Well-posedness of heat equation with distributional right hand side

Question

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?

Answer 1

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 $s$.