zbMATH — the first resource for mathematics

On the Cauchy problem for some nonhomogeneous parabolic differential-difference equations. (English. Russian original) Zbl 1059.35156
Math. Notes 74, No. 4, 510-519 (2003); translation from Mat. Zametki 74, No. 4, 538-549 (2003).
From the introduction: In a previous paper the author studied the Cauchy problem for the difference-differential parabolic equation \[ \frac {\partial u}{\partial t}=\Delta u+\sum^m_{k=1} a_ku(x-b_kh,t),\quad x\in\mathbb{R}^n,\;t>0,\tag{1} \] where \(a\) and \(b\) are arbitrary parameters from \(\mathbb{R}^m\) and \(h\) is a fixed vector of unit length in \(\mathbb{R}^n\). In the present paper, we consider imhomogeneous equations. We prove the classical solvability of the Cauchy problem, obtain an integral representation of the solution, and establish uniqueness classes. To do this, we use the fundamental solution of (1).

35R10 Partial functional-differential equations
35K15 Initial value problems for second-order parabolic equations
35C15 Integral representations of solutions to PDEs
Full Text: DOI