A quasilinear approach to fully nonlinear parabolic (S)PDEs on $\mathbb{R}^d$

Abstract: We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable blow-up criterion. The key idea is the use of the paradifferential operator calculus in order to reduce the fully nonlinear problem into an abstract quasilinear (stochastic) parabolic equation. This allows us to use some recent results on abstract quasilinear (stochastic) evolution equations in Banach spaces. To do so, we analyse the properties of the paradifferential operator, in light of known results on the boundedness of the $\mathcal{H}^{\infty}$-calculus for pseudodifferential operator. Finally, we extend the theory just developed to cover high order fully nonlinear parabolic (S)PDEs.