Consider the non-linear differential operator
$$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$
For $U\subset\mathbb{R}^2$ open and bounded with smooth boundary $\Gamma:=\partial U$ and $T>0$, can you guarantee that the (smooth classical) solution to the Cauchy-problem
$$\left\{\begin{aligned}\label{lem:ContrastiveDiffusions:eq2} \partial_t\varphi \ &= \ \mathfrak{L}\varphi\quad &&\text{in } \ (0,T)\times U,\\ \partial_x\partial_y\varphi \ &= \ 0 \quad &&\text{on } \ \{T\}\times\Gamma, \end{aligned}\right.$$
with initial condition $\lim_{t\rightarrow 0+}\varphi = \delta(x-y)$, is unique?
(Given that the answer is not immediate, references to the literature are appreciated.)
