Nonexistence of Global Positive Solutions of Weakly Coupled Systems of Semilinear Parabolic Equations with Time-Periodic Coefficients

Abstract

In the Cartesian product \((\mathbb {R}^n\setminus B)\times \mathbb {R} \), where \(B \) is a ball in \(\mathbb {R}^n \), \(n\geq 3\), we consider a system of two semilinear parabolic equations with bounded measurable time-periodic coefficients and with power nonlinearities. Exact conditions on the nonlinearity exponents under which this system does not have global positive time-periodic solutions are derived.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1

    Serrin, J. and Zou, H., Nonexistence of positive solutions of Lane–Emden system, Differ. Integr. Equat., 1996, vol. 9, pp. 635–653.

    MATH  Google Scholar 

  2. 2

    Gidas, B. and Spruck, J., Global and local behavior of positive solutions of linear elliptic equations, Comm. Pure Appl. Math., 1981, vol. 34, pp. 525–598.

    MathSciNet  MATH  Article  Google Scholar 

  3. 3

    Bidaut-Veron, M.F. and Pohozaev, S., Nonexistence results and estimates for some nonlinear elliptic problems, Anal. Math., 2001, vol. 84, pp. 1–49.

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

    Kon’kov, A.A., On global solutions of the radial \(p \)-Laplace equation, Nonlin. Anal. Theory Methods Appl., 2009, vol. 70, pp. 3437–3451.

    MathSciNet  MATH  Google Scholar 

  5. 5

    Kon’kov, A.A., On solutions of quasi-linear elliptic inequalities containing terms with lower-order derivatives, Nonlin. Anal., 2013, vol. 90, pp. 121–134.

    MathSciNet  MATH  Article  Google Scholar 

  6. 6

    Kondratiev, V., Liskevich, V., and Sobol, Z., Second-order semilinear elliptic inequalities in exterior domains, J. Differ. Equat., 2003, vol. 187, pp. 429–455.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7

    Levine, H.A., The role of critical exponents in blowup theorems, SIAM Rev., 1990, vol. 32, no. 2, pp. 262–288.

    MathSciNet  MATH  Article  Google Scholar 

  8. 8

    Bagirov, Sh.G., The absence of global solutions of a system of semilinear parabolic equations with a singular potential, Proc. Inst. Math. Mech. (PIMM) Natl. Acad. Sci. Azerb., 2017, vol. 43, no. 2, pp. 296–304.

    MathSciNet  MATH  Google Scholar 

  9. 9

    Fujita, H., On the blowing-up of solutions of the Cauchy problem for \(u_{t}=\Delta u+u^{1+\alpha }\), J. Fac. Sci. Univ. Tokyo. Sect. I , 1966, vol. 13, pp. 109–124.

    MathSciNet  MATH  Google Scholar 

  10. 10

    Hayakawa, K., On nonexistence of global solutions of some semi-linear parabolic equations, Proc. Jpn. Acad., 1973, vol. 49, pp. 503–505.

    MATH  Article  Google Scholar 

  11. 11

    Kobayashi, K., Siaro, T., and Tanaka, H., On the blowing up problem of semi linear heat equations, J. Math. Soc. Jpn., 1977, vol. 29, pp. 407–424.

    Article  Google Scholar 

  12. 12

    Pinsky, R.G., Existence and nonexistence of global solutions for \(u_{t}-\Delta u=a(x) u^{q}\) in \(\mathbb {R}^d \), J. Differ. Equat., 1997, vol. 133, pp. 152–177.

    MATH  Google Scholar 

  13. 13

    Deng, K. and Levine, H.A., The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 2000, vol. 243, no. 1, pp. 85–126.

    MathSciNet  MATH  Article  Google Scholar 

  14. 14

    Mitidieri, E. and Pohozaev, S.I., A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 2001, vol. 234, pp. 1–383.

    MATH  Google Scholar 

  15. 15

    Escobedo, M. and Herrero, M.A., Boundedness and blow up for a semilinear reaction–diffusion system, J. Differ. Equat., 1991, vol. 89, pp. 176–202.

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    Mochizuki, K. and Huang, Q., Existence and behavior of solutions for a weakly coupled system of reaction–diffusion equations, Methods Appl. Anal., 1998, vol. 5, pp. 109–124.

    MathSciNet  MATH  Google Scholar 

  17. 17

    Caristi, G., Existence and nonexistence of global solutions of degenerate and singular parabolic system, Abstr. Appl. Anal., 2000, vol. 5, no. 4, pp. 265–284.

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    Levine, H.A., A Fujita type global existence–global nonexistence theorem for a weakly coupled system of reaction–diffusion equations, Zeitschr. Ang. Math. Phys., 1992, vol. 42, pp. 408–430.

    MathSciNet  MATH  Article  Google Scholar 

  19. 19

    Fila, M., Levine, A., and Uda, Y.A., Fujita-type global existence–global nonexistence theorem for a system of reaction diffusion equations with differing diffusivities,Math. Methods Appl. Sci., 1994, vol. 17, pp. 807–835.

    MathSciNet  MATH  Article  Google Scholar 

  20. 20

    Uda, Y., The critical exponent for a weakly coupled system of the generalized Fujita type reaction–diffusion equations, Zeitschr. Ang. Math. Phys., 1995, vol. 46, no. 3, pp. 366–383.

    MathSciNet  MATH  Article  Google Scholar 

  21. 21

    Seidman, T.I., Periodic solutions of a non-linear parabolic equation,J. Differ. Equat., 1975, vol. 19, no. 2, pp. 242–257.

    MathSciNet  MATH  Article  Google Scholar 

  22. 22

    Beltramo, A. and Hess, P., On the principal eigenvalue of a periodic-parabolic operator, Commun. Partial Differ. Equat., 1984, vol. 9, no. 9, pp. 919–941.

    MathSciNet  MATH  Article  Google Scholar 

  23. 23

    Esteban, M.J., On periodic solutions of superlinear parabolic problems,Trans. Am. Math. Soc., 1986, vol. 293, no. 1, pp. 171–189.

    MathSciNet  MATH  Article  Google Scholar 

  24. 24

    Esteban, M.J., A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Am. Math. Soc., 1988, vol. 102, no. 1, pp. 131–136.

    MathSciNet  MATH  Article  Google Scholar 

  25. 25

    Quittner, P., Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, Nonlin. Differ. Equat. Appl., 2004, vol. 11, no. 2, pp. 237–258.

    MathSciNet  MATH  Google Scholar 

  26. 26

    Giga, Y. and Mizoguchi, N., On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type, J. Math. Anal. Appl., 1996, vol. 201, no. 2, pp. 396–416.

    MathSciNet  MATH  Article  Google Scholar 

  27. 27

    Hirano, N. and Mizoguchi, N., Positive unstable periodic solutions for superlinear parabolic equations, Proc. Am. Math. Soc., 1995, vol. 123, no. 5, pp. 1487–1495.

    MathSciNet  MATH  Article  Google Scholar 

  28. 28

    Huska, J., Periodic solutions in semilinear parabolic problems, Acta Math. Univ. Comenianae, 2002, vol. 71, no. 1, pp. 19–26.

    MathSciNet  MATH  Google Scholar 

  29. 29

    Bagyrov, Sh.G., Absence of positive solutions of a second-order semilinear parabolic equation with time-periodic coefficients, Differ. Equations, 2014, vol. 50, no. 4, pp. 548–553.

    MathSciNet  MATH  Article  Google Scholar 

  30. 30

    Bagyrov, Sh.G., On the existence of a positive solution of a nonlinear second-order parabolic equation with time-periodic coefficients, Differ. Equations, 2007, vol. 43, no. 4, pp. 581–585.

    MathSciNet  MATH  Article  Google Scholar 

  31. 31

    Bagyrov, Sh.G., On non-existence of positive periodic solution for second order semilinear parabolic equation, Azerb J. Math., 2018, vol. 8, no. 2, pp. 163–180.

    MathSciNet  MATH  Google Scholar 

  32. 32

    Mitidieri, E. and Pohozaev, S.I., Nonexistence of Positive Solutions for Quasilinear Elliptic Problems on \(\mathbb {R}^N\),Proc. Steklov Inst. Math., 1999, vol. 227, pp. 186–216.

    MATH  Google Scholar 

  33. 33

    Kurta, V.V., On the nonexistence of positive solutions to semilinear elliptic equations, Proc. Steklov. Inst. Math., 1999, vol. 227, pp. 155–162.

    MATH  Google Scholar 

  34. 34

    Aronson, D.G., Bounds for the fundamental solution of a parabolic equation,Bull. Am. Math. Soc., 1967, vol. 73, pp. 890–896.

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sh. G. Bagyrov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bagyrov, S.G. Nonexistence of Global Positive Solutions of Weakly Coupled Systems of Semilinear Parabolic Equations with Time-Periodic Coefficients. Diff Equat 56, 721–733 (2020). https://doi.org/10.1134/S0012266120060051

Download citation