The existence of periodic solutions to doubly degenerate Allen-Cahn equation with Neumann boundary condition

Document Type : Research Paper


1 Department of Mathematics, College of education for pure science, Tikrit University, Iraq

2 Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq

3 Department of Mathematics, College of Education, Al- Hamdaniya University, Mosul, Iraq


This work is concerned with the periodic solution of a doubly degenerate Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Firstly, we define a new associated auxiliary problem. Secondly, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, where this limit point represents a nontrivial nonnegative time-periodic solution of the main studied problem. It is observed that the topological degree theorem technique plays an important role in proving the desired results. Furthermore, this technique can be applied to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.


[1] W. Allegretto and P. Nistri, Existence and optimal control for periodic parabolic equations with nonlocal term,
IMA J. Math. Control Inform., 16(1) (1999), 43-58, DOI: 10.1093/imamci/16.1.43.
[2] A. Calsina and, C. Perello, Equations for biological evolution, Royal Soc. Edin. 125(5) (1995) 939–958.
[3] E. Dibenedetto, Degenerate Parabolic Equations, Springer, New York, 1993.
[4] R.A. Hameed, J. Sun and B. Wu, Existence of periodic solutions of a p-Laplacian-Neumann problem, Bound.
Value Prob. 171 (2013) 1–11.
[5] R.A. Hameed, B. Wu and J. Sun, Periodic solution of a quasilinear parabolic equation with nonlocal terms and
Neumann boundary conditions, Bound. Value Prob. 34 (2013) 1–11.[6] R.A. Hameed and W.M. Taha, Periodic solution for a class of doubly degenerate parabolic equation with Neumann
problem, J. Uiv. Anbar Pure Sci. 9(3),(2015) 35–43.
[7] R. Huang, Y. Wang and Y. Ke, Existence of the non-trivial nonnegative periodic solutions for a class of degenerate
parabolic equations with nonlocal terms, Discrete Contin. Dyn. Syst. 5(4) (2005) 1005–1014.
[8] Y. Ke, R. Huang and J. Sun, Periodic solutions for a degenerate parabolic equation, Appl. Math. Lett. 22(6)
(2009) 910–915.
[9] O. Ladyzenskaja, V. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Amer. Math. Soc. 1968.
[10] M.A. Rasheed, On blow-up solutions of a parabolic system coupled in both equations and boundary conditions,
Baghdad Sci. J. 18(2) (2021) 315–321.
[11] M.A. Rasheed, R.A. Hameed, S.K. Obeid and A.F. Jameel, On numerical blow-up solutions of semilinear heat
equations, Iraqi J. of Sci. 61(8) (2020) 2077–2086.
[12] H. Rizqan and D. Dhaigude, Nonlinear fractional differential equations with advanced arguments, Int. J. Nonlinear
Anal. Appl. 12(2) (2021) 1413–1423.
[13] Y. Wang and J. Yin, Periodic solutions for a class of degenerate parabolic equations with Neumann boundary
conditions, Nonlinear Anal. Real World Appl. 12(4) (2011) 2069–2076.
[14] Z.Q. Wu, J.N. Zhao, J.X. Yin and H.L. Li, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001.
[15] L. Zenkoufi, Existence of a positive solution for a boundary value problem of some nonlinear fractional differential
equation, Int. J. Nonlinear Anal. Appl. 11(2) (2020) 499–514.
[16] Q. Zhou, Y.Y. Ke, Y.F. Wang and J.X. Yin, Periodic p-Laplacian with nonlocal terms, Nonlinear Anal. 66(2)
(2007) 442–453.
Volume 13, Issue 1
March 2022
Pages 397-408
  • Receive Date: 10 August 2021
  • Revise Date: 03 September 2021
  • Accept Date: 26 September 2021