Abstract
We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the modeling of dynamic processes in materials with memory. The proof relies heavily on a regularity result on the interior Hölder continuity of weak solutions to time fractional diffusion equations, which has been proved recently by the author. We further establish an L 2 decay estimate for the special case with vanishing external source term and homogeneous Dirichlet boundary condition.
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References
Caputo M.: Diffusion of fluids in porous media with memory. Geothermics 28, 113–130 (1999)
Clément Ph., Li S.: Abstract parabolic quasilinear evolution equations and applications to a groundwater problem. Adv. Math. Sci. Appl. 3, 17–32 (1994)
Clément, Ph.; Gripenberg, G.; Londen, S.-O.: Regularity properties of solutions of fractional evolution equations. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 235–246, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001.
Clément Ph., Londen S.-O., Simonett G.: Quasilinear evolutionary equations and continuous interpolation spaces. J. Differ. Equ. 196, 418–447 (2004)
Clément Ph., Prüss J.: Global existence for a semilinear parabolic Volterra equation. Math. Z. 209, 17–26 (1992)
DiBenedetto E.: Degenerate parabolic equations. Springer, New York (1993)
Gripenberg G.: Volterra integro-differential equations with accretive nonlinearity. J. Differ. Equ. 60, 57–79 (1985)
Gripenberg, G.; Londen, S.-O.; Staffans, O.: Volterra integral and functional equations. Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990.
Jakubowski, V. G.: Nonlinear elliptic-parabolic integro-differential equations with L 1-data: existence, uniqueness, asymptotics. Dissertation, University of Essen, 2001.
Jakubowski, V. G.; Wittbold, P.: On a nonlinear elliptic-parabolic integro-differential equation with L 1-data. J. Differ. Equ. 197 (2004), 427–445.
Kilbas, A.A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations. Elsevier, 2006.
Kochubei A.N.: The Cauchy problem for evolution equations of fractional order. Differ. Equ. 25, 967–974 (1989)
Krägeloh A.M.: Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups. J. Math. Anal. Appl. 283, 459–467 (2003)
Meerschaert M.M., Nane E., Vellaisamy P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37,979–1007 (2009)
Metzler R., Klafter J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
Metzler, R.; Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37 (2004), R161–R208.
Nakagawa, J., Sakamoto, K., Yamamoto, M.: Overview to mathematical analysis for fractional diffusion equations—new mathematical aspects motivated by industrial collaboration. J. Math-for-Ind. 2 (2010A-10), 99–108.
Pollard, H.: The completely monotonic character of the Mittag-Leffler function E a (−x). Bull. Am. Math. Soc. 54 (1948), 1115–1116.
Prüss, J.: Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser, Basel, 1993.
Prüss J.: Maximal regularity for evolution equations in L p -spaces. Conf. Semin. Mat. Univ. Bari 285, 1–39 (2002)
Sobolevskii, P.E.: Coerciveness inequalities for abstract parabolic equations. Soviet Math. (Doklady) 5 (1964), 894–897.
Vergara, V.; Zacher, R.: Lyapunov functions and convergence to steady state for differential equations of fractional order. Math. Z. 259 (2008), 287–309.
Zacher, R.: A De Giorgi-Nash type theorem for time fractional diffusion equations. To appear in Math. Ann.
Zacher R.: Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients. J. Math. Anal. Appl. 348, 137–149 (2008)
Zacher, R.: De Giorgi-Nash-Moser estimates for evolutionary partial integro-differential equations. Habilitation Thesis. MLU Halle, 2010. Online available at http://digital.bibliothek.uni-halle.de/hs/content/titleinfo/825947.
Zacher, R.: Maximal regularity of type L p for abstract parabolic Volterra equations. J. Evol. Equ. 5 (2005), 79–103.
Zacher R.: Quasilinear parabolic integro-differential equations with nonlinear boundary conditions. Differ. Integral Equ. 19, 1129–1156 (2006)
Zacher R.: Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj 52, 1–18 (2009)
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Zacher, R. Global strong solvability of a quasilinear subdiffusion problem. J. Evol. Equ. 12, 813–831 (2012). https://doi.org/10.1007/s00028-012-0156-0
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DOI: https://doi.org/10.1007/s00028-012-0156-0