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:


  • 1
    $\begingroup$ It is surely a (nonlinear) integrodifferential equation since $$\Delta^{-1}u(x)\triangleq \int\limits_{\Bbb R^n}s(x-y)u(y)\mathrm{d}y\quad x\in\Bbb R^n,$$ where $s(x-y)$ is the fundamental solution of the Laplace operator with Dirac measure concentrated at the point $x\in\Bbb R^n$. I have not seen anything similar. $\endgroup$ – Daniele Tampieri Jun 7 at 15:05

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