On a generalization of the Poisson formula for the solution of the heat flow equation

Abstract

Here & denotes the variable (1,2, * ) which ranges over the n-dimensional Euclidean space En and has the norm = 2 ? ? *+) 1I2, the number t has the range 0 < t < to where to is a positive number or + oo, a is a positive constant, and the symbol A denotes the n-dimensional Laplace operator. Under suitable assumptions on 0(t, t) and o(), a solution of the system (la, b) is given in the region R_ {(t, &): O<t<to, CEE }, by (2) G (t, 16-I,,(t, 0), where