyle u(t,y) \in\left\{{ u_0(t,y) + \sum_{i=1}^N\mu_i\int_t^0u(t_i+s,y)\,{\mathrm d}s:\mu_i\in A_i,\, i=1,...,N}\right\}\quad&[-r,0]\times\mathbb{R}^n
\end{cases}
\end{equation*}$
where $ a\in \mathbb{R}^n $, the functions $\Phi$, $ h $$ :\mathbb{R}\to\mathbb{R} $ are continuous and bounded and the map $ \ell:[0,T]\times\mathbb{R}\to\mathbb{R} $ satisfies suitable assumptions. Here, the multivalued condition is defined by the function $ u_0:[-r,0]\times \mathbb{R}^n\to\mathbb{R} $, the fixed instants $ 0 $ |
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