Name for a Particular (Parabolic) PDE

Question

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. Also if someone could point me to an introductory resource( at the graduate level) for parabolic PDE, that would be nice: I have experience with elliptic equations (variational formulations, regularity theory like DeGiorgi Nash, Viscosity Solutions) but am rather inexperienced when it comes to Parabolic equations).

Edit: I have realized itis 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: $$ S(t) = \int_{\mathbb{R}^{n}}u(t,x)\log (u(t,x))\mathrm{d}x $$ is a decreasing function of time.

Answer 1

A good survey of recent work on this is given by Gualdani and Zamponi in

"A review for an isotropic Landau model," in PDE models for multi-agent phenomena, Springer INDAM. (arxiv link: https://arxiv.org/abs/1708.02097).

Other important papers in this area (discussed also in the above article) include the work of Krieger-Strain, Gressman-Krieger-Strain, and Gualdani-Guillen.

Answer 2

You should consider posting this in mathoverflow. I have never seen an equation like this. It is nasty. Anyway, the first place to learn a bit of parabolic equations is Evans "Partial differential equations".

A more advanced (and more difficult to read) book is Lieberman "Second Order Parabolic Differential Equations". It is meant to be like Gilbarg and Trudinger "Elliptic Partial Differential Equations of Second Order", which is a classical book in elliptic equations.

Then there are a lot of very good books on abstract evolution equations (which might help, given your equation). A classical book is Dautray and Lions "Evolutions Problems" Volume 5 (so yes, there are five volumes, but they are written in a beautiful way and you might learn a lot).

A book which might help for your equation is Hu "Blow-up theory for semilinear parabolic equations".

Now coming to your equation% $$ u_{t}=b\Delta u+u^{2}, $$ this is parabolic if $b(x,t)\geq\delta>0$ for all $x,t$. If $b\geq0$ then it would be a degenerate parabolic equation. Degenerate parabolic equations are difficult to treat. If $b<0$ then you have backward parabolic equations. These are nasty because in general solutions do not exist. If $b$ changes sign then all hell breaks loose. I have no idea what can happen.

In your case $b=-(\Delta^{-1}u)$ which makes this equation nonlocal. To determine the sign of $b$ you have to use $-\Delta b=u$. This means that you should use the maximum principle for elliptic equations. The problem is that it works well in bounded domains but you are in $\mathbb{R}^{n}$, where it can fail. Your best bet is to assume good hypotheses on $u_{0}$ in such a way $-\Delta b_{0}=u_{0}$ implies that $b_{0}(x)\geq\delta>0$. Then you might try to find a solution to your problem for $[0,T]$ and $T$ very small. I don't know.

Now in the case in which $b(x,t)\geq\delta>0$, then the equation $u_{t}=b\Delta u+u^{2}$ is a semilinear parabolic equation. When $b=1$ it has been studied by Fujita. You will find a chapter in the book of Hu "Blow-up theory for semilinear parabolic equations" (see Section 5.4). Depending on the value of $n$ and $u_{0}$ you might have global existence or blow-up of solutions.

In conclusion, your equation is non-standard. I don't think it has a name. Maybe "nonlocal semilinear equation of parabolic type". Where did it come from?

Anyway, try math overflow. You might find experts in parabolic equations. I am not one.