Uniqueness of the heat equation for initial data in $L^\infty$ For $1 \leq p < \infty$ and initial-data $u _ 0 \in L ^p ( \mathbb{R} ^d)$ due to P. Li (Uniqueness of $L^1$ solutions for the Laplace equation and the heat equation on Riemannian manifolds) there exists a unique solution $u$ of the heat equation $u_t - \Delta u = 0$ in $(0, \infty) \times \mathbb{ R }^d$ which satisfies
\begin{equation*}
  u \in C ( [0 , \infty ) ; L^p ( \mathbb{ R }^d) ) \quad \text{and} \quad u ( 0 , \cdot ) = u_0.
 \end{equation*}
This solution is given by $u ( t ) = e ^{t \Delta} u_0$, the convolution with the fundamental solution.
My question is, whether there is known literature for the uniqueness in the case of non-negative $u_0 \in L^\infty ( \mathbb{ R }^d )$ for the heat equation in $(0, \infty) \times \mathbb{ R }^d$, where we replace $u \in C ( [0 , \infty ) ; L^p ( \mathbb{ R }^d) )$ by $u \in L^\infty ( (0, \infty) \times \mathbb{ R }^d)$ and $u(t,x) \to u_0(x)$ for almost every $x \in \mathbb{ R }^d$ as $t \to 0$. Is weak star convergency of $u(t, \cdot)$ to  $u_0$ in $L^\infty(\mathbb{R}^d)$ as $t \to 0$ sufficient?
 A: Let $u$ and $v$ be two solutions with the stated properties. Let $\eta$ be a standard mollifier on $\mathbb{R}^d$ and let $u_\varepsilon :=u \ast \eta_\epsilon$ and $v_\varepsilon := v \ast \eta_\epsilon$, where we only convolve with respect to the $x$-variable.
Claim: We have $u_\varepsilon, v_\varepsilon \in C ^\infty((0, \infty) \times \mathbb{R} ^d) \cap C ( [0, \infty) \times \mathbb{R} ^d)$, $\partial_t u_\varepsilon - \Delta u_\varepsilon = \partial_t v_\varepsilon - \Delta v_\varepsilon = 0$ and $u_\varepsilon(0) = v_\varepsilon(0) = u_0 \ast \eta_\epsilon \in C_b ( \mathbb{R} ^d)$.
Proof of the Claim: As $u$ and $v$ are bounded, we can pass the differentiation into the integrals and get
\begin{equation*}
  \partial_t u_\varepsilon - \Delta u_\varepsilon = (u_t - \Delta u) \ast \eta_\epsilon = 0
 \end{equation*}
and similarly $\partial_t v_\varepsilon - \Delta v_\varepsilon = 0$. Furthermore, $u_\varepsilon, v_\varepsilon \in C ^\infty((0, \infty) \times \mathbb{R} ^d)$ and $u_0 \ast \eta_\epsilon \in C_b ( \mathbb{R} ^d)$. Thus, we are left to show continuity in $t = 0$. This follows by the weak$^*$ convergence: Let $x \in \mathbb{R} ^d$. Then
\begin{align*}
  u_\varepsilon (t,x) - u_\varepsilon (0,x) 
  & = \int_{ \mathbb{R} ^d }\! ( u(t,y) - u_0 (y) ) \eta_\epsilon (x - y) \, dy
  \\
  & \to 0
 \end{align*}
as $\eta_\epsilon (x - \cdot) \in L^1(\mathbb{R} ^d)$ and $u(t,\cdot) \to u_0$ weak$^*$ in $L^\infty ( \mathbb{R}^d )$ as $t \to 0$. In the same fashion, we see that $v_\varepsilon$ is continuous in $t = 0$, which proofs the claim.
The heat equation for regular initial data in $C_b(\mathbb{R}^d)$ has a unique bounded classical solution, see Thm. 6 in Evans, Chapter 2.3. Hence, $u_\varepsilon = v_\varepsilon$ for all $\varepsilon > 0$. As $u_\varepsilon \to u$ and $v_\varepsilon \to v$ locally uniformly in $(0,\infty) \times \mathbb{R} ^d$ as $\varepsilon \downarrow 0$, we get $u = v$ in $(0,\infty) \times \mathbb{R} ^d$.
