Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients

Abstract We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.

[1]  Alexis Vasseur,et al.  A Parabolic Problem with a Fractional Time Derivative , 2015, 1501.07211.

[2]  P. Rybka,et al.  Weak solutions of fractional differential equations in non cylindrical domain , 2016, 1605.06775.

[3]  Rudolf Gorenflo,et al.  Operator theoretic treatment of linear Abel integral equations of first kind , 1999 .

[4]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[5]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[6]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[7]  Zhiyuan Li,et al.  Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients , 2013, Appl. Math. Comput..

[8]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .

[9]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[10]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[11]  Masahiro Yamamoto,et al.  Time-fractional diffusion equation in the fractional Sobolev spaces , 2015 .

[12]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[13]  Rico Zacher.  Weak Solutions of Abstract Evolutionary Integro-Differential Equations in Hilbert Spaces , 2009 .

[14]  William Rundell,et al.  Strong maximum principle for fractional diffusion equations and an application to an inverse source problem , 2015, 1507.00845.

[15]  G. Simonett,et al.  Quasilinear evolutionary equations and continuous interpolation spaces , 2004 .

[16]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 2014 .

[17]  Rico Zacher,et al.  Maximal regularity of type Lp for abstract parabolic Volterra equations , 2005 .