## Journal of Integral Equations and Applications

### Solvability of linear boundary value problems for subdiffusion equations with memory

#### Abstract

For $\nu \in (0,1)$, the nonautonomous integro-differential equation $\mathbf {D}_{t}^{\nu }u-\mathcal {L}_{1}u-\int _{0}^{t}\mathcal {K}_{1}(t-s)\mathcal {L}_{2}u(\cdot ,s)\,ds =f(x,t)$ is considered here, where $\mathbf {D}_{t}^{\nu }$ is the Caputo fractional derivative and $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ are uniformly elliptic operators with smooth coefficients dependent on time. The global classical solvability of the associated initial-boundary value problems is addressed.

#### Article information

Source
J. Integral Equations Applications, Volume 30, Number 3 (2018), 417-445.

Dates
First available in Project Euclid: 8 November 2018

https://projecteuclid.org/euclid.jiea/1541668120

Digital Object Identifier
doi:10.1216/JIE-2018-30-3-417

Mathematical Reviews number (MathSciNet)
MR3874008

Zentralblatt MATH identifier
06979947

#### Citation

Krasnoschok, Mykola; Pata, Vittorino; Vasylyeva, Nataliya. Solvability of linear boundary value problems for subdiffusion equations with memory. J. Integral Equations Applications 30 (2018), no. 3, 417--445. doi:10.1216/JIE-2018-30-3-417. https://projecteuclid.org/euclid.jiea/1541668120

#### References

• A. Alsaedi, M. Kirane and R. Lassoued, Global existence and asymptotic behavior for a time fractional reaction-diffusion system, Comp. Math. Appl. 73 (2017), 951–958.
• E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Mat. 131 (2015), 1–31.
• J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990), 127–293.
• P. Cannarsa and D. Sforza, Global solution of abstract semilinear parabolic equations with memory effect, NoDEA Nonlin. Diff. Eqs. Appl. 10 (2003), 399–430.
• M. Caputo, Models of flux in porous media with memory, Water Resources Res. 36 (2000), 693–705.
• P. Clément, G. Gripenberg and S-O. Londen, Schauder estimates for equations with fractional derivative, Trans. Amer. Math. Soc. 352 (2000), 2239–2260.
• P. Clément, S-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Diff. Eqs. 196 (2004), 418–447.
• P. Clément and J. Prüss, Global existence for semilinear parabolic Volterra equation, Math. Z. 209 (1992), 17–26.
• B.D. Coleman and M.H. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199–208.
• M.G. Crandall, S-O. Londen and J.A. Nohel, An abstract nonlinear Volterra integrodifferential equations, J. Math. Anal. Appl. 64 (1978), 701–735.
• C.M. Dafermos, The mixed initial boundary value problem for the equations of nonlinear one dimensional visco-elasticity, J. Diff. Eqs. 6 (1969), 71–86.
• G. Da Prato and M. Iannelli, Existence and regularity for a class of integrodifferential equation of parabolic type, J. Math. Anal. Appl. 112 (1985), 36–55.
• G. Da Prato, M. Iannelli and E. Sinestrari, Regularity of solutions of a class of linear integrodifferential equations in Banach spaces, J. Int. Eqs. 8 (1985), 27–40.
• W. Desch and R.K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert spaces, J. Diff. Eqs. 70 (1987), 366–389.
• K. Diethelm, The analysis of fractional differential equations, Springer, Berlin, 2010.
• H. Engler, Strong solutions of quasilinear integro-differential equations with singular kernels in several space dimensions, Electr. J. Diff. Eqs. 1995 (1995), 1–16.
• Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990), 309–321, 797–804.
• V.D. Gejji and H. Jafari, Boundary value problems for fractional diffusion-wave equation, Australian J. Math. Anal. Appl. 3 (2006), 1–8.
• G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation, J. Math. Anal. Appl. 333 (2007), 839–862.
• C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlin. Diff. Eqs. Appl. 5 (1998), 333–354.
• G. Gripenberg, Nonlinear Volterra equations of parabolic type due to singular kernels, J. Diff. Eqs. 112 (1994), 154-169.
• G. Gripenberg, S.-O. Londen and J. Prüss, On a fractional partial differential equations with dominating linear part, Math. Meth. Appl. Sci. 20 (1997), 1427–1448.
• R. Hifler, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
• J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other nonlocal in time subdiffusion equations in $R^{d}$, Math. Ann. 366 (2016), 941–979.
• A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud. 204, Elsevier Science B.V., Amsterdam, 2006.
• A.N. Kochubei, A Cauchy problem for evolution equations of fractional order, J. Diff. Eqs. 25 (1989), 967–974.
• ––––, Fractional order diffusion, J. Diff. Eqs. 26 (1990), 485–492.
• M. Krasnoschok, Solvability in Hölder space of an initial-boundary value problem for the time-fractional diffusion equation, J. Math. Phys. Anal. Geom. 12 (2016), 48–77.
• M. Krasnoschok and N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces, Nonlin. Stud. 20 (2013), 589–619.
• O.A. Ladyzhenskaia, V.A. Solonnikov and N.N. Ural'tseva, Linear and quasilinear parabolic equations, Academic Press, New York, 1968.
• T.A.M. Langlands and B.I. Henry, Fractional chemotaxis diffusion equation, Phys. Rev. 81 (2010), 051102.
• A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progr. Nonlin. Diff. Eqs. Appl. 16, Birkhäuser Verlag, Basel, 1995.
• A. Lunardi and E. Sinestrari, $C^{\alpha}$-regularity for nonautonomous linear integrodifferential equations of parabolic type, J. Diff. Eqs. 63 (1986), 88–116.
• F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and fractional calculus in continuum mechanics, A. Garpinteri and F. Mainardi, eds., Springer-Verlag, New York, 1997.
• R. Metzler and J. Klafter, The restaurant at the end of the random walk: Resent developments in the description of anomalous transport by fractional dynamics, J. Phys. 37 (2004), 161–208.
• G.M. Mophou and G.M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Appl. Anal. 91 (2012), 15–34.
• M. Muslim and A.K. Nandakumaran, Existence and approximations of solutions to some fractional order functional integral equations, J. Int. Eqs. Appl. 22 (2010), 95–114.
• R. Ponce, Hölder continuous solutions for fractional differential equatinos and maximal regularity, J. Diff. Eqs. 255 (2013), 3284–3304.
• R. Ponce and V. Poblete, Maximal $L_{p}-$ regularity for fractional differential equations on the line, Math. Nach., doi 10.1002/mana.201600175, 2017.
• A.C. Pipkin, Lectrures on viscoelasticity theory, Appl. Math. Sci. 7 (1986).
• J. Prüss, Evolutionary integral equations and applications, Mono. Math 87 (1993).
• A.V. Pskhu, Partial differential equations of the fractional order, Nauka, Moscow, 2005.
• M. Renardy, W.J. Hrasa and J.A. Nohel, Mathematical problems in viscoelacticity, Longman, Harlow, 1987.
• K. Ritchie, X.-Y. Shan, J. Kondo, K. Iwasawa, T. Fujiwara and A. Kusumi, Detection of non-Brownian diffusion in the cell membrance in single molecule tracking, Biophys. J. 88 (2005), 2266–2277.
• K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), 426–447.
• F. Shen, W. Tan, Y. Zhao and T. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlin. Anal. 7 (2006), 1072–1080.
• O. Staffans, Semigroups generated by a convolution equation, infinite dimensional systems, Lect. Notes Math. 1076 (1984).
• W.C. Tan, W.X. Pan and M.Y. Xu, A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. Non-Lin. Mech. 38 (2003), 645–650.
• V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods, SIAM. J. Math. Anal. 47 (2015), 210–239.
• E. Weeks and D.Weitz, Subdiffusion and the cage effect studied near the colloidal glass transition, Chem. Phys. 284 (2002), 361–367.
• M. Weiss, H. Hashimoto and T. Nilsson, Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy, Biophys. J. 84 (2003), 4043–4052.
• H.-M. Yin, On parabolic Voltera equations in several space dimensions, SIAM J. Math. Anal. 22 (1991), 1723–1737.