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<p>This research paper investigates a category of nonlinear parabolic equations characterized by double variable exponent flux operators, analyzed under time-periodic conditions. The primary objective is to establish the well-posedness of these equations by addressing the existence and uniqueness of solutions. By employing the framework of Lebesgue-Sobolev spaces with variable exponents and Bochner spaces, a suitable functional structure is developed to support the analysis. Two key findings are presented concerning weak solutions. For cases where the source term does not depend on the solution, a general abstract method is utilized, leveraging the time-periodic condition to provide both existence and uniqueness. For scenarios involving a nonlinear source term strongly linked to the solution, the existence and uniqueness of weak solutions are achieved without requiring any sign constraints on the nonlinearity. The methodology heavily relies on Leray-Schauder alternative, supported by innovative technical estimates.</p>

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Weak Solvability of Some Periodic Evolution Problems Driven by Anisotropic (p(x), q(x))-Growth Operators

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Abderrahim Charkaoui,
  • Ghita El Guermai

摘要

This research paper investigates a category of nonlinear parabolic equations characterized by double variable exponent flux operators, analyzed under time-periodic conditions. The primary objective is to establish the well-posedness of these equations by addressing the existence and uniqueness of solutions. By employing the framework of Lebesgue-Sobolev spaces with variable exponents and Bochner spaces, a suitable functional structure is developed to support the analysis. Two key findings are presented concerning weak solutions. For cases where the source term does not depend on the solution, a general abstract method is utilized, leveraging the time-periodic condition to provide both existence and uniqueness. For scenarios involving a nonlinear source term strongly linked to the solution, the existence and uniqueness of weak solutions are achieved without requiring any sign constraints on the nonlinearity. The methodology heavily relies on Leray-Schauder alternative, supported by innovative technical estimates.