Analytical studies on the global existence and blow-up of solutions for a free boundary problem of two-dimensional diffusion equations of moving fractional order

Abstract

This paper particularly addresses and discusses some analytical studies on the existence and uniqueness of global or blow-up solutions under the traveling profile forms for a free boundary problem of two-dimensional diffusion equations of moving fractional order. It does so by applying the properties of Schauder's and Banach's fixed point theorems. For application purposes, some examples of explicit solutions are provided to demonstrate the usefulness of our main results.

Keywords

• Fractional diffusion
• moving fractional order
• free boundary
• blow-up
• global existence
• uniqueness

Kaynakça

1. Y. Arioua, B. Basti and N. Benhamidouche, Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative, Appl. Math. E-Notes, 19 (2019), 397--412.
2. B. Basti, Y. Arioua and N. Benhamidouche, Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations, J. of Math. and Applications, 42 (2019), 35--61.
3. B. Basti, Y. Arioua and N. Benhamidouche, Existence results for nonlinear Katugampola fractional differential equations with an integral condition, Acta Mathematica Universitatis Comenianae, 89 (2020), 243--260.
4. B. Basti and N. Benhamidouche, Existence results of self-similar solutions to the Caputo-type's space-fractional heat equation, Surveys in Mathematics and its Applications, 15 (2020), 153--168.
5. B. Basti and N. Benhamidouche, Global existence and blow-up of generalized self-similar solutions to nonlinear degenerate diffusion equation not in divergence form, Appl. Math. E-Notes, 20 (2020), 367--387.
6. N. Benhamidouche, Exact solutions to some nonlinear PDEs, travelling profiles method, Electronic Journal of Qualitative Theory of Differential Equation 15 (2008), 1--7.
7. [1] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.
8. [2] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.

9. [3] A.A. Kilbas, H.H. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006.
10. [4] K. Diethelm, The Analysis of Fractional Differential Equations, Springer Berlin, 2010.
11. [5] A.D. Polyanin, V.F. Zaitsev, Handbook of Nonlinear Partial Equation, Chapman&Hall/CRC, Boca Raton (2004).
12. [6] B. Basti and N. Benhamidouche, Existence results of self-similar solutions to the Caputo-type's space-fractional heat equation, Surv. Math. Appl., 15 (2020), 153-168.
13. [7] E. Buckwar and Y. Luchko, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. Math. Anal. Appl. 227(1) (1998), 81-97.
14. [8] Y. Luchko and R. Gorenfl, Scale-invariant solutions of a partial differential equation of fractional order, Fract. Calc. Appl. Anal. 1(1) (1998), 63-78.
15. [9] R. Metzler and T.F. Nonnemacher, Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chem. Phys. 284 (2002), 67-90.
16. [10] F. Nouioua and B. Basti, Global existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditions, Ann. Univ. Paedagog. Crac. Stud. Math. 20 (2020), 43-56.
17. [11] L. Vázquez, J.J. Trujillo and M.P. Velasco, Fractional heat equation and the second law of thermodynamics, Fract. Cal. Appl. Anal. 14 (2011), 334-342.
18. [12] Y. Arioua, B. Basti and N. Benhamidouche, Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative, Appl. Math. E-Notes, 19 (2019), 397-412.
19. [13] B. Basti, Y. Arioua and N. Benhamidouche, Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations, J. Math. Appl., 42 (2019), 35-61.
20. [14] B. Basti, Y. Arioua and N. Benhamidouche, Existence results for nonlinear Katugampola fractional differential equations with an integral condition, Acta Math. Univ. Comenian., 89(2) (2020), 243-260.
21. [15] B. Basti, N. Hammami, I. Berrabah, F. Nouioua, R. Djemiat, N. Benhamidouche, Stability analysis and existence of solutions for a modified SIRD model of COVID-19 with fractional derivatives, Symmetry, 13(8) (2021), 1431.
22. [16] B. Basti and N. Benhamidouche, Global existence and blow-up of generalized self-similar solutions to nonlinear degenerate diffusion equation not in divergence form, Appl. Math. E-Notes, 20 (2020), 367-387.
23. [17] N. Benhamidouche, Exact solutions to some nonlinear PDEs, travelling profiles method, Electron. J. Qual. Theory Differ. Equ. 15 (2008), 1-7.
24. [18] E.M.E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G'/G)-expansion method, J. of appl. math. & informatics 28 (2010), 383-395.
25. [19] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.