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<p>The present paper is concerned with the Cauchy–Dirichlet problem for <i>fractional</i> (and non-fractional) <i>nonlinear diffusion equations</i> posed in bounded domains. Main results consist of well-posedness in an <i>energy class</i> with no sign restriction and convergence of such (possibly sign-changing) energy solutions to asymptotic profiles after a proper rescaling. They are proved in a variational scheme only, without any use of semigroup theories nor classical quasilinear parabolic theories. Proofs are self-contained and performed in a totally unified fashion for both fractional and non-fractional cases as well as for both porous medium and fast diffusion cases.</p>

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Energy Solutions of the Cauchy–Dirichlet Problem for Fractional Nonlinear Diffusion Equations

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  • 发表日期：

    2025年7月31日

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  • Goro Akagi,
  • Florian Salin

摘要

The present paper is concerned with the Cauchy–Dirichlet problem for fractional (and non-fractional) nonlinear diffusion equations posed in bounded domains. Main results consist of well-posedness in an energy class with no sign restriction and convergence of such (possibly sign-changing) energy solutions to asymptotic profiles after a proper rescaling. They are proved in a variational scheme only, without any use of semigroup theories nor classical quasilinear parabolic theories. Proofs are self-contained and performed in a totally unified fashion for both fractional and non-fractional cases as well as for both porous medium and fast diffusion cases.