I'm reading up on the KPZ equation through the article by Bertini and Giacomin from 1997 and some lecture notes by Jeremy Quastel, the equation in 1+1 dimensions is stated as (for $h_t$ the height of the surface and $\mathcal{W}_t$ space-time white noise)
$$ \partial_t h_t = \frac{1}{2} \Delta h_t - \frac{1}{2} (\nabla h_t)^2 + \mathcal{W}_t. $$
As indicated in the notes here by Quastel (https://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf, section 1.4 to 1.6) the nonlinear term is a problem that would actually make the equation ill-posed, but I don't really see how the nonlinear term would give the problems here. The only thing I can think of is integrability issues. The second derivative term requires more regularity, and would that not give more problems?
The lecture notes say that in order for the equation to make sense, we would need an "infinite renormalization", that is
$$ \partial_t h_t = \frac{1}{2} \Delta h_t - \frac{1}{2} ( (\nabla h_t)^2 - \infty) + \mathcal{W}_t. $$
Now this is only formally, but I don't see how the equation as it is (without the $\infty$) would make any less sense than the PDE without the nonlinear term. What's the issue with simply squaring a term that makes the entire PDE ill-posed? Could someone help me out?
Cheers.
The 'problems' they are referring to are problems in solving the equation. There are many, many methods for solving linear PDEs, both for specific and general cases. For example, if we were to remove the squaring so that term was linear, then we could solve on a rectangular domain using seperation of variables. Nonlinearity in the equation (or boundary conditions) is almost always the reason why equations occurring in physics are hard to solve exactly, and in many cases people resort to numerics.