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We study existence, uniqueness and regularity properties of classicalsolutions to viscous Hamilton-Jacobi equations with Caputo time-fractionalderivative. Our study relies on a combination of a gradient bound for thetime-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint methodand sharp estimates in Sobolev and H\"older spaces for the corresponding linearproblem.

Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative