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<p>We study a time-fractional superdiffusion equation involving a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\upvarphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">φ</mi> </math></EquationSource> </InlineEquation>-Caputo derivative and a fractional Laplacian within a Lorentz-space framework. First, we analyze the associated linear problem. By combining the Fourier transform in space with the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\upvarphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">φ</mi> </math></EquationSource> </InlineEquation>-Laplace transform in time, we derive explicit representation formulas for the Green kernels. These kernels are expressed in terms of Fox <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation>-functions, which allow us to obtain sharp pointwise bounds and decay estimates in Lorentz spaces. We then consider the corresponding semilinear equation with a singular source term. Using the linear estimates, we establish local well-posedness of mild solutions in weighted Lorentz spaces, together with continuous dependence on the initial data and a blow-up alternative. Finally, we discuss the scaling structure induced by the decay estimates and identify <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( p_c(\uprho )=1+\frac{2\updelta \,\uprho }{\upalpha d} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">δ</mi> <mspace width="0.166667em" /> <mi mathvariant="normal">ρ</mi> </mrow> <mrow> <mi mathvariant="normal">α</mi> <mi>d</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> as a natural candidate critical exponent in this setting.</p>

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Well-posedness and decay for a \(\upvarphi \)-caputo time-fractional superdiffusion equation in Lorentz spaces

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Brahim Benaissi,
  • Yacine Arioua

摘要

We study a time-fractional superdiffusion equation involving a \(\upvarphi \) φ -Caputo derivative and a fractional Laplacian within a Lorentz-space framework. First, we analyze the associated linear problem. By combining the Fourier transform in space with the \(\upvarphi \) φ -Laplace transform in time, we derive explicit representation formulas for the Green kernels. These kernels are expressed in terms of Fox \(H\) H -functions, which allow us to obtain sharp pointwise bounds and decay estimates in Lorentz spaces. We then consider the corresponding semilinear equation with a singular source term. Using the linear estimates, we establish local well-posedness of mild solutions in weighted Lorentz spaces, together with continuous dependence on the initial data and a blow-up alternative. Finally, we discuss the scaling structure induced by the decay estimates and identify \( p_c(\uprho )=1+\frac{2\updelta \,\uprho }{\upalpha d} \) p c ( ρ ) = 1 + 2 δ ρ α d as a natural candidate critical exponent in this setting.