Let us consider quasilinear second order PDEs.
The following can have a weak formulation which depends only on $u$, not $u',u''$, so we can consider $L^p$ weak solution of this problem. $$u_t-u_{xx}=uu_x+u^2$$
However, what about the following? $$u_t-u_{xx}=(u_x)^2$$
It seems, at least, that such pde would not have weak formulation depending only on $u$.
Question) Can we define what $L^p$ weak solution is on the second one? (in any reasonable way)