I have the following equation in $\mathbb{R}^2$,
$ \partial_t u(x,t) - \Delta u(x,t) = \delta_0, $
with $u(x,0) = 0.$
I received this equation and I instantly got suspicious. I have tried to prove that anything that satisfies the PDE will have a singularity in 0 at all times, implying that actually there isn't a solution for this initial data.
My intuition comes from the fact that if a Dirac delta arises from differentiating, it should arise in at least one of the derivatives. If it there is one on the time derivative, it would mean that the solution is discontinuous in time, and this seems highly unlikely. In the case of the space derivatives, I think it should look like the Green function of the Laplacian (the steady state actually IS), which is singular at 0.
I have tried reducing to dimension 1 and obtaining an explicit solution by means of Fourier transform, and the solution gets so unmanageable that I abandoned that path.
Do you have any clues on this?
A solution exists indeed. Let us define the operator $L = -\Delta$ and rewrite the heat equation with source $f$ as $$ \nabla_t u + L[u] = f$$ with initial conditions $u(x,0)=0$
A formal solution can be written in terms of exponential operator as $$ u = \int_0 ^t e^{-(t-s)L } [f(s)] \mathrm{d}s$$ where the operator $e^{-tL}$ is given by an integral operator having the Green's function as kernel $$ G[f] = \int_{R^n} G(x, \eta) f(\eta) \mathrm{d} \eta$$
Now we specialise the $f$ to a Dirac delta and using its "chief"property under integration, find the solution $$ u(x,t)=\int_0 ^t G(x, t-s) \mathrm{d}s$$ and further, as $$ G(x,t) = \frac{1}{(4 \pi t)^{\frac{n}{2}}} e^{-\frac{x^2}{4t}}$$ the solution $$ u(x,t) =\frac{1}{(4 \pi )^{\frac{n}{2}}} \int_0 ^t \frac{e^{-\frac{x^2}{4s}}}{s^{\frac{n}{2}}} \mathrm{d}s$$ is recovered, which I believe might be further simplified by using the Gamma function, if one needs to do thermal calculations. The main point is that by all means, a solution (of relevant practical importance, for what matters) exists.
There are other notable PDEs with Dirac deltas as sources, if you wish to further challenge your intuition: another example of remarkable technical interest is the Kelvin problem, the PDE describing the displacements in an infinite elastic body subject to a point force. Also very interesting and intuition challenging is the case of a point force on an elastic half-plane, https://en.wikipedia.org/wiki/Flamant_solution, as displacements are unbounded.