Uniform stability for fractional Cauchy problems and applications
Keywords
Fractional Cauchy problem, uniform stability, $\mu$-pseudo almost automorphic functions, resolvent operator families, mild solutionsAbstract
In this paper we give uniform stable spatial bounds for the resolvent operator families of the abstract fractional Cauchy problem on $\mathbb{R}_+$. Such bounds allow to prove existence and uniqueness of $\mu$-pseudo almost automorphic $\epsilon$-mild regular solutions to the nonlinear fractional Cauchy problem in the whole real line. Finally, we apply our main results to the fractional heat equation with critical nonlinearities.References
L. Abadias, E. Alvarez and C. Lizama, Regularity properties of mild solutions for a class of volterra equation with critical nonlinearities, J. Integral Equations Appl. 30 (2018), no. 2, 219–256.
L. Abadias, C. Lizama and P. J. Miana, Sharp extensions and algebraic properties for solutions families of vector-valued differential equations, Banach J. Math. Anal. 10 (2016), 169–208.
L. Abadias and P.J. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces. Art. (2015), ID 158145, pp. 9.
W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, second edition, Monographs in Mathematics, vol. 96, Birkhäuser, 2011.
J.M. Arrieta and A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations, Trans. Amer. Math. Soc. 352 (1999), no. 1, 285–310.
E. Bajlekova, Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal. 3 (2000), no. 3, 213–230.
E. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, University Press Facilities, Eindhoven University of Technology, 2001.
J. Blot, P. Cieutat, G.M. N’Guérékata and D. Pennequin, Superposition operators between various almost periodic function spaces and applications, Commun. Math. Anal. 6 (2009), no. 1, 42–70.
J. Blot, G.M. Mophou, G.M. N’Guérékata and D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 71 (2009), 903–909.
J. Blot, P. Cieutat and K. Ezzinbi, Measure theory and pseudo almost automorphic functions: New developments and applications, Nonlinear Anal. 75 (2012), no. 4, 2426–2447.
S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. USA 35 (1949), 368–370.
E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr, Models based on Mittag–Leffler functions for anomalous relaxation in dielectrics, Eur. Phys. J. Special Topics 193 (2011), 161–171; revised version in http://arxiv.org/abs/1106.1761.
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum. 80 (2010), 121–142.
P. Cieutat, S. Fatajou and G.M. N’Guérékata, Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Appl. Anal. 89 (2010), no. 1, 11–27.
B. De Andrade, A.N. Carvalho, P.M. Carvalho-Neto and P. Marı́n-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 439–467.
B. De Andrade, C. Cuevas, J. Liang and H. Soto, Periodicity and ergodicity for abstract evolution equations with critical nonlinearities, Adv. Difference Equ. 2015 (2015), 1–20.
K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer–Verlag, New York, 2000.
R. Figueiredo Camargo, A.O. Chiacchio, R. Charnet and E. Capelas de Oliveira, Solution of the fractional Langevin equation and the Mittag–Leffler functions, J. Math. Phys. 50 (2009), 1–8.
R. Gorenflo, A.A. Kilbas, F. Mainardi and S.V. Rogosin, Mittag–Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014.
G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer, New York, 2001.
G.M. N’Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal. 68 (2008), 2658–2667.
B.H. Guswanto and T. Suzuki, Existence and uniqueness of mild solutions for fractional semilinear differential equations, Electron. J. Differential Equations 168 (2015), 1–16.
M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications. vol. 169, Birkhäuser, Basel, 2006.
A. Hanyga and M. Seredyńska, On a mathematical framework for the constitutive equations of anisotropic dielectric relaxation, J. Stat. Phys. 131 (2008), 269–303.
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer–Verlag, Berlin, 1981.
B. Kaltenbacher and I. Lasiecka, Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Control Cybernet. 40 (2011), no. 4, 971–988.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
M. Kostić, (a, k)-regularized C-resolvent families: regularity and local properties, Abstr. Appl. Anal. (2009), Article ID 858242, pp. 27.
Y.N. Li and H.R. Sun, Integrated fractional resolvent operator function and fractional abstract Cauchy problem, Abstr. Appl. Anal. (2014), Article ID 430418, pp. 9.
K. Li and J. Peng, Fractional resolvents and fractional evolution equations, Appl. Math. Lett. 25 (2012), 808–812.
C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl. 243 (2000), 278–292.
C. Lizama and E. Vergara, Uniform stability of resolvent families, Proc. Amer. Math. Soc. 132 (2004), no. 1, 175–181.
F. Mainardi, Fractional calculus and waves in linear viscoelasticity, an introduction to mathematical models, Imperial College Press, 2010.
F. Mainardi, G. Pagnini and R. Gorenflo, Mellin transform and subodination laws in fractional diffusion processes, Fract. Calc. Appl. Anal. 6 (2003), no. 4, 441–459.
Z.-D. Mei, J.-G. Peng and Y. Zhang, A characteristic of fractional resolvents, Fract. Calc. Appl. Anal. 16 (2013), no. 4, 777–790.
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
J. Mu, Y. Zhou and L. Peng, Periodic solutions and S-asymptotically periodic solutions to fractional evolution equations, Discrete Dyn. Nat. Soc. (2017), Article ID 1364532, pp. 12.
D. Mugnolo, Asymptotic of semigroups generated by operator matrices, Arab. J. Math. 3 (2014), 419–435.
R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Sol. (B) 33 (1986), 425–430.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer–Verlag, New York, 1983.
T.R. Prabhakar, A singular integral equation with a generalized Mittag–Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15.
I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 98, Acamedic Press, San Diego, California, USA, 1999.
J. Prüss, Evolutionary integral equations and applications, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 2012.
J. Van Neerven, The asymptotic behaviour of semigroups of linear operators, Operator Theory Advances and Applications, vol. 88, Birkhäuser, 1996.
E.M. Wright, On the coefficients of power series having exponential singularities, J. London Math. Soc. 8 (1933), 71–79.
T.-J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (2008), 518–524.
K. Yosida, Functional Analysis, fifth edition, A Series of Comprehensive Studies in Mathematics, vol. 123, Springer, 1978.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0