Recall the statement of the weak maximum principle for the heat equation on $\mathbb{R}^n.$
If $u: \mathbb{R}^n \times[0, T] \rightarrow \mathbb{R}$, where $u \in > C^{2,1}\left(\mathbb{R}^n \times(0, T]\right) \cap > C^0\left(\mathbb{R}^n \times[0, T]\right)$, is a solution to the heat equation on $\mathbb{R}^n \times(0, T)$ such that $$ u(x, t) \leq A > e^{B|x|^2} \quad \text { on } \mathbb{R}^n \times[0, T] $$ for some constants $A, B<\infty$, then $$ \sup _{(x, t) \in \mathbb{R}^n > \times[0, T]} u(x, t)=\sup _{x \in \mathbb{R}^n} u(x, 0).$$
Is it possible to weaken the exponential growth assumption if I assume the initial data $u_0(x)=u(x,0)$ to be strictly positive and in $L^1(\mathbb{R}^n)$ for instance? In that case, I can use the integral representation of $u$ to get $$u(x,t)=K_t*u_0$$ where $K_t$ is the Gaussian heat kernel to get an $L^\infty$ bound of the solution in terms of the $L^1$ bound of the initial data which implies that $u(x,t)$ has exponential growth for $t>\delta>0.$ And so we can apply the weak maximum principle.
Questions:
- Is the above reasoning correct?
- Can this argument be generalized to complete noncompact Riemannian Manifolds with Ricci curvature $\operatorname{Ric}\geq -K$?
- Would some version of this argument also work when the maximum principle is applied to two tensors?