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<p>In this work, we give sufficient conditions for the existence and uniqueness of the heat equation involving the operator <Equation ID="Equ44"> <EquationSource Format="TEX">\( \Delta _{\mathcal {G}}=\dfrac{1}{2}\left( \Delta _{x}+|x|^2\Delta _{y}\right) \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi mathvariant="script">G</mi> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mfenced close=")" open="("> <msub> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </msub> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msub> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </msub> </mfenced> </mrow> </math></EquationSource> </Equation>in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.</p>

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On the heat equation involving a Grushin operator in Marcinkiewicz spaces

  • 期刊论文

  • 发表日期：

    2025年7月29日

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  • Alessia E. Kogoj,
  • Maria E. Lima,
  • Arlúcio Viana

摘要

In this work, we give sufficient conditions for the existence and uniqueness of the heat equation involving the operator \( \Delta _{\mathcal {G}}=\dfrac{1}{2}\left( \Delta _{x}+|x|^2\Delta _{y}\right) \) Δ G = 1 2 Δ x + | x | 2 Δ y in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.