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I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$): $$x^3 f_{xxxt}+ f =0$$ Does anyone know if this type of PDE already appeared in the literature? ...

Let $\Omega\subset \mathbb R^n$ be a convex subset. All the objects below will be defined on this set. Let us assume $P(x,D)$ to be a differentiable operator order $m$ and of square size, that is ...