Existence of smooth solutions to the heat equation with variable coefficients

Question

Consider $$\partial_{t} u - a \partial_{x}^{2}u = f, $$ in $[0,1]\times [0,T]$, where $a, f : \mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ are smooth. Additionally, the initial data $$u(x,0) = u_{0} $$ is smooth and we further endow $u$ with boundary conditions $$u(0,t) = u(1,t). $$

Question: How can you prove the existence of a smooth solution to this problem?

I have read somewhere that this follows from the theory of linear parabolic equations, but I don't know which theory exactly. I did see a proof in Evans for the homogeneous heat equation in $\mathbb{R}$ but that's about it.

Answer 1

One way to proceed would be to explicitly construct the solution to this equation (which is relatively simple since it's linear). To satisfy the periodicity condition on the interval $[0,1]$ one needs to expand the solution in a Fourier series. Following through with this procedure, one finds that the function given by

$$u(x,t)=\int_0^1 dy ~u_0(y)\Theta(x-y,t)+\int_0^1 dy\int_0^{t}dt' f(y,t')\Theta(x-y,t-t')~~,~~ t>0$$

satisfies the equation and the initial conditions. Here $\Theta(x,t)$ is a kernel with the properties $$1.~~\Theta(x,t)\in C^\infty([0,1]\times(0,T])$$ $$2.~~~~~~~~~~~~~~~~~~\int_0^\infty \Theta(x,t)dx=1$$

Can you show properties 1. and 2. and replicate the proof of Evans?

HINT: $$\Theta(x,t)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}e^{-4\pi^2 a n^2 t }e^{i2\pi n x}$$