# American Institute of Mathematical Sciences

December  2021, 29(6): 3581-3607. doi: 10.3934/era.2021052

## Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients

 1 Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City, Vietnam 2 Faculty of Natural Sciences, Duy Tan University, Da Nang, Vietnam 3 Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India 4 Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

* Corresponding author: Anh Tuan Nguyen, email: nguyenanhtuan@tdmu.edu.vn

Received  March 2021 Revised  May 2021 Published  December 2021 Early access  July 2021

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. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052
##### 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). Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.  Google Scholar [5] F. Al-Musalhi, N. Al-Salti and E. 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Benchohrab, $\psi$-Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory, Results in Nonlinear Analysis, 3 (2020), 167-178.   Google Scholar [10] I. Dimovski, On an operational calculus for a differential operator, C.R. Acad. Bulg. Sci., 21 (1968), 513-516.   Google Scholar [11] I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulgare Sci., 19 (1966), 1111-1114.   Google Scholar [12] R. Garra, A. Giusti, F. Mainardi and G. Pagnini, Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17 (2014), 424-439.  doi: 10.2478/s13540-014-0178-0.  Google Scholar [13] M. Ginoa, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A, Stat. Mech. Appl., 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar [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.  Google Scholar [15] R. Gorenflo, Y. Luchko and F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2 (1999), 383-414.   Google Scholar [16] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Res., 34 (1998), 1027-1033.  doi: 10.1029/98WR00214.  Google Scholar [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840. Springer, 1981.  Google Scholar [18] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar [19] V. Kiryakova, From the hyper-Bessel operators of Dimovski to the generalized fractional calculus, Fract. Calc. Appl. Anal., 17 (2014), 977-1000.  doi: 10.2478/s13540-014-0210-4.  Google Scholar [20] V. Kiryakova and B. Al-Saqabi, Explicit solutions to hyper-Bessel integral equations of second kind, Comput. Math. Appl., 37 (1999), 75-86.  doi: 10.1016/S0898-1221(98)00243-0.  Google Scholar [21] W. Lamb and A. C. McBride, On relating two approaches to fractional calculus, J. Math. Anal. Appl., 132 (1988), 590-610.  doi: 10.1016/0022-247X(88)90086-8.  Google Scholar [22] W. Lian, J. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.  Google Scholar [23] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [24] 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 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar [25] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar [26] A. C. McBride, A theory of fractional integration for generalized functions, SIAM J. Math. Anal., 6 (1975), 583-599.  doi: 10.1137/0506052.  Google Scholar [27] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar [28] A. Mura, M. S. Taqqu and F. Mainardi, Non-Markovian diffusion equations and processes: Analysis and simulations, Phys. A, 387 (2008), 5033-5064.  doi: 10.1016/j.physa.2008.04.035.  Google Scholar [29] E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206-249.  doi: 10.1214/08-AOP401.  Google Scholar [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.  Google Scholar [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. Google Scholar [32] L. Shen, S. Wang and Y. Wang, The well-posedness and regularity of a rotating blades equation, Electron. Res. Arch., 28 (2020), 691-719.  doi: 10.3934/era.2020036.  Google Scholar [33] D. D. Trong, E. Nane, D. M. Nguyen and N. H. Tuan, Continuity of solutions of a class of fractional equations, Potential Anal., 49 (2018), 423-478.  doi: 10.1007/s11118-017-9663-5.  Google Scholar [34] N. H. Tuan, L. N. Huynh, D. Baleanu and 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 (2020), 2858-2882.  doi: 10.1002/mma.6087.  Google Scholar [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.  Google Scholar [36] N. H. Tuan, V. V. Au and R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Comm. Pure Appl. Anal., 20 (2021), 583-621.  doi: 10.3934/cpaa.2020282.  Google Scholar [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.  Google Scholar [38] J. R. L. Webb, Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl., 471 (2019), 692-711.  doi: 10.1016/j.jmaa.2018.11.004.  Google Scholar [39] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Functional Analysis, 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar [40] R. Xu, X. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] K. Zhang, The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane, Math. Methods Appl. Sci., 41 (2018), 2429-2441.  doi: 10.1002/mma.4750.  Google Scholar [44] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar

show all references

##### 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). Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.  Google Scholar [5] F. Al-Musalhi, N. Al-Salti and E. Karimov, Initial boundary value problems for a fractional differential equation with hyper-Bessel operator, Fract. Calc. Appl. Anal., 21 (2018), 200-219.  doi: 10.1515/fca-2018-0013.  Google Scholar [6] H. Allouba and W. Zheng, Brownian-time processes: The PDE connection and the half-derivative generator, Ann. Probab., 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772.  Google Scholar [7] E. Alvarez, C. G. Gal, V. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016.  Google Scholar [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.  Google Scholar [9] Z. Baitichea, C. Derbazia and M. Benchohrab, $\psi$-Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory, Results in Nonlinear Analysis, 3 (2020), 167-178.   Google Scholar [10] I. Dimovski, On an operational calculus for a differential operator, C.R. Acad. Bulg. Sci., 21 (1968), 513-516.   Google Scholar [11] I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulgare Sci., 19 (1966), 1111-1114.   Google Scholar [12] R. Garra, A. Giusti, F. Mainardi and G. Pagnini, Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17 (2014), 424-439.  doi: 10.2478/s13540-014-0178-0.  Google Scholar [13] M. Ginoa, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A, Stat. Mech. Appl., 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar [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.  Google Scholar [15] R. Gorenflo, Y. Luchko and F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2 (1999), 383-414.   Google Scholar [16] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Res., 34 (1998), 1027-1033.  doi: 10.1029/98WR00214.  Google Scholar [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840. Springer, 1981.  Google Scholar [18] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar [19] V. Kiryakova, From the hyper-Bessel operators of Dimovski to the generalized fractional calculus, Fract. Calc. Appl. Anal., 17 (2014), 977-1000.  doi: 10.2478/s13540-014-0210-4.  Google Scholar [20] V. Kiryakova and B. Al-Saqabi, Explicit solutions to hyper-Bessel integral equations of second kind, Comput. Math. Appl., 37 (1999), 75-86.  doi: 10.1016/S0898-1221(98)00243-0.  Google Scholar [21] W. Lamb and A. C. McBride, On relating two approaches to fractional calculus, J. Math. Anal. Appl., 132 (1988), 590-610.  doi: 10.1016/0022-247X(88)90086-8.  Google Scholar [22] W. Lian, J. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.  Google Scholar [23] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [24] 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 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar [25] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar [26] A. C. McBride, A theory of fractional integration for generalized functions, SIAM J. Math. Anal., 6 (1975), 583-599.  doi: 10.1137/0506052.  Google Scholar [27] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar [28] A. Mura, M. S. Taqqu and F. Mainardi, Non-Markovian diffusion equations and processes: Analysis and simulations, Phys. A, 387 (2008), 5033-5064.  doi: 10.1016/j.physa.2008.04.035.  Google Scholar [29] E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206-249.  doi: 10.1214/08-AOP401.  Google Scholar [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.  Google Scholar [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. Google Scholar [32] L. Shen, S. Wang and Y. Wang, The well-posedness and regularity of a rotating blades equation, Electron. Res. Arch., 28 (2020), 691-719.  doi: 10.3934/era.2020036.  Google Scholar [33] D. D. Trong, E. Nane, D. M. Nguyen and N. H. Tuan, Continuity of solutions of a class of fractional equations, Potential Anal., 49 (2018), 423-478.  doi: 10.1007/s11118-017-9663-5.  Google Scholar [34] N. H. Tuan, L. N. Huynh, D. Baleanu and 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 (2020), 2858-2882.  doi: 10.1002/mma.6087.  Google Scholar [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.  Google Scholar [36] N. H. Tuan, V. V. Au and R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Comm. Pure Appl. Anal., 20 (2021), 583-621.  doi: 10.3934/cpaa.2020282.  Google Scholar [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.  Google Scholar [38] J. R. L. Webb, Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl., 471 (2019), 692-711.  doi: 10.1016/j.jmaa.2018.11.004.  Google Scholar [39] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Functional Analysis, 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar [40] R. Xu, X. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] K. Zhang, The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane, Math. Methods Appl. Sci., 41 (2018), 2429-2441.  doi: 10.1002/mma.4750.  Google Scholar [44] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar
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