A P ] 1 1 A ug 2 01 9 STRONG DISSIPATIVITY OF GENERALIZED TIME-FRACTIONAL DERIVATIVES AND QUASI-LINEAR ( STOCHASTIC ) PARTIAL DIFFERENTIAL EQUATIONS

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted L-path spaces is proved. In particular, the classical Caputo derivative is included as a special case. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of type d dt (k ∗ u)(t) +A(t, u(t)) = f(t), 0 < t < T, with (in general nonlinear) operators A(t, ·) satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on [0,∞) with lim s→∞ k(s) = 0. Analogous results for the case, where f is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasilinear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators or the time-fractional (stochastic) p-Laplace equation are covered.

[1]  Wei Liu,et al.  Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives , 2017, SIAM J. Math. Anal..

[2]  Zhen-Qing Chen.  Time fractional equations and probabilistic representation , 2017, 1703.01739.

[3]  Le Chen.  NONLINEAR STOCHASTIC TIME-FRACTIONAL DIFFUSION EQUATIONS ON R: MOMENTS, HÖLDER REGULARITY AND INTERMITTENCY , 2017 .

[4]  Yuri Luchko,et al.  General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems , 2016 .

[5]  Yimin Xiao,et al.  L-Kuramoto-Sivashinsky SPDEs vs. time-fractional SPIDEs: exact continuity and gradient moduli, 1/2-derivative criticality, and laws , 2016, 1603.01172.

[6]  Kristian Kirsch,et al.  Methods Of Modern Mathematical Physics , 2016 .

[7]  Le Chen,et al.  Space-time fractional diffusions in Gaussian noisy environment , 2015, 1508.00252.

[8]  Erkan Nane,et al.  Asymptotic properties of some space-time fractional stochastic equations , 2015 .

[9]  Sungbin Lim,et al.  An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients , 2015, 1505.00504.

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

[11]  Vicente Vergara,et al.  Optimal Decay Estimates for Time-Fractional and Other NonLocal Subdiffusion Equations via Energy Methods , 2013, SIAM J. Math. Anal..

[12]  Erkan Nane,et al.  Space-time fractional stochastic partial differential equations , 2014, 1409.7366.

[13]  Zhen-Qing Chen,et al.  Fractional time stochastic partial differential equations , 2014, 1404.1546.

[14]  Viorel Barbu,et al.  An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise , 2014, 1402.4940.

[15]  Benjamin Gess,et al.  Random Attractors for Degenerate Stochastic Partial Differential Equations , 2012, 1206.2329.

[16]  Wei Liu,et al.  Local and global well-posedness of SPDE with generalized coercivity conditions☆ , 2012, 1202.0019.

[17]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[18]  W. Marsden.  I and J , 2012 .

[19]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[20]  M. Meerschaert,et al.  Stochastic Models for Fractional Calculus , 2011 .

[21]  R. Herrmann.  Fractional Calculus: An Introduction for Physicists , 2011 .

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

[23]  Viorel Barbu,et al.  Nonlinear Differential Equations of Monotone Types in Banach Spaces , 2010 .

[24]  René L. Schilling,et al.  Bernstein Functions: Theory and Applications , 2010 .

[25]  Luisa Beghin,et al.  Fractional diffusion equations and processes with randomly varying time. , 2011, 1102.4729.

[26]  Teodor M. Atanackovic,et al.  Time distributed-order diffusion-wave equation. I. Volterra-type equation , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[28]  Mark M. Meerschaert,et al.  Brownian subordinators and fractional Cauchy problems , 2007, 0705.0168.

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

[30]  Damian Craiem,et al.  Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries , 2008, Physics in medicine and biology.

[31]  Anatoly N. Kochubei,et al.  Distributed-order calculus: An operator-theoretic interpretation , 2007, 0710.1710.

[32]  R. Gorenflo,et al.  Time-fractional Diffusion of Distributed Order , 2007, cond-mat/0701132.

[33]  R. Gorenflo,et al.  The Two Forms of Fractional Relaxation of Distributed Order , 2007, cond-mat/0701131.

[34]  Feng-Yu Wang,et al.  Stochastic generalized porous media and fast diffusion equations , 2006, math/0602369.

[35]  R. Gorenflo,et al.  FRACTIONAL RELAXATION OF DISTRIBUTED ORDER , 2006 .

[36]  Mark M. Meerschaert,et al.  Limit theorems for continuous-time random walks with infinite mean waiting times , 2004, Journal of Applied Probability.

[37]  Volker G. Jakubowski,et al.  On a nonlinear elliptic–parabolic integro-differential equation with L1-data , 2004 .

[38]  J. Klafter,et al.  Fractional Fokker-Planck equation for ultraslow kinetics , 2003, cond-mat/0301487.

[39]  Stefan Samko,et al.  INTEGRAL EQUATIONS OF THE FIRST KIND OF SONINE TYPE , 2003 .

[40]  Mark M. Meerschaert,et al.  STOCHASTIC SOLUTIONS FOR FRACTIONAL CAUCHY PROBLEMS , 2003 .

[41]  Hans-Peter Scheffler,et al.  Stochastic solution of space-time fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[43]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[44]  R. Wolpert.  Lévy Processes , 2000 .

[45]  J. Klafter,et al.  Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach , 1999 .

[46]  Wilhelm Stannat,et al.  The theory of generalized Dirichlet forms and its applications in analysis and stochastics , 1999 .

[47]  Zhi-Ming Ma,et al.  Introduction to the theory of (non-symmetric) Dirichlet forms , 1992 .

[48]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[49]  S. Edwards,et al.  The Theory of Polymer Dynamics , 1986 .

[50]  F. Browder,et al.  Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[51]  B. Ross,et al.  The development of fractional calculus 1695–1900 , 1977 .

[52]  Haim Brezis,et al.  Équations et inéquations non linéaires dans les espaces vectoriels en dualité , 1968 .

[53]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[54]  M. Huggins.  Viscoelastic Properties of Polymers. , 1961 .

[55]  N. Sonine.  Sur la généralisation d’une formule d’Abel , 1884 .