Abstract
We study the following time-fractional nonlinear superdiffusion equation
where \(1<\alpha <2\), \(p>1\), \(u_0,u_1\in L^q(\mathbb {R}^N)\) (\(q>1\)) and \({_0^CD_t^\alpha u}\) denotes the Caputo fractional derivative of order \(\alpha .\) The critical exponents of this problem are determined when \(u_1\equiv 0\) and \(u_1\not \equiv 0, \) respectively.
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References
M.F. de Almeida, L.C.F. Ferreira, Self-similarity, symmetries and asymptotic behavior in Morrey spaces for a fractional wave equation, Differ. Integr. Equ. 25 (2012), 957–976.
E. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Technische Universiteit Eindhoven, 2001.
C.M. Carracedo, M.S. Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, 187 Elsevier, 2001.
P.M. de Carvalho-Neto, G. Planas, Mild solutions to the time fractional Navier-Stokes equations in \(\mathbb{R}^N\), J. Differ. Equ. 259 (2015), 2948–2980.
T. Cazenave, F. Dickstein and F.B. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal. 68 (2008), 862–874.
P. Clément, S.O. Londen, G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differ. Equ. 196 (2004), 418–447.
S.D. Eidelman, A.N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differ. Equ. 199 (2004), 211–255.
H. Engler, Asymptotic self-similarity for solutions of partial integro-differential equations, Z. Anal. Anwend. 26 (2007), 417–438.
A.Z. Fino, M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Quart. Appl. Math. 70 (2012), 133–157.
Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990), 309–321.
Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation II, Osaka J. Math. 27 (1990), 797–804.
B. Guo, X. Pu, F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific, 2015.
G. Gripenberg, Weak solutions of hyperbolic-parabolic Volterra equations, Trans. Amer. Math. Soc. 343 (1994), 675–694.
H. Hirata, C.X. Miao, Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative, Adv. Differential Equ. 7 (2002), 217–236.
T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Commun. Pure Appl. Math. 33 (1980), 501–505.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, vol 204. Elsevier Science B.V., Amsterdam, 2006.
I. Kim, K.H. Kim, S. Lim, An \(L_q(L_p)\)-theory for the time fractional evolution equations with variable coefficients, Adv. Math. 306 (2017), 123–176.
M. Li, C. Chen, F.B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal. 259 (2010), 2702–2726.
K. Li, J. Peng, J. Xia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal. 263 (2012), 476–510.
Y.N. Li, H.R. Sun, Z.S. Feng, Fractional abstract Cauchy problem with order \(\alpha \in (1,2)\), Dyna. PDE 13(2) (2016), 155–177.
R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), 161–208.
R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.
M.M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab. 37 (2009), 979–1007.
C.X. Miao, H. Yang, The self-similar solution to some nonlinear integrodifferential equations corresonding to fractional order time derivate, Acta. Math. Sin. 21 (2005), 1337–1350.
E. Mitidieri, S.I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1–383.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
R. Ponce, V. Poblete, Maximal \(L_p\)-regularity for fractional differential equations on the line, Math. Nachr. 290 (2017), 2009–2023.
J.Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993.
P. Quittner, P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007.
K. Sakamoto, 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.
R.N. Wang, D.H. Chen, T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equ. 252 (2012), 202–235.
J.Y. Wang, Y. Zhou, Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal. 74 (2011), 5929–5942.
B.T. Yordanov, Q.S. Zhang, Finite time blow-up for critical wave equations in high dimensions, J. Funct. Anal. 231 (2006), 361–374.
R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, Adv. Differential Equ. 14 (2009), 749–770.
R. Zacher, Maximal regularity of type \(L_p\) for abstract parabolic Volterra equations, J. Evol. Equ. 5 (2005), 79–103.
R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations, Math. Ann. 356 (2013), 99–146.
Q.G. Zhang, H.R. Sun, The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation, Topol. Methods Nonlinear Anal. 46(1) (2015), 69–92.
Acknowledgements
The authors are very grateful to the anonymous referee for valuable comments and suggestions. This work is partially supported by National Natural Science Foundation of China (Grant Nos. 11526108, 11601216).
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Zhang, Q., Li, Y. Global well-posedness and blow-up solutions of the Cauchy problem for a time-fractional superdiffusion equation. J. Evol. Equ. 19, 271–303 (2019). https://doi.org/10.1007/s00028-018-0475-x
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DOI: https://doi.org/10.1007/s00028-018-0475-x