S-asymptotically ω periodic solutions of heat equation


  • Abdelkader Bouadi
  • Souhila Boudjema




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


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.


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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