Regularity of solutions to the parabolic fractional obstacle problem (Q2842808)

There is considered the obstacle problem NEWLINE\[NEWLINE \min\{u_t + (-\Delta)^s u, u-\psi\} = 0 \;\text{on}\;[0,T]\times \mathbb{R}^n, \;\;u(0) = \psi \;\text{on}\;\mathbb{R}^n, \;s\in(0,1).\tag{1}NEWLINE\]NEWLINE The main result is as follows. Let \(\psi \in C^2(\mathbb{R}^n)\), \ \(\parallel\nabla\psi\parallel_{L^\infty(\mathbb{R}^n)} + \parallel D^2 \psi\parallel_{L^\infty(\mathbb{R}^n)} + \parallel(-\Delta)^s \psi\parallel_{C^{1-s}_x(\mathbb{R}^n)} <+\infty\) and \(u\) be the unique continuous viscosity solution of (1).NEWLINENEWLINEThen \(u\) is globally Lipschitz on \([0,T]\times \mathbb{R}^n\) and NEWLINE\[NEWLINEu_t, (-\Delta)^s u \in \log \text{Lip}_t C^{1-s}_x(\mathbb{R}^n)\big( (0,T]\times \mathbb{R}^n \big), \;\text{if} \;0 < s\leq 1/3, \tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_t \in C^{\frac{1-s}{2s} - 0^+, 1-s}_{t \;\;\;\;\;\;\;\;\;x}\big( (0,T]\times \mathbb{R}^n \big), \;(-\Delta)^s u \in C^{\frac{1-s}{2s}, 1-s}_{t \;\;\;\;\;\;x}\big( (0,T]\times \mathbb{R}^n \big), \;\text{if} \;1/3 < s <1, \tag{3}NEWLINE\]NEWLINE here the norm of the space \(\log \text{Lip}_t C^{\beta}_x( [a,b]\times \mathbb{R}^n)\) is determined by NEWLINE\[NEWLINE\parallel w \parallel_{L^\infty( [a,b]\times\mathbb{R}^n)} + \sup_{[a,b]\times\mathbb{R}^n}\frac{|w(t,x) - w(t',x')|}{|t-t'|(1+\big|\log|t-t'|\big|) + |x - x'|^\beta }, \;\beta\in (0,1), NEWLINE\]NEWLINE the Hölder index \(\frac{1-s}{2s} - 0^+\) in (3) and the set \((0,T]\times \mathbb{R}^n\) in (2), (3) are understood as \(\frac{1-s}{2s} - \varepsilon \) and \([\varepsilon,T]\times \mathbb{R}^n\) for all\, \(\varepsilon > 0\) respectively.