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<p>In this paper, we establish a Liouville type theorem for the homogeneous dual fractional parabolic equation</p><p><Equation ID="Equa"> <EquationSource Format="TEX">\(\partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = 0\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R},\)</EquationSource> </Equation></p><p>where 0 &lt; <i>α, s</i> &lt; 1. By employing the Fourier analysis method, we prove that all solutions in the sense of distributions must be affine. Consequently, when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0&lt;s\leq\frac{1}{2}\)</EquationSource> </InlineEquation>, these solutions are constant. In the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\frac{1}{2}&lt;s&lt; 1\)</EquationSource> </InlineEquation>, under the asymptotic assumption</p><p><Equation ID="Equb"> <EquationSource Format="TEX">\(\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^\gamma}\geq 0 \; (\mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leq\gamma\leq 1,\)</EquationSource> </Equation></p><p>all solutions must be constant. Our result includes the previous Liouville theorems on <i>s</i>-harmonic functions [3] as special cases and it is still novel even restricted to the one-sided Marchaud fractional equation. Our methods can be applied to a variety of dual nonlocal parabolic problems.</p><p>As an application of the above Liouville theorem, we establish an equivalence between the pseudo-differential equations involving Marchaud fractional derivatives and the corresponding integral equations.</p>

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Liouville type theorems for dual nonlocal evolution equations involving Marchaud derivatives

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Yahong Guo,
  • Lingwei Ma,
  • Zhenqiu Guo

摘要

In this paper, we establish a Liouville type theorem for the homogeneous dual fractional parabolic equation

\(\partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = 0\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R},\)

where 0 < α, s < 1. By employing the Fourier analysis method, we prove that all solutions in the sense of distributions must be affine. Consequently, when \(0<s\leq\frac{1}{2}\) , these solutions are constant. In the case \(\frac{1}{2}<s< 1\) , under the asymptotic assumption

\(\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^\gamma}\geq 0 \; (\mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leq\gamma\leq 1,\)

all solutions must be constant. Our result includes the previous Liouville theorems on s-harmonic functions [3] as special cases and it is still novel even restricted to the one-sided Marchaud fractional equation. Our methods can be applied to a variety of dual nonlocal parabolic problems.

As an application of the above Liouville theorem, we establish an equivalence between the pseudo-differential equations involving Marchaud fractional derivatives and the corresponding integral equations.