Consider a distributional solution $u(t,\cdot) \in C^0([0,T],\mathcal{D}'(\mathbb R^n))$ to the linear heat equation
$$
\left\{
\begin{align*}
u_t - \Delta u &= 0, \\
u(0,\cdot) &= 0
\end{align*}\right.\quad x \in \mathbb R^n, t \in (0,T]
$$
The notation $C^0([0,T],\mathcal{D}'(\mathbb R^n))$ means that $t \mapsto \langle u(t,\cdot), \phi \rangle$ is continuous on $[0,T]$ for any $\phi \in C^{\infty}_c(\mathbb R^n)$.
The initial condition asks for this function to be zero at time $t = 0$ for any test function $\phi$. One can prove (see Claude Zuily, "Eléments de distributions et d'équations aux dérivées partielles", Chapter 4, Prop. 3.3, ) that $t \mapsto \langle u(t,\cdot), \phi(t,\cdot) \rangle$ is continuous for any $\phi \in C^{\infty}_c([0,T] \times \mathbb R^n)$. Hence, I'm assuming that $u \in \mathcal{D}'([0,T] \times \mathbb R^n)$ is a distribution defined as
$$
\langle u, \phi \rangle_{\mathcal{D}([0,T] \times \mathbb R^n)} = \int_0^{T} \langle u(t,\cdot), \phi(t,\cdot) \rangle_{\mathcal{D}(\mathbb R^n)} dt
$$
which satisfies
$$
\langle \partial_t u - \Delta u, \phi \rangle = 0
$$
for any $\phi \in C^{\infty}_c((0,T) \times \mathbb R^n)$.
Anyway, we can throw away most of this formalism because hypoellipticity of the heat operator implies that $u \in C^{\infty}((0,T) \times \mathbb R^n)$. The only remaining assumption which is not covered by hypoellipticity is the weak continuity at $t = 0$, $$ \lim \limits_{t \to 0} \langle u(t, \cdot), \phi \rangle = \lim \limits_{t \to 0} \int_{\mathbb R^n} u(t,x) \phi(x)dx = 0 \quad \forall \phi \in C^{\infty}_c(\mathbb R^n) $$
The solution to this heat equation is not unique. For example, in his PDE book, Evans shows that the trivial solution is the unique solution under the additional assumptions that $u \in C^0([0,T] \times \mathbb R^n)$ and $|u(t,x)| \leq Ae^{a|x|^2}$ on $[0,T] \times \mathbb R^n$ for some $A, a > 0$.
Moreover, one can exhibit non-physical solutions which grow exponentially fast. But does there exist solutions which are not $C^0([0,T) \times \mathbb R^n)$ ?
