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Boundedness and Almost Periodicity of Solutions for a Class of Semilinear Parabolic Equations with Boundary Degeneracy

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Abstract

In this paper the authors investigate the boundedness and almost periodicity of solutions of semilinear parabolic equations with boundary degeneracy. The equations may be weakly degenerate or strongly degenerate on the lateral boundary. The authors prove the existence, uniqueness and global exponential stability of bounded entire solutions, and also establish the existence theorem of almost periodic solutions if the data are almost periodic.

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References

  1. Amar, M. and Gianni, R., Almost-periodic solutions of elliptic and parabolic equations in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 132(6), 2002, 1275–1306.

    Article  MathSciNet  Google Scholar 

  2. Asanova, A. T., A bounded almost periodic solution of a semilinear parabolic equation, Izv. Minister. Nauki Akad. Nauk Resp. Kaz. Ser. Fiz.-Mat., 5, 1996, 17–22 (in Russian).

    MathSciNet  Google Scholar 

  3. Cai, J. J. and Lou, B. D., Convergence in a quasilinear parabolic equation with time almost periodic boundary conditions, Nonlinear Anal., 75, 2012, 6312–6324.

    Article  MathSciNet  Google Scholar 

  4. Cannarsa, P., Martinez, P. and Vancostenoble, J., Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal., 3(4), 2004, 607–635.

    Article  MathSciNet  Google Scholar 

  5. Cannarsa, P., Martinez, P. and Vancostenoble, J., Null controllability of degenerate heat equations, Adv. Differential Equations, 10(2), 2005, 153–190.

    MathSciNet  MATH  Google Scholar 

  6. Corduneanu, C., Almost Periodic Oscillations and Wave, Springer-Verlag, New York, 2009.

    Book  Google Scholar 

  7. Damlamian, A. and Kenmochi, N., Asymptotic behavior of solutions to a multiphase Stefan problem, Japan J. Appl. Math., 3, 1986, 15–36.

    Article  MathSciNet  Google Scholar 

  8. Damlamian, A. and Kenmochi, N., Periodicity and almost periodicity of solutions to a multi-phase Stefan problem in several space variables, Nonlinear Anal., 12(9), 1988, 921–934.

    Article  MathSciNet  Google Scholar 

  9. Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, 19, Amer. Math. Soc. Providence, RI, 1998.

    Google Scholar 

  10. Keldys, M. V., On certain cases of degeneration of equations of elliptic type on the boundary of a domain, Dokl. Akad. Nauk SSSR, 77, 1951, 181–183.

    Google Scholar 

  11. Kubo, M., Periodic and almost periodic stability of solutions to degenerate parabolic equations, Hiroshima Math. J., 19, 1989, 499–514.

    Article  MathSciNet  Google Scholar 

  12. Kufner, A., Weighted Sobolev Spaces, John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1985.

    MATH  Google Scholar 

  13. Levenshtam, V. B., On the unique solvability of parabolic equations with almost periodic coefficients in Holder spaces, Mat. Zametki, 73(6), 2003, 861–877 (in Russian).

    Article  Google Scholar 

  14. Levitan, B. M. and Zhikov, V. V., Almost Periodic Functions and Differential Equations, Combridge University Press, Combridge, 1982.

    MATH  Google Scholar 

  15. Nakao, M., On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19, 1975, 371–385.

    Article  MathSciNet  Google Scholar 

  16. Nakao, M. and Nanbu, T., On the existence of global, bounded, periodic and almost-periodic solutions of nonlinear parabolic equations, Math. Rep. Kyushu Univ., 10(2), 1975, 99–112.

    MathSciNet  MATH  Google Scholar 

  17. Oleinik, O. A. and Radkevic, E. V., Second-Order Equations with Nonnegative Characteristic Forms, Amer. Math. Soc., Pleum Press, New York, 1973.

    Book  Google Scholar 

  18. Rossi, L., Liouville type results for periodic and almost periodic linear operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(6), 2009, 2481–2502.

    Article  MathSciNet  Google Scholar 

  19. Shen, W. X. and Yi, Y. F., Asymptotic almost periodicity of scalar parabolic equations with almost periodic time dependence, J. Differential Equations, 122(2), 1995, 373–397.

    Article  MathSciNet  Google Scholar 

  20. Shen, W. X. and Yi, Y. F., Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (647), 1998.

  21. Vuillermot, P. A., Global exponential attractors for a class of almost-periodic parabolic equations in ℝN, Proc. Amer. Math. Soc., 116, 1992, 775–782.

    MathSciNet  MATH  Google Scholar 

  22. Wang, C. P., Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ., 10, 2010, 163–193.

    Article  MathSciNet  Google Scholar 

  23. Ward, J. R., Bounded and almost periodic solutions of semilinear parabolic equations, Rocky Mountain J. Math., 18, 1988, 479–494.

    Article  MathSciNet  Google Scholar 

  24. Yamazaki, N., Almost periodicity of solutions to free boundary problems, Dynamical Systems and Differential Equations (Kennesaw, GA, 2000), Discrete Contin. Dynam. Systems, Added Volume, 2001, 386–397.

  25. Yin, J. X. and Wang, C. P., Evolutionary weighted p-Laplacian with boundary degeneracy, J. Differential Equations, 237(2), 2007, 421–445.

    Article  MathSciNet  Google Scholar 

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Acknowledgment

The authors would like to express their thanks to the referee for his/her careful reading of the manuscript and insightful comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Yi Xie & Peidong Lei

Authors
  1. Yi Xie
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  2. Peidong Lei
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Corresponding author

Correspondence to Peidong Lei.

Additional information

This work was supported by the Guangdong Provincial Natural Science Foundation of China (No. S2013010015800) and the KLSCK of the Ministry of Education of China (No. 93K172012K03).

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Xie, Y., Lei, P. Boundedness and Almost Periodicity of Solutions for a Class of Semilinear Parabolic Equations with Boundary Degeneracy. Chin. Ann. Math. Ser. B 41, 303–324 (2020). https://doi.org/10.1007/s11401-020-0200-2

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  • DOI: https://doi.org/10.1007/s11401-020-0200-2

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