I am trying to solve an inhomogeneous heat equation where the source term depends linearly on the temperature of the system.
$\partial_t u(x,t) - D \partial_{x,x}u(x,t) = -\gamma u(x,t) H(t) \delta(x)$
Where $H(t)$ is the Heaviside function and $\delta(x)$ is the Dirac delta.
The initial condition is $u(x,0) = const.$
The domain of the problem is the entire real line.
At the boundary $\partial_x u(\pm \infty, t) =0 $.
I attempted to solve this equation in momentum space but have made little progress towards a solution. I can numerically solve this equation but I would really benefit from a closed form solution.
Are there any techniques that may be useful for find the solution?