Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations

The purpose is to study the Cauchy problem for non-linear in time and space pseudo- differential equations. These include the fractional in time versions of Hamilton-Jacobi-Bellman (HJB) equations governing the limits of controlled scaled Continuous Time Random Walks (CTRWs). As a preliminary step which is of independent interest we analyse the corresponding linear equation proving its well-posedness and smoothing properties.

[1]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[2]  M. Veretennikova.  Controlled continuous time random walks and their position dependent extensions , 2014 .

[3]  Vassili N. Kolokoltsov,et al.  A fractional Hamilton Jacobi Bellman equation for scaled limits of controlled Continuous Time Random Walks , 2014 .

[4]  The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space , 2013 .

[5]  A. Pskhu.  Multi-time fractional diffusion equation , 2013 .

[6]  Yong Zhou,et al.  Cauchy problem for fractional evolution equations with Caputo derivative , 2013 .

[7]  Changpin Li,et al.  The asymptotics of the solutions to the anomalous diffusion equations , 2013, Comput. Math. Appl..

[8]  José António Tenreiro Machado,et al.  Fractional differentiation and its applications I , 2013, Comput. Math. Appl..

[9]  C. Lizama,et al.  Mild solutions for abstract fractional differential equations , 2013 .

[10]  Ravi P. Agarwal,et al.  Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions , 2013 .

[11]  Yong Zhou,et al.  Abstract Cauchy problem for fractional differential equations , 2013 .

[12]  V. Uchaikin.  Fractional Derivatives for Physicists and Engineers , 2013 .

[13]  M. Fu,et al.  Positive Solutions for Fractional Differential Equations from Real Estate Asset Securitization via New Fixed Point Theorem , 2012 .

[14]  J. T. Tenreiro Machado,et al.  Fractional order inductive phenomena based on the skin effect , 2012 .

[15]  Mark M. Meerschaert,et al.  Space-time fractional diffusion on bounded domains , 2012 .

[16]  Li Kexue,et al.  Laplace transform and fractional differential equations , 2011 .

[17]  M. Matar.  On existence and uniqueness of the mild solution for fractional semilinear integro-differential equations , 2011 .

[18]  V. Kolokoltsov.  Markov Processes, Semigroups and Generators , 2011 .

[19]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[20]  Jigen Peng,et al.  Laplace transform and fractional differential equations , 2011, Appl. Math. Lett..

[21]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[22]  Mark M. Meerschaert,et al.  Fractional Cauchy problems on bounded domains , 2008, 0802.0673.

[23]  Vassili N. Kolokoltsov,et al.  Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics , 2007, 0706.1928.

[24]  S. Grebenshchikov.  Stochastic Models of Structural Plasma Turbulence (VSP, Leiden, 2006), Ed. by V. Yu. Korolev and N. N. Skvortsova , 2007 .

[25]  N. N. Skvortsova,et al.  Stochastic Models of Structural Plasma Turbulence , 2006 .

[26]  Margarita Rivero,et al.  CAPUTO LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS , 2006 .

[27]  R. Gorenflo,et al.  ON SOLVABILITY OF LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES∗ , 2005 .

[28]  Enrico Scalas,et al.  Speculative option valuation and the fractional diffusion equation , 2005 .

[29]  A. Kochubei,et al.  Cauchy problem for fractional diffusion equations , 2003, math/0310271.

[30]  C. Heyde,et al.  Dynamic models of long-memory processes driven by Lévy noise , 2002, Journal of Applied Probability.

[31]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[32]  Emilia Bazhlekova,et al.  Fractional evolution equations in Banach spaces , 2001 .

[33]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[34]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[35]  Francesco Mainardi,et al.  The time fractional diffusion-wave equation , 1995 .

[36]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[37]  V. Zolotarev.  One-dimensional stable distributions , 1986 .