Global existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditions

Nouioua, Farid; Basti, Bilal

doi:10.2478/aupcsm-2021-0003

Abstract

This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder’s and Banach’s fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

References

[1] Arioua, Yacine, and Bilal Basti, and Nouredine Benhamidouche. “Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative.” Appl. Math. E-Notes 19 (2019): 397-412. Cited on 44.
Search in Google Scholar Back to article
[2] Basti, Bilal, and Yacine Arioua, and Nouredine Benhamidouche. “Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations.” J. Math. Appl. 42 (2019): 35-61. Cited on 44.
Search in Google Scholar Back to article
[3] Basti, Bilal, and Yacine Arioua, and Nouredine Benhamidouche. “Existence results for nonlinear Katugampola fractional differential equations with an integral condition.” Acta Math. Univ. Comenian. (N.S.) 89 (2020): 243-260. Cited on 44.10.7862/rf.2019.3
Search in Google Scholar Back to article
[4] Basti, Bilal, and Nouredine Benhamidouche. “Existence results of self-similar solutions to the Caputo-type’s space-fractional heat equation.” Surv. Math. Appl. 15 (2020): 153-168. Cited on 44, 45 and 46.
Search in Google Scholar Back to article
[5] Basti, Bilal, and Nouredine Benhamidouche. “Global existence and blow-up of generalized self-similar solutions to nonlinear degenerate dffusion equation not in divergence form.” Appl. Math. E-Notes 20 (2020): 367-387. Cited on 44.
Search in Google Scholar Back to article
[6] Buckwar, Evelyn, and Yurii F. Luchko. “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations.” J. Math. Anal. Appl. 227, no. 1 (1998): 81-97. Cited on 44.10.1006/jmaa.1998.6078
Search in Google Scholar Back to article
[7] Diethelm, Kai. The Analysis of Fractional Differential Equations, Vol. 2004 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2010. Cited on 43.10.1007/978-3-642-14574-2_8
Search in Google Scholar Back to article
[8] Granas, Andrzej and James Dugundji. Fixed Point Theory. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003. Cited on 46 and 49.10.1007/978-0-387-21593-8
Search in Google Scholar Back to article
[9] Kilbas, Anatoly A., and Hari M. Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Diffrential Equations. vol. 204 of North-Holland Mathematics Studies. Elsevier Science, 2006. Cited on 43, 44 and 45.
Search in Google Scholar Back to article
[10] Luchko, Yurii F., and Rudolf Gorenflo. “Scale-invariant solutions of a partial differential equation of fractional order.” Fract. Calc. Appl. Anal. 1, no. 1 (1998): 63-78. Cited on 44.
Search in Google Scholar Back to article
[11] Luchko, Yurii F., et al. “Fractional models, non-locality, and complex systems.” Comput. Math. Appl. 59, no. 3 (2010): 1048-1056. Cited on 44.10.1016/j.camwa.2009.05.018
Search in Google Scholar Back to article
[12] Metzler, Ralf, and Theo F. Nonnemacher. “Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation.” Chem. Phys. 284, no. 1-2 (2002): 67-90. Cited on 44.10.1016/S0301-0104(02)00537-2
Search in Google Scholar Back to article
[13] Miller, Kenneth S., and Bertram Ross. An Introduction to the Fractional Calculus and Differential Equations. A Wiley-Interscience Publication. New York– Chichester–Brisbane–Singapore: John Wiley & Sons Inc., 1993. Cited on 44.
Search in Google Scholar Back to article
[14] Pierantozzi, Teresa, and Luis Vázquez Martínez. “An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like.” J. Math. Phys. 46, no. 11 (2005): Art no. 113512. Cited on 44.10.1063/1.2121167
Search in Google Scholar Back to article
[15] Podlubny, Igor. Fractional Differential Equations. Vol. 198 of Mathematics in Science and Engineering. New York: Academic Press, 1999. Cited on 43.
Search in Google Scholar Back to article
[16] Samko, Stefan Grigor’evich, and Anatoliĭ Aleksandrovich Kilbas, and Oleg Igorevich Marichev. Fractional Integral and Derivatives (Theory and Applications). Switzerland: Gordon and Breach, 1993. Cited on 43.
Search in Google Scholar Back to article
[17] Vázquez Martínez, Luis, and Juan J. Trujillo, and María Pilar Velasco. “Fractional heat equation and the second law of thermodynamics.” Fract. Calc. Appl. Anal. 14, no. 3 (2011): 334-342. Cited on 44.10.2478/s13540-011-0021-9
Search in Google Scholar Back to article

Global existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditions

Abstract

Paradigm

My account