a.- Show that the solution to the heat equation with initial data is given by the convolution of the initial data with the fundamental solution:
where the fundamental solution is given by:
b.- Show that the solution is smooth in both and for all and that the solution is infinitely differentiable in both and for all . Check that the same holds for the solutions found on the previous exercise.
c.- Argue that even when the initial data is of compact support the solution extends to infinity for any .
d.- The solution decays exponentially fast as .