On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives



Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron’s method, unique existence for the Cauchy–Dirichlet and Cauchy–Neumann problems are proved.


Caputo time fractional derivatives Initial-boundary value problems Second order fully nonlinear equations Viscosity solutions 

Mathematics Subject Classification

35D40 35K20 35R11 



Part of the result in this paper was established while the author was visiting the University of Warsaw during May and June 2017. Its hospitality is gratefully acknowledged. The author would like to thank Dr. Yikan Liu for letting him know references on linear equations with Caputo time fractional derivatives. This work is supported by Grant-in-aid for Scientific Research of JSPS Fellows No. 16J03422 and partly the Program for Leading Graduate Schools, MEXT, Japan.


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Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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