doi: 10.3934/era.2021052
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, 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. AgarwalM. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.  Google Scholar
[5]

F. Al-MusalhiN. 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. AlvarezC. G. GalV. 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. BaiticheaC. 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. GarraA. GiustiF. 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. GinoaS. 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. GorenfloY. 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. LianJ. 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. MuraM. 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, 332–348. Google Scholar

[32]

L. ShenS. 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. TrongE. NaneD. 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. TuanL. N. HuynhD. 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. TuanV. 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. XuX. 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. AgarwalM. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.  Google Scholar
[5]

F. Al-MusalhiN. 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. AlvarezC. G. GalV. 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. BaiticheaC. 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. GarraA. GiustiF. 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. GinoaS. 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. GorenfloY. 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. LianJ. 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. MuraM. 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, 332–348. Google Scholar

[32]

L. ShenS. 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. TrongE. NaneD. 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. TuanL. N. HuynhD. 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. TuanV. 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. XuX. 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

[1]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[2]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

[3]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[4]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[5]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[6]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[7]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[8]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605

[9]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[10]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203

[11]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025

[12]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

[13]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

[14]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112

[15]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[16]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[17]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227

[18]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[19]

Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013

[20]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

2020 Impact Factor: 1.833

Metrics

  • PDF downloads (214)
  • HTML views (143)
  • Cited by (0)

Other articles
by authors

[Back to Top]