\(C^{\alpha}\)-regularity for non-autonomous linear integrodifferential equations of parabolic type (Q1078467)

The authors study a linear partial Volterra integrodifferential equation of the following type: \[ (*)\quad u_ t(t,x)=A(t,x,D)u(t,x)+\int^{1}_{0}B(t,s,x,D)u(s,x)ds+f(t,x);\quad (t,x)\in [0,T]\times {\bar \Omega} \] supplemented with an initial condition \(y(0,x)=u_ 0(x)\) and homogeneous boundary condition of Dirichlet type. Here \(\Omega\) is a bounded open subset of \({\mathbb{R}}^ n\) with regular boundary \(\partial\Omega\); \(A(t,x,D)\) is a linear elliptic partial differential operator with coefficients depending on \(t\in [0,T]\) and \(x\in {\bar \Omega}\) and \(B(t,s,x,D)\) is a linear partial differential operator of the same order of A, and its coefficients depend on \((t,s,x)\) with \(0\leq s\leq t\leq T\) and \(x\in {\bar \Omega}\). f and \(u_ 0\) are given functions. The aim is to find a space of regular functions such that in this space problem (*) is well posed in the sense that for its solution one can prove existence, uniqueness, and continuous dependence on the data f and \(u_ 0\).