# Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold

Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a smooth bundle map. Then I believe the following is true:

Given any smooth compactly supported section $u_0$ of $E$, there exists a unique solution $u: [0,\infty) \times M \rightarrow E$ of the linear heat equation $$\partial_t u = g^{ij}\nabla_i\nabla_j u + Ku$$ such that $u$ is smooth for $t > 0$ and the $L^p$ norm of $u(t,\cdot)$ is bounded for each $t \ge 0$ and $1 \le p \le \infty$.

If I'm wrong, please explain. Otherwise, is there a reference for this somewhere? If not the exact theorem, a proof of a similar theorem that can be adapted to a proof of this?

ADDED: I'm mostly interested in proving the existence statement and preferably using a standard PDE approach. It appears to me that there is a straightforward argument starting by approximating the equation by the standard constant coefficient heat equation on a sufficiently small co-ordinate chart and patching together local solutions to the constant coefficient equation to get a global section. You then use a priori estimates to show that the global section is an approximate solution to the original system. This can then be iterated into an exact solution. But the details are a bit tedious, so it would be nice if I could find it all written down somewhere. And it would be nice to be clear about what, if any, assumptions on the Riemannian metric are needed to make this all work.

ADDED: I'm being a little slow here. Since everything is linear and homogeneous, it suffices to prove existence for initial data supported on a sufficiently small co-ordinate chart. If you start with compactly supported initial data, then you can always write it as a finite sum, where each term is supported in a sufficiently small open set.

ADDED: I took another quick look at a partition of unity argument for reducing the question to solving a parabolic PDE on an open set in $R^n$. It looks like this really does work pretty easily. But it still seems like something that someone should write up carefully and publish in a book somewhere.

I've played with such things 15 years ago. Here's what I remember...

Sure you need some semi-boundedness condition on K. Uniqueness should then follow via semigroup domination (aka Kato's inequality) from scalar case.

I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108

Shigekawa had rediscovered Barry Simon's semigroup domination criterion (i.e. Kato's inequality) and generalized it to the vector valued things.