Consider the following partial differential equation,
\begin{equation}
\tag{1}
\frac{\partial\rho}{\partial t}=\frac{1}{k}\ln\left(1+e^{k\alpha\nabla^{2}\rho}\right)-\alpha\nabla^{2}\rho.
\end{equation}
Assuming a one dimensional system, we may set $\nabla^{2}\rho$ to $\frac{\partial^{2}\rho}{\partial x^{2}}$. Separation of variables doesn't seem to apply here. However, a Fourier transform yields
\begin{equation}
\tag{2}
\frac{d}{dt}\hat{p}=\frac{1}{k}\ln\left(1+e^{-k\alpha\omega^{2}\hat{p}}\right)+\alpha\omega^{2}\hat{p},
\end{equation}
such that
\begin{equation}
\begin{split}
\mathcal{F}\left(\rho\left(x,t\right)\right)&=\hat{\rho}\left(\omega,t\right),\\
\mathcal{F}\left(\rho_{\times}\right)&=i\omega\hat{\rho}\left(\omega,t\right),\\
\mathcal{F}\left(\rho_{\times\times}\right)&=-\omega^{2}\hat{\rho}\left(\omega,t\right),
\end{split}
\end{equation}
for the system in eq. $1$ denoted by $\rho_{t}=\frac{1}{k}\ln\left(1+e^{k\alpha\rho_{\times\times}}\right)-\alpha\rho_{\times\times}$. How could one solve eq. $2$ or use another method for eq. $1$? I would be grateful for any help.
Best Answer
Solved in comments by simplifying the problem as a variant of the heat equation. However, any new thoughts on the original nonlinear PDE are welcome.