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 $$ \sup_{(x,t) \in \mathbb{R}\times [0,\infty)} \| e^{\lambda (\|x\|+|t|)}p(t,x) \|<\infty $$