Some Elementary Properties of the Fundamental Solution of Parabolic Equations
Ronald B. Guenther · Mathematics Magazine · 1966
where x = (xi, * , xn) is an n-dimensional point and t a point on the real line. Before giving a precise description of the results obtained here and comparing them with those of other authors, we shall introduce some notation and make certain definitions. Let Rn denote n-dimensional Euclidean space and let x, (, etc. be elements of Rnwith coordinates (xi, , xn), (t1, . . . , n) X etc., and let lxi =X ( + X2)12. Let I= [to, T] be a closed interval in R1 with 0<to< T, and let t, r, etc., be points of I. Let R be the topological product of Rn with I. It is customary to speak of a strip, e.g. r <t <h, and mean the topological product of Rn with the interval r < t < h. Finally, we let d = d . . . d n. A function u(x, t) is said to be a solution to (1) in R, if it is continuous in R and if in the strip to <t _ T, it is twice continuously differentiable with respect to the x-variables and once with respect to the t-variables and satisfies (1) there. Fundamental solutions arise naturally in trying to find a representation for the solution to the problem