Reference request: Well-posedness of the semilinear heat equation with initial data $u_0=\delta$

Question

Consider the $1$-dimensional semi-linear reaction-diffusion equation:

\begin{align} \dot{u}-u^{\prime \prime}&=H(u)\;, \quad x\in \mathbb R, \;t>0 \\ u(\cdot,0)&=\delta(x)\ \end{align}

where $H$ denotes the Heaviside function and $\delta$ stands for the standard dirac measure.

Is there any reference for the well-posedness of this problem, in the sense that there exists a unique weak solution? So far, I wasn't able to find anything useful. However, I find it hard to believe that this particular problem has not been studied yet.

Any help is much appreciated. Thanks in advance!

Answer 1

You should look into parabolic obstacle problems and, in particular, the obstacle problem for the heat equation.

The Heaviside function $H(u)$ is just the characteristic function $\chi_{\{u>0\}}$. Thus, your equation is of the form $$\partial_t - \Delta u = \chi_{\{u>0\}}.$$ Equations like this make me think free boundary problem. In particular, this equation arises in the obstacle problem for the heat equation. There are lots of references on parabolic obstacle problems.

Usually free boundary problems are posed on domains with boundary (as opposed to the whole space). I'll try to see what I can get that's close to your equation.

Edit: Here are a few references:

1. Blanchet, Dolbeault and Monneau. On the One-Dimensional Parabolic Obstacle Problem with Variable Coefficients.
2. Blanchet. On the Regularity of the Free Boundary in the Parabolic Obstacle Problem - Application to American Options.
3. Apushkinskaya. Free Boundary Problems - Regularity Properties Near the Fixed Boundary.
4. Petrosyan and Shahgholian. Parabolic Obstacle Problems Applied to Finance.

The above are quite a bit more general than the problem you state. I'll keep searching and will update if I find anything else of interest.

Edit: This problem makes me think of the no-sign obstacle problem, which makes an appearance, for example, in potential theory and the problem of harmonic continuation. There is an analogue in this case in parabolic potential theory. What sorts of boundaries allow caloric continuation of the heat potential from free space into the space occupied by the density function? For example, let $U_f$ be the heat potential with density $f$: $$U^f(x,t) = \int_{\mathbb{R}^d\times\mathbb{R}}G(x-y,s-t) \ dy ds,$$ where $G(\cdot,\cdot)$ is the heat kernel. Letting $H := \partial_t - \Delta$ denote the heat operator, we have $$HU^f = c_df.$$ Consider $f = \frac{1}{c_d}\chi_\Omega$ for some domain $\Omega$ and let $U^\Omega$ denote the corresponding potential. Then, $$HU^\Omega = \chi_\Omega.$$ Suppose there exist a caloric continuation $v$ of $U^\Omega$. Then, $v$ satisfies $$\begin{cases} Hv = 0 &\text{ in } Q_r(P,t_0)\\ v = U^\Omega &\text{ in } Q_r(P,t_0) - \Omega \end{cases}$$ for some $(P,t_0) \in \partial\Omega$, where, for some $r>0$, $Q_r(P,t_0) := B_r(P) \times (t_0-r^2,t_0+r^2)$.

Further, the difference $u = U^\Omega - v$ satisfies $$\begin{cases} Hu = \chi_\Omega &\text{ in } Q_r\\ u = |\nabla u| = 0 &\text{ in } Q_r-\Omega \end{cases}.$$