Variation of the Diffusion Equation

Question

Let $\mathcal{L}=D\dfrac{\partial^{2}}{\partial x^{2}}-v\dfrac{\partial}{\partial x}+\beta$ be a differential operator describing diffusion ($D$) with drift ($v$) and a source ($\beta$). As part of a calculation I did, I ended up with a non-linear variant of the diffusion/heat equation

$$\dfrac{\partial \psi}{\partial t}=\psi\mathcal{L}\psi+D^{\prime}\left(\dfrac{\partial\psi}{\partial x}\right)^{2}$$

which can be also written as

$$\dfrac{\partial \psi}{\partial t}=\mathcal{L}^{\prime}\psi^{2}+\left(D^{\prime}-D\right)\left(\dfrac{\partial\psi}{\partial x}\right)^{2}$$

where $\mathcal{L}^{\prime}=\dfrac{1}{2}D\dfrac{\partial^{2}}{\partial x^{2}}-v\dfrac{\partial}{\partial x}+\beta$. Note that $D^{\prime}$ is just another coefficient, not related to $D$. I am wondering if this equation is known and has a solution (or at least some illuminating analysis of the resulting scaling behaviors). A look at Wikipedia's list of non-linear PDEs isn't much of a help, and I mostly find equations with a linear laplacian term $D\dfrac{\partial^{2}\psi}{\partial x^{2}}$ and non-linearity in the reaction part.

Answer 1

A good starting point is probably to look at literature for $$ \partial_t u = \Delta (u^m), \quad m > 1, $$ which is known as the porous medium equation (for you $m = 2$ is of course of most interest).

I'm no expert at all, but let me quickly argue why the theory for these equations is complicated: For $m = 2$ one can rewrite the equation as you did, $$ \partial_t u = 2 u \Delta u + 2 |\nabla u|^2 $$ and one can think of the factor $2 u$ in front of $\Delta u$ as a diffusion coefficient.

• For $u < 0$ we have negative diffusion, which corresponds, ignoring $| \nabla u |^2$, to solving the heat equation backward in time (for given final value find the initial value - this doesn't really work for a diffusive process).
• For $u = 0$ the equation is no longer parabolic, and the corresponding equation is known as the eikonal equation $\partial_t u = 2 | \nabla u |^2$.
• For $u > 0$ one gets almost normal diffusion, but with speed of diffusion related to the size of $u$ and degenerating for $u$ small.

The kind of nonlinear diffusion equations you mentioned are in contrast well-behaved since the, mathematically very nice, diffusion is always active at a stable rate.