Using Leray-Schauder topological degree to solve a linear diffusion parabolic equation with periodic initial conditions

Document Type : Research Paper


1 Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, 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

4 Department of Mathematics, College of Sciences, Kirkuk University, Kirkuk, Iraq


Throughout this manuscript, we show time periodic solutions to a linear diffusion parabolic equation with Diriclet condition. Based on the topological degree theorem, we prove a time periodic solutions of the system such that we found the fixed point when the domain of the solution is sufficiently small.


[1] N. Alaa and M. Iguernane, Weak periodic solutions of some quasilinear parabolic equations with data measures,
J. Inequal. Pure Appl. Math. 3(3) (2002) Article 46.
[2] R.A. Hameed, J. Sun and B. Wu, Existence of periodic solutions of a p-Laplacian-Neumann problem, Boundary
Value Prob. 171 (2013) 1–11.
[3] R.A. Hameed, B. Wu and J. Sun, Periodic solution of a quasilinear parabolic equation with nonlocal terms and
Neumann boundary conditions”, Boundary Value Prob. 34(2013) 1–11.
[4] R.A. Hameed, M.A. Rasheed, H.S. Mustafa and F.N. Ghaffoori, The existence of periodic solutions to doubly
degenerate Allen-Cahn equation with Neumann boundary condition, Int. J. Nonlinear Anal. Appl. 13(1) (2022)
[5] L.G. Khoma and N.G. Khoma, Generalized periodic solutions of quasilinear equations, Ukr. Math. J. 48(3) (1996)
[6] G.M. Lieberman, Time-periodic solutions of quasilinear parabolic differential equations: I Dirichlet boundary
conditions, J. Math. Anal. Appl. 264(2) (2001) 617–638.
[7] Z.H. Liu, Periodic solutions for double degenerate quasilinear parabolic equations, Nonlinear Anal. 51(7) (2002)
[8] B.P. Liu and C.V. Pao, Periodic solutions of coupled semilinear parabolic boundary value problems, Nonlinear
Anal. 6(3) (1982) 237–252.
[9] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type,
Translation of Mathematical Monographs, vol. 23. Amer. Math. Soc. Providence, 1968.
[10] I.I. Smulev, Periodic solutions of the first boundary value problem for parabolic equations, Amer. Math. Soc.
Transl. Set 79(2) (1969) 215–229.
[11] D.G. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, 2001.
Volume 13, Issue 1
March 2022
Pages 1629-1635
  • Receive Date: 16 September 2021
  • Revise Date: 10 October 2021
  • Accept Date: 29 November 2021
  • First Publish Date: 29 November 2021