= Solution
For each injection time $\tau$, let $v_\tau$ solve the homogeneous <Cauchy problem for a partial differential equation> with initial profile $v_\tau(x,\tau)=h(x,\tau)$. Its representation by the <fundamental solution of a linear differential operator> is
$$
v_\tau(x,t)=\int_\mathbb R F(x-y,t-\tau)h(y,\tau)\,dy\qquad(t>\tau).
$$
The causal <Green function> and <Duhamel principle> give
$$
\boxed{\phi(x,t)=\int_0^t v_\tau(x,t)\,d\tau
=\int_0^t\int_\mathbb R F(x-y,t-\tau)h(y,\tau)\,dy\,d\tau}.
$$
For <smooth> forcing with sufficient decay, differentiating this formula gives
$$
\phi_t=h(x,t)+\int_0^t\partial_t v_\tau\,d\tau
=h(x,t)+K\phi_{xx},\qquad \phi(x,0)=0.
$$
The boundary term uses the <Dirac delta function> limit of $F$, not a finite pointwise value of $F(x,0)$. The same formula extends to weaker forcing in an appropriate integrable or <distribution> sense. Physically, each short time interval adds $h(\cdot,\tau)d\tau$ to the temperature profile; that addition subsequently undergoes homogeneous <diffusion equation>. The total solution is the superposition of all such diffusing injections.
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