The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to boundary conditions
$$
\Theta=0,~~ (x,y,t)\in\partial\Omega\times[0,T],
$$
and initial condition
$$
\Theta(x,y,0)=\Theta\_0(x,y),~~ (x,y,t)\in\Omega.
$$
where $\delta(x)$ is Dirac`s delta function. Here $p$, $u$, $v$ are piecewise continuous functions, and $\Theta\_0$ is bounded and may have finite discontinuities.


In which space (preferably Sobolev) the solution $\Theta(x,y,t)$ exists and is unique?



Well-posedness of heat equation with distributional right hand side
Let $E\_t:=e^{t\alpha^2\Delta}$ be the solution operator to the heat equation
$$ \frac{\partial \Theta}{\partial t} = \alpha^2 \Delta \Theta$$
subject to the initial and boundary contitions above. Then the solution to your equation is given by


$$ \Theta\_t = \Theta\_0 +\int\_0^t E\_{t-s} T \,\mathrm{d} s.$$


Here the term $T$ is your distributional right-hand side.


Regarding well-posedness: Notice that $\Theta\_t$ is at least a distributional solution: For any test-function $\varphi$, we have
$$\partial\_t \langle \Theta\_t, \varphi\rangle = \langle T, \varphi\rangle + \int\_0^t \langle T, E\_{t-s} \Delta \varphi\rangle \mathrm{d} s$$
so that $\Theta\_t$ solves the equation in the distributional sense.


Regarding the Sobolev regularity: For $T$ regular enough, you have
$$|\langle \Delta E\_t T, \varphi\rangle| \leq \|\Delta E\_t T\|\_{H^{-s}}\|\varphi\|\_{H^s} \leq C \|E\_t T\|\_{H^{2-s}}\|\varphi\|\_{H^s}$$
The heat operator has the short-time asymptotics
$$\|E\_t \|\_{H^{r}, H^l} \leq C t^{(r-l)/2}.$$
for $l>r$. Here $l=2-s$ so that we obtain
$$|\langle \Delta E\_t T, \varphi\rangle| \leq C^\prime t^{(r-2+s)/2}\|T\|\_{H^r}\|\varphi\|\_{H^{s}} = C^\prime t^{-1+r/2+s/2}\|T\|\_{H^r}\|\varphi\|\_{H^{s}}$$
In order for this to be integrable at zero, you want the exponent to be greater than $-1$. In your special case, we have $T \in H^r$ for every $r=-n/2-\varepsilon$ with $\varepsilon>0$ so that we can choose $s>n/2$ (here $n$ is the dimension of your domain).


What does this tell us? The functional
$$\langle \Theta\_t, - \rangle$$
is bounded for any $s>n/2$, hence $\Theta\_t \in H^{-s}$ for every such $s$.


