On a class of parabolic integro-differential equations
We consider the domain \(Q= (0,1)\times (0,T]\), its parabolic boundary \(F=\overline Q/Q\) and sufficiently smooth functions \(c: Q\to\mathbb{R}\), \(k:(0,1)\times (0,1)\to \mathbb{R}\), \(f: Q\to\mathbb{R}\), \(\varphi: F\to\mathbb{R}\), \(h: Q\times\mathbb{R}\to \mathbb{R}\). Let \(L\) be the one-dimensional heat operator, \(C\) the multiplication operator associated with \(c\), and \(K\), \(H\) the operators defined by \[ (Ku)(x, t)= \int^1_0 k(s,x) u(s,t) ds\quad\text{and}\quad (Hu)(x, t)= h(x,t,u(x, t)). \] The author studies, in suitable Hölder spaces, existence and uniqueness of the solutions of the linear problem \(Lu= (C+ K)u+ f\) [resp. the nonlinear problem \(Lu= (C+ KH)u+ f\)] in \(Q\), with the boundary condition \(u=\varphi\) on \(F\). The resolution of these problems is based on the heat potential \(L^{-1}\) associated with the equation \(Lu= f\) in the finite set \(Q\) and the Dirichlet boundary condition on \(F\). A large part of the paper is devoted to a minitious investigation of useful properties of \(L^{-1}\).
parabolic integro-differential equations
scientific article; zbMATH DE number 1453211