Suppose $u(x,t)$ satisfies the heat equation $u_t=u_{xx}.$ Assume $u(x,0) > 0$ for all $x$; then, by the maximum principle, $u(x,t)>0$ for all $x,t$.
Define $\phi(x,t):=\log u(x,t)$. Then, by a simple calculation, $\phi(x,t)$ satisfies the PDE $$\phi_t=\phi_{xx}+\phi_x^2.$$ Does this nonlinear PDE have a name? Or, does it belong to a larger class of PDEs that people study? Any relevant search terms would be helpful.