Well-posedness of Hamilton-Jacobi equations with Caputo’s time-fractional derivative

For the diffusion phenomena of substances in the domain with a fractal structure, it has been confirmed through experiments that it is better to consider diffusion equations replaced the integer order time derivatives with Caputo’s time-fractional derivatives than normal equations. Such equations are called anomalous diffusion equations and have received much attention in recent years not only in applied fields but also mathematically. Caputo's time-fractional derivative is beginning to be considered not only for linear but also for fully nonlinear equations which is appearing in mathematical finance for example. However, the well-posedness has not been fully discussed except for some of equations. In this talk, we consider Hamilton-Jacobi equations with Caputo’s time-fractional derivative and introduce a proper extended notion of viscosity solutions. As for the integer order, the comparison principle (for uniqueness) as well as the Perron’s method (for existence) is established. Stability with respect to the order of time derivative as well as the standard one is also proved. We are going to touch several possible extensions of these results, especially to second-order equations. This talk is partially based on a joint work with Professor Yoshikazu Giga (U. Tokyo, Japan)