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.
Citation: Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients[J]. Electronic Research Archive, 2021, 29(6): 3581-3607. doi: 10.3934/era.2021052
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.
[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). |
[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. |
[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. doi: 10.1186/s13662-020-03076-z |
[4] | (2001) Fixed Point Theory and Applications. Cambridge: Cambridge University Press. |
[5] | Initial boundary value problems for a fractional differential equation with hyper-Bessel operator. Fract. Calc. Appl. Anal. (2018) 21: 200-219. |
[6] | Brownian-time processes: The PDE connection and the half-derivative generator. Ann. Probab. (2001) 29: 1780-1795. |
[7] | Well-posedness results for a class of semi-linear super-diffusive equations. Nonlinear Anal. (2019) 181: 24-61. |
[8] | 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. doi: 10.1007/s00033-020-01348-y |
[9] | $\psi$-Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory. Results in Nonlinear Analysis (2020) 3: 167-178. |
[10] | On an operational calculus for a differential operator. C.R. Acad. Bulg. Sci. (1968) 21: 513-516. |
[11] | Operational calculus for a class of differential operators. C. R. Acad. Bulgare Sci. (1966) 19: 1111-1114. |
[12] | Fractional relaxation with time-varying coefficient. Fract. Calc. Appl. Anal. (2014) 17: 424-439. |
[13] | Fractional diffusion equation and relaxation in complex viscoelastic materials. Phys. A, Stat. Mech. Appl. (1992) 191: 449-453. |
[14] | R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. doi: 10.1007/978-3-662-43930-2 |
[15] | Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. (1999) 2: 383-414. |
[16] | Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resources Res. (1998) 34: 1027-1033. |
[17] | D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840. Springer, 1981. |
[18] | R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747 |
[19] | From the hyper-Bessel operators of Dimovski to the generalized fractional calculus. Fract. Calc. Appl. Anal. (2014) 17: 977-1000. |
[20] | Explicit solutions to hyper-Bessel integral equations of second kind. Comput. Math. Appl. (1999) 37: 75-86. |
[21] | On relating two approaches to fractional calculus. J. Math. Anal. Appl. (1988) 132: 590-610. |
[22] | Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J. Differential Equations (2020) 269: 4914-4959. |
[23] | Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632. |
[24] | The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electron. Res. Arch. (2020) 28: 263-289. |
[25] | Fractional Brownian motions, fractional noises and applications. SIAM Rev. (1968) 10: 422-437. |
[26] | A theory of fractional integration for generalized functions. SIAM J. Math. Anal. (1975) 6: 583-599. |
[27] | (2000) Strongly Elliptic Systems and Boundary Integral Equations. Cambridge: Cambridge University Press. |
[28] | Non-Markovian diffusion equations and processes: Analysis and simulations. Phys. A (2008) 387: 5033-5064. |
[29] | Fractional diffusion equations and processes with randomly varying time. Ann. Probab. (2009) 37: 206-249. |
[30] | 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. |
[31] | 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, 332Ã¢â‚¬â€œ348. |
[32] | The well-posedness and regularity of a rotating blades equation. Electron. Res. Arch. (2020) 28: 691-719. |
[33] | Continuity of solutions of a class of fractional equations. Potential Anal. (2018) 49: 423-478. |
[34] | On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator. Math. Methods Appl. Sci. (2020) 43: 2858-2882. |
[35] | 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. doi: 10.1007/s11784-020-00813-5 |
[36] | Semilinear Caputo time-fractional pseudo-parabolic equations. Comm. Pure Appl. Anal. (2021) 20: 583-621. |
[37] | 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. doi: 10.1016/j.jmaa.2020.124481 |
[38] | Weakly singular Gronwall inequalities and applications to fractional differential equations. J. Math. Anal. Appl. (2019) 471: 692-711. |
[39] | Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Functional Analysis (2013) 264: 2732-2763. |
[40] | Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy. Appl. Math. Lett. (2018) 83: 176-181. |
[41] | 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. doi: 10.1155/2013/769724 |
[42] | 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. doi: 10.1155/2018/3931297 |
[43] | The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane. Math. Methods Appl. Sci. (2018) 41: 2429-2441. |
[44] | Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069 |