Time-variable singularities for solutions of the heat equation

D. V. Widder
1972 Proceedings of the American Mathematical Society  
A solution u(x, t) of the two-dimensional heat equation uxx=u, may have the representation foo u(x, 0 = 1 k(x -y,t)da(y) where k(x, í) = (47rí)-1'2 exp[-x2/(4r)l, valid in some strip 0 1/c, for each fixed real x0. It is shown here that if <x(y) is nondecreasing and not absolutely continuous then u(x0, t) must have a singularity at /=0. Examples show that both restrictions on a(_y) are necessary for that conclusion. It is shown further under the same hypothesis on <x(y), that for each fixed
more » ... ive t0<c, u(x, t") is an entire function of x of order 2 and of type l/(4f0). Compare the function k(x, t) itself for a check on both conclusions.
doi:10.1090/s0002-9939-1972-0294906-7 fatcat:6xxg4re22ncstbpefdt7pj4lsy