This is a cross post from MSE. The original question can be found here:https://math.stackexchange.com/questions/3248114/name-for-a-particular-parabolic-pde

Consider the following initial value problem: $$ \begin{cases} u(0,x) = u_0(x) & \text{in }\mathbb{R}^n\\ u_t = -[(\Delta)^{-1}u]\Delta u + u^2 & \text{in }(0,\infty) \times \mathbb{R}^n \end{cases} $$ Here the operator $\Delta^{-1}$ is given by $\Delta^{-1} u = a(t,x)$ such that $\Delta_x a = u$. (for all times $t$)

**Question:** Is there a name for this particular equation, or a class of equations in which it falls? I know that it is a parabolic equation, but not much else. Apologies in advance if this seems like an elementary question. I have also heard that this equation is a toy model of the "Landau Equation", although, on first sight, the Landau equation looks horribly complex (to a student like myself).

**Edit:** I have realized it is also necessary to consider the regularity of $u_0$. Let us assume that $u_0 \geq 0$ and $u_0 \in C^{\infty}_c(\mathbb{R}^n)$ (smooth and compactly supported).

According to my professor, it is also the case that the quantity: $$ \int_{\mathbb{R}^n} u(t,x) \mathrm{d}x $$ is conserved (i.e constant in time), and the quantity (which may be interpreted as entropy): $$ S(t) = \int_{\mathbb{R}^{n}}u(t,x)\log (u(t,x))\mathrm{d}x $$ is a decreasing function of time.

**Edit 2** After some time, a user on MSE has answered the question. This is known as the "isotropic Landau equation". The answer can be found here:
https://math.stackexchange.com/a/3254369/294517

and the paper it links to gives a survey on certain results regarding the equation: