The author considers a linear parabolic differential equation \[ (1)\quad Lu\equiv \sum^{n}_{i,j=1}a_{ij}u_{x_ ix_ j}+\sum^{n}_{i=1}b_ iu_{x_ i}+cu-u_ t=f\quad in\quad D:=\Omega \times (0,T] \] where \(a_{ij}\), \(b_ i\), c and f depend on \(x\in \Omega \subset {\mathbb{R}}^ n\) and \(t\in (0,T]\), together with a Dirichlet boundary condition \((2)\quad u(x,t)=(x,t)\) on \(\Gamma:=\partial \Omega \times [0,t]\) and the following condition for u(x,0), \((3)\quad u(x,0)+F(x,u(\cdot,\cdot))=\psi (x)\) in \(\Omega\), where F maps \({\bar \Omega}\times C(\bar D)\) into \({\mathbb{R}}\). Examples for F are \((a)\quad F(x,u)=\int_{D}u(y,s)d\mu^ x(y,s),\) e.g., \((b)\quad F=\sum^{N}_{i=1}\beta_ i(x)u(x,T_ i);\) here \(\{\mu^ x\}\) is a family of signed Borel measures.
Under appropriate assumptions regarding \(\{\mu^ x\}\) and the coefficients, in particular \(c\leq 0\), it is proved that a unique solution \(u\in C^{2,1}(D)\cap C(\bar D)\) for (1), (2), (3a) exists.
Similar theorems with a more general F and for \(D={\mathbb{R}}^ n\times (0,T]\) are also given, as well as an integral representation of solutions. As to method: Using Green’s function, the problem is transformed into an integral equation which has a unique solution. ”Most of the theorems of this paper extend the results of the author” [see Nagoya Math. J. 93, 109-131 (1984; Zbl 0506.35048)], where the case (b) was treated.