Received: 13th June 2022; Revised: 17th June 2022, 26th June 2022, 1st October 2022; Accepted: 13th October 2022


  • H T Liu Ph.D., Professor, Department of Information Management, Tatung University, Taipei, Taiwan, China




Green function, Heat operator, Fractional Diffusion Equations, Sequential Concentrated Source, Quenching, Quenching Points


This article studies the diffusion problems with a concentrated source which is provided at a sequential time steps in 1 dimensional space.  The problems are considered for both Gaussian and fractional diffusion operators.  For the fractional diffusion case, Riemann-Liouville operator with fractional order is used to describe the model with diffusion rate slower than normal time scale, which is known as subdiffusive problems.  Due to this subdiffusive property, the existence and nonexistence behavior of the solution will be studied. Since the forcing term will experience a concentrated source at a sequence of time steps, the frequency, the time difference and strength of the source may affect the growth rate of the solution.   Criteria for these effects which may cause for the quenching behavior of the solution will be given. The existence of the solution is investigated. The monotone behavior in spatial will be given. The quenching behavior of the solution will be studied.  The location of the quenching set will be discussed.


A. A. Greenenko, A. V. Chechkin and N. F. Shulga (2004), Anomalous diffusion and Levy flights in channelling, Physics Letters A 324, 82-85. https://doi.org/10.1016/j.physleta.2004.02.053

B. M. Schula and M. Schulz (2006), Numerical investigations of anomalous diffusion effects in glasses, Journal of Non-Crystalline Solids 352, 4884-4887. https://doi.org/10.1016/j.jnoncrysol.2006.04.027

C. Y. Chan (1993), Computation of the critical domain for quenching in an elliptic plate, Neural Parallel Sci. Comput. 1 153-162.

C. Y. Chan (2011), A quenching criterion for a multi-dimensional parabolic problem due to a concentrated nonlinear source, J. of Comp. and Appl. Math., 3724-3727. https://doi.org/10.1016/j.cam.2011.01.017

C. Y. Chan and C. S. Chen (1989), A numerical method for semilinear singular parabolic quenching problems, Quart. Appl. Math., 47, 45--57. https://doi.org/10.1090/qam/987894

C. Y. Chan and H. T. Liu (2016). A maximum principle for fractional diffusion differential equations, Quart. Appl. Math., 74, 421-427. https://doi.org/10.1090/qam/1433

C. Y. Chan and H. T. Liu (2018). Existence of solution for the problem with a concentrated source in a subdiffusive medium, Journal of Integral Equations and Applications, 30, No.1 41-65. https://doi.org/10.1216/JIE-2018-30-1-41

C. Y. Chan and M. K. Kwong (1989), Existence results of steady-states of semilinear reaction-diffusion equations and their applications, J. Differential Equations, 77, 304-321. https://doi.org/10.1016/0022-0396(89)90146-0

C. Y. Chan and P. C. Kong (1995), A thermal explosion model, Appl. Math. Comput., 71, 201-210. https://doi.org/10.1016/0096-3003(94)00154-V

C. Y. Chan and P. Tragoonsirisak (2008), A multi-dimensional quenching problem due to a concentrated nonlinear source in R^N, Nonlinear Anal., 69, 1494-1514. https://doi.org/10.1016/j.na.2007.07.001

H. J. Haubold, A. M. Mathai and R. K. Saxena (2011), Mittag-Leffler functions and their applications, J. Appl. Math., Art. ID 298628, 1-51. https://doi.org/10.1155/2011/298628

H. Kawarada (1975), On solution of initial-boundary problem for $u_t=u_{xx} +1/(1-u) $, Publ. Res. Inst. Math. Sci., Kyoto Univ., 10, 729-736. https://doi.org/10.2977/prims/1195191889

H. T. Liu (2016), Strong maximum principle for fractional diffusion differential equation, Dynam. Systems and Appl., 26, 365-376.

H. T. Liu and Chien-Wei Chang (2016), Impulsive Effects on the Existence of Solution for a Fractional Diffusion Equation, Dynamic Systems and Applications, 25, 493-500.

H. T. Liu and Wei-Cheng Huang (2018), Existence of Solution for the Problem with a Concentrated Source in a Subdiffusive Medium, 6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017), AIP Conference Proceedings, 1926:020027.

H.T. Liu (2019), Blow-up behavior of the Solution for the problem in a subdiffusive mediums, Math. Meth. Appl. Sci., 42, No.16, 5383-5389. https://doi.org/10.1002/mma.5393

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

J. Trujillo, Fractional models: Sub and super-diffusive, and undifferentiable solutions, in Innovation in Engineering Computational Technology, Sax-Coburg Publ., Striling, Scotland, 2006.

M. M. Meerschaert and C. Tadjeran (2004), Finite difference approximations of fractional advection-dispersion flow equations, Journal of Comp. Appl. Math., 172 65-77. https://doi.org/10.1016/j.cam.2004.01.033

M. M. Wyss and W. Wyss (2001), Evolution, its fractional extension and generalization, Fract. Calc. Appl. Anal., 4, 273-284.

R. Metzler and J. Klafter (2000), The random walk's guide to anomalous diffusion, A fractional dynamics approach, Phys. Rep., 339, 1-77. https://doi.org/10.1016/S0370-1573(00)00070-3

R. Metzler and J. Klafter (2004), The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37, 161-208. https://doi.org/10.1088/0305-4470/37/31/R01

V. V. Ah, J. M. Angulo and M. D. Ruiz-Medina (2005), Diffusion on multifactals, Nonlinear Analysis 63, e2043-e2056. https://doi.org/10.1016/j.na.2005.02.107

W. E. Olmstead and C. A. Roberts (2008), Thermal blow-up in a subdiffusive medium, SIAM J. Appl. Math., 69, 514-523. https://doi.org/10.1137/080714075

W. Y. Chan (2017), Determining the critical domain of quenching problems for coupled nonlinear parabolic differential equations, Proceedings of Dynamic Systems and Applications.

W. Y. Chan and H. T. Liu (2017), Finding the Critical Domain of Multi-Dimensional Quenching Problems, Neural, Parallel, and Scientific Computations, 25, 19-28.




How to Cite

Liu, H. T. (2022). QUENCHING BEHAVIOR OF THE SOLUTION FOR THE PROBLEMS WITH SEQUENTIAL CONCENTRATED SOURCES: Received: 13th June 2022; Revised: 17th June 2022, 26th June 2022, 1st October 2022; Accepted: 13th October 2022. MATTER: International Journal of Science and Technology, 8, 176–186. https://doi.org/10.20319/mijst.2022.8.176186