A non-linear stable non-Gaussian process in fractional time
KeywordsNon-Gaussian process, fundamental solutions, Banach contraction principle, large-time behavior of solutions
AbstractA subdiffusion problem in which the diffusion term is related to a stable stochastic process is introduced. Linear models of these systems have been studied in a general way, but non-linear models require a more specific analysis. The model presented in this work corresponds to a non-linear evolution equation with fractional time derivative and a pseudo-differential operator acting on the spatial variable. This type of equations has a couple of fundamental solutions, whose estimates for the $L_p$-norm are found to obtain three main results concerning mild and global solutions. The existence and uniqueness of a mild solution is based on the conditions required in some parameters, one of which is the order of stability of the stochastic process. The existence and uniqueness of a global solution is found for the case of small initial conditions and another for non-negative initial conditions. The relationship between the Fourier analysis and Markov processes, together with the theory of fixed points in Banach spaces, is particularly exploited. In addition, the present work includes the asymptotic behavior of global solutions as a linear combination of the fundamental solutions with $L_p$-decay.
D. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890–896.
J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer–Verlag, 1989.
N.G. De Bruijin, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958.
E. Davies, Linear Operators and their Spectra, Cambridge University Press, 2007.
J. Duoandikoetxea and E. Zuazua, Moments, masses de Diracet dècomposition de fonctions, C.R. Acad. Sci. Sèr. 1 Math. 315 (1992), 693–698.
S.D. Eidelman, S.D. Ivasyshen and A.N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Springer Basel AG, 2004.
S.D. Eidelman and A.N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), 211–255.
A. Friedman, Partial Differential Equations of Parabolic Type, Robert E. Krieger Publishing Company, 1983.
T. Ghoul, V. Nguyen and H. Zaag, Construction of type I blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear Anal. 9 (2020), 388–412.
N. Hayashi, E.I. Kaikina, P.I. Naumkin and I.A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Springer–Verlag, 2006.
A. Ilyin, A. Kalashnikov and O. Oleynik, Second order linear equations of parabolic type, J. Math. Sci. 108 (2002), 435–542.
N. Jacob, Pseudo-differential operators and Markov processes, Volume I, Imperial College Press, 2001.
I. Johnston and V. Kolokoltsov, Green’s function estimates for time-fractional evolution equations, Fractal Fract. 3 (2019), 36 pp.
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for timefractional and other non-local in time subdiffusion equation in Rd , Math. Ann. 366 (2016), 941–979.
J. Kemppainen, J. Siljander and R. Zacher, Representation of solutions and largetime behavior for fully nonlocal diffusion equations, J. Differential Equations 263 (2017), 149–201.
V.N. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. 80 (2000), 725–768.
V.N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Springer, 2000.
V.N. Kolokoltsov, The Lèvy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups, Probab. Theory Relat. Fields 151 (2011), 95–123.
V.N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, 2011.
V.N. Kolokoltsov, Differential Equations on Measures and Functional Spaces, Birkhäuser Advanced Texts Basler Lehrbücher e-book, 2019.
V. Kolokoltsov and M. Veretennikova, Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations, Fract. Differ. Calc. 4 (2014), 1–30.
F. Mainardi and R. Gorenflo, On Mittag–Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math. 118 (2000), 283–299.
R. Metzler, J. Jeon, A. Cherstvy and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Physical Chemistry Chemical Physics 16 (2014), 24128–24164.
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser–Verlag, 1993.
W. Rudin, Análisis Funcional, Editorial Reverté S.A., 2002.
R.L. Schilling, Dirichlet operators and the Positive Maximum Principle, Integral Equations Operator Theory 41 (2001), 74–92.
V. Uchaikin and V. Zolotarev, Chance and Stability: Stable Distributions and their Applications, Monographs Modern Probability and Statistics, 1999.
R. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker Inc., 1977.
R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl. 348 (2008), 137–149.
V.M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986.
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