S-asymptotically ω periodic solutions of heat equation

Authors

  • Abdelkader Bouadi
  • Souhila Boudjema

DOI:

https://doi.org/10.54021/seesv5n1-107

Keywords:

S-asymptotically ω-periodic functions, heat equation, implicit function theorem, dependence results

Abstract

We mean by an S-asymptotically ω-periodic  function any continuous and bounded function from the real axis to a Banach space that converges to a periodic function as t tends to infinity. We study in this paper the existence, uniqueness and differentiable dependence on the S-asymptotically ω-periodic mild solution of a heat equation on the space of continuous functions from a non-empty n-dimensional  bounded domain with a Lipschitz boundary to the real axis. The dependence of the solution concerns the initial conditions, more precisely when the initial conditions is a S-asymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of heat equation. To show our main result in this work we introduce the properties of the superposition operator, or also called Nemytskii operator, in the space of S-asymptotically ω-periodic functions. The notion of derivation for the last operator will also be highlighted. We also use in this paper semi-groups which have become important tools for differential equations. In this study, our focus shifted from seeking an s-asymptotically w-periodic solution for our heat equation, which is a problem in dynamic systems, to one in functional analysis. More precisely, our strategy consists of applying the implicit function theorem on a certain operator that we constructed on our workspaces in order to achieve the objective described in our main theorem. In fact our theorem gives conditions to ensure that around a mild S-asymptotically ω-periodic solution  of our heat equation with an initial value,  there exists a regular (in the usual sens) mild S-asymptotically ω-periodic solution which depends on a neighboring initial condition.

References

Blot, J.; Boudjema, S. Small almost periodic and almost automorphic oscillations in forced Liénard equations, J. Abstr. Differ. Equ. Appl. 1(1), 75-85 (2010).

Blot, J.; Boudjema, S.; Cieutat, P. Dependence results for S-asymptotically Periodic Solutions of Evolution Equations, Nonlinear studies. 20(3), 295-307 (2013).

Blot, J.; Cieutat, P.; N'Guérékata, G.M. S-asymptotically ω-periodic functions and applications to evolution equations, Afr. Diaspora J. Math. 12(2), 113-121 (2011).

Cartan, H. Cours de calcul différentiel, Hermann, Paris (1977).

Cazenave, T.; Haraux, A. Introduction aux problèmes d'évolution semilinéaires, Ellipses-Edition, Paris (1990).

Dimbour, W.; Mophou, G., N'Guérékata, G.M. S-asymptotically periodic solutions for partial differential equations with finite delay, Electron. J. Differential Equations 117, 1-12 (2011).

Dimbour, W.; N'Guérékata, G.M. S-asymptotically ω-periodic solutions to some classes of partial evolution equations, Appl. Math. Comput. 218(14) , 7622-7628 (2012). http://dx.doi.org/10.1016/j.amc.2012.01.029

Grimmer, R.C. Asymptotically almost periodic solutions of differential equations, SIAM J. Appl. Math. 17, 109-115 (1996). http://dx.doi.org/10.1137/0117012

Henriquez, H.R.; Pierri, M., Táboas, P. On S-asymptotically ω-periodic functions on Banach spaces and applications, J. Math. Anal. Appl. 343(2), 1119-1130 (2008). http://dx.doi.org/10.1016/j.jmaa.2008.02.023

Liang, B.; Zhong, C. Asymptotically periodic solutions of a class of order nonlinear differential equations, Proc. Amer. Math. Soc. 99(4), 693-699 (1987). http://dx.doi.org/10.2307/2046478

Pazy, A. Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983).

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Published

2024-05-24

How to Cite

Bouadi, A., & Boudjema, S. (2024). S-asymptotically ω periodic solutions of heat equation. STUDIES IN ENGINEERING AND EXACT SCIENCES, 5(1), 2159–2174. https://doi.org/10.54021/seesv5n1-107