Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\mathbb{R}\times [0,\infty);(0,\infty))$.  By standard methods one finds a candidate solution of the form
$$
u=K\star p(0,\cdot);
$$
where $K$ is the heat Kernel.  However, since the domain is unbounded the the maximum principle fails and there may be more than one such solution.

Therefore, is there a solution $\hat{u}$ to the IV heat equation which minimizes the metric-like projection:
$$
\sup_{(x,t) \in \mathbb{R}\times [0,\infty)} \|
e^{\lambda \|x\|+|t|}\left(
  u(t,x)- p(t,x)
\right)
\|,
$$
for some $\lambda>0$ satisfying
$$
\sup_{(x,t) \in \mathbb{R}\times [0,\infty)} \|
e^{\lambda (\|x\|+|t|)}p(t,x)
\|<\infty
$$