Laplace-Operator of distribution-valued function (heat equation)

Question

I'm having trouble making sense of an exercise involving this definition of the heat equation:

$u'(t) = \Delta (u(t))$, $u(0) = \delta_0$ for $t > 0$ where $u : [0, \infty) \rightarrow S'(\mathbb{R})$ is a distribution-valued function which is sequentially continuous and $u'(t) = \lim_{h \rightarrow 0} \frac{u(t+h) - u(h)}{h}$ ist well defined for $t >0$.

Specifically, I don't understand what $\Delta$ means in this context: since $u$ depends only on $t$, shouldn't it be identically zero?

Answer 1

Actually $u(t) \in S'(\mathbb{R})$ for each $t$. So in that sense $\Delta(u(t))$ can be computed.