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<p>This paper examines a class of nonlinear parabolic equations driven by the <i>p</i>(<i>x</i>)-Laplacian operator and subject to vanishing initial conditions. Our focus is on investigating the well-posedness, specifically the existence and uniqueness, of solutions to these models. Through the consideration of Lebesgue and Sobolev spaces with variable exponents, we build a suitable functional framework for our analysis. Thereafter, we establish two interesting results regarding the existence and uniqueness of weak solution. In instances where the source term is independent of the solution, we employ an abstract approach, exploiting the vanishing initial condition to establish both existence and uniqueness. Conversely, when the source term is nonlinear and strongly dependent on the solution, we demonstrate the existence and uniqueness of a weak solution without imposing any sign restriction on the nonlinearity. Our approach fundamentally depends on employing Schaefer’s fixed-point theorem, underpinned by novel technical estimates.</p>

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Schaefer’s fixed point approach for some evolution equations governed by p(x)-Laplacian operator with vanishing initial datum

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Jinxia Cen,
  • Abderrahim Charkaoui,
  • Ghita El Guermai

摘要

This paper examines a class of nonlinear parabolic equations driven by the p(x)-Laplacian operator and subject to vanishing initial conditions. Our focus is on investigating the well-posedness, specifically the existence and uniqueness, of solutions to these models. Through the consideration of Lebesgue and Sobolev spaces with variable exponents, we build a suitable functional framework for our analysis. Thereafter, we establish two interesting results regarding the existence and uniqueness of weak solution. In instances where the source term is independent of the solution, we employ an abstract approach, exploiting the vanishing initial condition to establish both existence and uniqueness. Conversely, when the source term is nonlinear and strongly dependent on the solution, we demonstrate the existence and uniqueness of a weak solution without imposing any sign restriction on the nonlinearity. Our approach fundamentally depends on employing Schaefer’s fixed-point theorem, underpinned by novel technical estimates.