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<p>We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\log (-\Delta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and a second-order differential operator whose leading coefficients are not positive semi-definite.</p>

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An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

  • 期刊论文

  • 发表日期：

    2025年9月19日

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  • Jae-Hwan Choi,
  • Ildoo Kim

摘要

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator \(\log (-\Delta )\) log ( - Δ ) and a second-order differential operator whose leading coefficients are not positive semi-definite.