On the existence of classical solutions for differential-functional IBVP (Q5926235)

Let \(\Omega\subset \mathbb{R}^n\) be any open bounded domain, and \(T> 0\), \(a_0\), \(\tau\in\mathbb{R}_+= [0,+\infty)\). Define NEWLINE\[NEWLINE\begin{aligned} &\Omega_\tau= \{x\in\mathbb{R}^n: \text{dist}\{x,\Omega\}\leq \tau\},\quad\partial_0\Omega= \Omega_\tau/\Omega,\\ &\Theta_0= [-a_0, 0]\times \Omega_\tau,\quad \partial_0\Theta= (0,T)\times \partial_0\Omega,\quad \Gamma= \Theta_0\cup \partial_0\Theta,\quad E=\Gamma\cup\Theta,\\ &D= [-a_0, 0]\times B(\tau),\quad B(\tau)= \{x\in\mathbb{R}^n:|x|\leq r\}.\end{aligned}NEWLINE\]NEWLINE For every \(z: E\to\mathbb{R}\), \((t,x)\in\Theta\), define \(z_{(t,x)}= z(t+ s,x+ y)\), \((s,y)\in D\) and call this restriction operator \(z\to z_{(t,x)}\) Hale's operator. Assume that \(L\) is a strictly uniformly parabolic operator in \(\Theta\): NEWLINE\[NEWLINELz(t,x)= D_t z(t,x)- \sum^n_{i,j=1} a_{i,j}(t, x) D^2_{x_ix_j} z(t,x)- \sum^n_{i=1} b_i(t, x)D_{x_i} z(t,x).NEWLINE\]NEWLINE The initial-boundary value problem NEWLINE\[NEWLINE\begin{cases} Lu(t,x)= f(t, x,u(t,x), u_{(t,x)}, D_xu(t,x))\quad &\text{in }\theta,\\ u(t,x)= \Phi(t,x)\quad &\text{in }\Gamma\end{cases}\tag{1}NEWLINE\]NEWLINE is considered. By the Leray-Schauder theorem, the authors show the existence of classical solutions of (1), which covers a large class of parabolic problems both with a deviated argument and integrodifferential equations.