On the existence of solutions to an inhomogeneous pantograph type equation with singular coefficients


We establish the existence of solutions to an initial boundary value problem that involves a certain class of nonhomogeneous functional partial differential equations of the pantograph type with singular time dependent coefficients. The problem is motivated from a cell division equation. The techniques for solving such problems are rare due to the nonlocal term and the singularity in coefficients.

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  1. 1.

    Arlotti, L., Banasiak, J.: Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss. J. Math. Anal. Appl. 293(2), 693–720 (2004)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Baccelli, F., McDonald, D.R., Reynier, J.: A mean field model for multiple TCP connections through a buffer implementing RED. Perform. Eval. 11, 77–97 (2002)

    Article  Google Scholar 

  3. 3.

    Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: A generalization of the Paley-Wiener theorem for Mellin transforms and metric characterization of function spaces. Fract. Calc. Appl. Anal. 20(5), 1216–1238 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Butzer, P.L., Stefan, J.: A self-contained approach to mellin transform analysis for square integrable functions; applications. Integral Transforms Spec. Funct. 8(3–4), 175–198 (1999)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Efendiev, M.A., van-Brunt, B., Wake, G.C., Zaidi, A.A.: A functional partial differential equation arising in a cell growth model with dispersion. Math. Methods Appl. Sci. 41(4), 1541–1553 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bizhanova, G.I.: A solution to the Cauchy problem for parabolic equation with singular coefficients. J. Math. Sci. 244(6), 946–958 (2020)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Efendiev, M., van-Brunt, B., Zaidi, A.A., Shah, T.H.: Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev. Math. Methods Appl. Sci. 41(17), 8059–8069 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Escobedo, M., Laurencot, P., Mischler, S., Perthame, B.: Gelation and mass conservation in coagulation-fragmentation models. J. Differ. Equ. 195(1), 143–174 (2003)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gul, S.: Functional differential equations arising in the study of a cell growth model. PhD Thesis (2019)

  10. 10.

    Hall, A.J., Wake, G.C.: A functional differential equation arising in modelling of cell growth. J. Aust. Math. Soc. Ser. B. 30, 424–435 (1989)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hall, A.J., Wake, G.C.: A functional differential equation determining steady size distributions for populations of cells growing exponentially. J. Aust. Math. Soc. Ser. B. 31, 344–353 (1990)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lo, C.F.: Exact solution of the functional Fokker-Planck equation for cell growth with asymmetric cell division. Phys. A 533, 122079 (2019). https://doi.org/10.1016/j.physa.2019.122079

    MathSciNet  Article  Google Scholar 

  13. 13.

    McGrady, E.D., Ziff, R.M.: Shattering transition in fragmentation. Phys. Rev. Lett. 58(9), 892–895 (1987)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Paley, R.E., Wiener, N.: Fourier transforms in the complex domain. Am. Math. Soc. 19 10–178 (1934)

  15. 15.

    Perthame, B., Ryzhik, L.: Exponential decay for the fragmentation or cell-division equation. J. Differ. Equ. 210, 155–177 (2005)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Sinko, J.W., Streifer, W.: A new model for age-size structure of a population. Ecology 48(6), 910–918 (1967)

    Article  Google Scholar 

  17. 17.

    Sinko, J.W., Streifer, W.: A model for populations reproducing by fission. Ecology 52, 330–335 (1971)

    Article  Google Scholar 

  18. 18.

    Titchmarsh, E.C.: Introduction to the theory of Fourier integrals. Clarendon Press, Oxford (1948)

    Google Scholar 

  19. 19.

    van Brunt B, B., Almalki, A., Lynch, T., Zaidi, A.: On a cell division equation with a linear growth rate. ANZIAM J 59, 293–312 (2018)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Wake, G.C., Cooper, S., Kim, H.K., van-Brunt, B.: Functional differential equations for cell-growth models with dispersion. Comm. Appl. Anal. 4, 561–574 (2000)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Zaidi, A.A., van-Brunt, B., Wake, G.C.: A model for asymmetrical cell division. Math. Biosci. Eng. 12(3), 491–501 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Zaidi, A.A., van-Brunt, B.: Probability density function solutions to a Bessel type pantograph equation. Appl. Anal. 95(11), 2565–2577 (2016)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Zaidi, A.A., van Brunt, B., Wake, G.C.: Solutions to an advanced functional partial differential equation of the pantograph type. Proc. R. Soc. A 471, 20140947 (2015)

    MathSciNet  Article  Google Scholar 

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We thank the referees for their valuable suggestions that led to improvements in the paper.


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Shah, S.T.H., Zaidi, A.A. On the existence of solutions to an inhomogeneous pantograph type equation with singular coefficients. J Elliptic Parabol Equ 6, 935–945 (2020). https://doi.org/10.1007/s41808-020-00089-3

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  • Functional partial differential equations
  • Parabolic partial differential equations
  • Partial differential equations with singular coefficients

Mathematics Subject Classification

  • 35K20
  • 34K06
  • 35K67