×

Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. (English) Zbl 1478.35218

Summary: The semi-linear problem of a fractional diffusion equation with the Caputo-like counterpart of a hyper-Bessel differential is considered. The results on existence, uniqueness and regularity estimates (local well-posedness) of the solutions are established in the case of linear source and the source functions that satisfy the globally Lipschitz conditions. Moreover, we prove that the problem exists a unique positive solution. In addition, the unique continuation of solutions and a finite-time blow-up are proposed with the reaction terms are logarithmic functions.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B44 Blow-up in context of PDEs
33E12 Mittag-Leffler functions and generalizations
44A20 Integral transforms of special functions
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] R. S. Adiguzel, U. Aksoy, E. Karapinar and I. M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods Appl. Sci, (2020). · Zbl 07388072
[2] H. Afshari, S. Kalantari and E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations, 2015 (2015), 12 pp. · Zbl 1328.47081
[3] H. Afshari and E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via \(\psi \)-Hilfer fractional derivative on \(b \)-metric spaces, Adv. Difference Equ., (2020), Paper No. 616, 11 pp.
[4] F. Al-Musalhi; N. Al-Salti; E. Karimov, Initial boundary value problems for a fractional differential equation with hyper-Bessel operator, Fract. Calc. Appl. Anal., 21, 200-219 (2018) · Zbl 1439.35515
[5] H. Allouba; W. Zheng, Brownian-time processes: The PDE connection and the half-derivative generator, Ann. Probab., 29, 1780-1795 (2001) · Zbl 1018.60066
[6] E. Alvarez; C. G. Gal; V. Keyantuo; M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181, 24-61 (2019) · Zbl 1411.35268
[7] B. de Andrade, V. Van Au, D. O’Regan and N. H. Tuan, Well-posedness results for a class of semilinear time fractional diffusion equations, Z. Angew. Math. Phys., 71 (2020), Paper No. 161, 24 pp.
[8] Z. Baitichea; C. Derbazia; M. Benchohrab, \( \psi \)-Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory, Results in Nonlinear Analysis, 3, 167-178 (2020)
[9] I. Dimovski, On an operational calculus for a differential operator, C.R. Acad. Bulg. Sci., 21, 513-516 (1968) · Zbl 0188.43002
[10] I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulgare Sci., 19, 1111-1114 (1966)
[11] R. Garra; A. Giusti; F. Mainardi; G. Pagnini, Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17, 424-439 (2014) · Zbl 1305.26018
[12] M. Ginoa; S. Cerbelli; H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A, Stat. Mech. Appl., 191, 449-453 (1992)
[13] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. · Zbl 1309.33001
[14] R. Gorenflo; Y. Luchko; F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2, 383-414 (1999) · Zbl 1027.33006
[15] Y. Hatano; N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Res., 34, 1027-1033 (1998)
[16] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840. Springer, 1981.
[17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. · Zbl 0998.26002
[18] V. Kiryakova, From the hyper-Bessel operators of Dimovski to the generalized fractional calculus, Fract. Calc. Appl. Anal., 17, 977-1000 (2014) · Zbl 1314.44003
[19] V. Kiryakova; B. Al-Saqabi, Explicit solutions to hyper-Bessel integral equations of second kind, Comput. Math. Appl., 37, 75-86 (1999) · Zbl 0936.45003
[20] W. Lamb; A. C. McBride, On relating two approaches to fractional calculus, J. Math. Anal. Appl., 132, 590-610 (1988) · Zbl 0651.26007
[21] W. Lian; J. Wang; R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269, 4914-4959 (2020) · Zbl 1448.35322
[22] W. Lian; R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9, 613-632 (2020) · Zbl 1421.35222
[23] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28, 263-289 (2020) · Zbl 1447.35060
[24] B. B. Mandelbrot; J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437 (1968) · Zbl 0179.47801
[25] A. C. McBride, A theory of fractional integration for generalized functions, SIAM J. Math. Anal., 6, 583-599 (1975) · Zbl 0302.46026
[26] A. Mura; M. S. Taqqu; F. Mainardi, Non-Markovian diffusion equations and processes: Analysis and simulations, Phys. A, 387, 5033-5064 (2008)
[27] E. Orsingher; L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37, 206-249 (2009) · Zbl 1173.60027
[28] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications, 198 1999, Elsevier, Amsterdam. · Zbl 0924.34008
[29] A. Salim, M. Benchohra, J. E. Lazreg and J. Henderson, Nonlinear implicit generalized Hilfer-Type fractional differential equations with non-instantaneous impulses in banach spaces, Adv. Theory Nonlinear Anal. Appl., 4, 332–348.
[30] L. Shen; S. Wang; Y. Wang, The well-posedness and regularity of a rotating blades equation, Electron. Res. Arch., 28, 691-719 (2020) · Zbl 1446.35200
[31] D. D. Trong; E. Nane; D. M. Nguyen; N. H. Tuan, Continuity of solutions of a class of fractional equations, Potential Anal., 49, 423-478 (2018) · Zbl 1407.35205
[32] N. H. Tuan; L. N. Huynh; D. Baleanu; N. H. Can, On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator, Math. Methods Appl. Sci., 43, 2858-2882 (2020) · Zbl 1447.35390
[33] N. H. Tuan, V. V. Au, V. V. Tri and D. O’Regan, On the well-posedness of a nonlinear pseudo-parabolic equation, J. Fix. Point Theory Appl., 22 (2020), Paper No. 77, 21 pp.
[34] N. H. Tuan; V. V. Au; R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Comm. Pure Appl. Anal., 20, 583-621 (2021) · Zbl 1460.35381
[35] N. H. Tuan, V. V. Au, R. Xu and R. Wang, On the initial and terminal value problem for a class of semilinear strongly material damped plate equations, J. Math. Anal. Appl., 492 (2020), 124481, 38 pp. · Zbl 1448.35339
[36] J. R. L. Webb, Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl., 471, 692-711 (2019) · Zbl 1404.26022
[37] R. Xu; J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Functional Analysis, 264, 2732-2763 (2013) · Zbl 1279.35065
[38] R. Xu; X. Wang; Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83, 176-181 (2018) · Zbl 06892576
[39] X.-J. Yang, D. Baleanu and J. A. Tenreiro Machado, Systems of Navier-Stokes equations on Cantor sets, Math. Probl. Eng., 2013 (2013), Art. ID 769724, 8 pp. · Zbl 1299.76047
[40] K. Zhang, Nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term, Adv. Math. Phys., 2018 (2018), Art. ID 3931297, 7 pp. · Zbl 1404.35067
[41] K. Zhang, The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane, Math. Methods Appl. Sci., 41, 2429-2441 (2018) · Zbl 1394.35288
[42] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.