Problem 2. Suppose f € C₁(Rn) is uniformly continuous, and u(t,-) = F(t,-) * f is a solution

to the heat equation

u-Au=0

with initial condition u(0, x) = f(x). Show that u is uniformly continuous in [0, ∞) x R", and

||u(t,·)||L(Rn) ≤ C/t for some constant C> 0.