2025 Volume 15 Issue 6
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Guangwei Du, Fushan Li, Yuying Zheng. LIOUVILLE THEOREM FOR INDEFINITE FRACTIONAL PARABOLIC EQUATION INVOLVING PSEUDO-RELATIVISTIC SCHRÖDINGER OPERATORS[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3677-3689. doi: 10.11948/20250044
Citation: Guangwei Du, Fushan Li, Yuying Zheng. LIOUVILLE THEOREM FOR INDEFINITE FRACTIONAL PARABOLIC EQUATION INVOLVING PSEUDO-RELATIVISTIC SCHRÖDINGER OPERATORS[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3677-3689. doi: 10.11948/20250044

LIOUVILLE THEOREM FOR INDEFINITE FRACTIONAL PARABOLIC EQUATION INVOLVING PSEUDO-RELATIVISTIC SCHRÖDINGER OPERATORS

  • In this paper, we study the following indefinite fractional parabolic equation involving pseudo-relativistic Schrödinger operators

    $\begin{equation*}\frac{\partial u}{\partial t}(x, t)+(-\Delta+m^2)^su(x, t)=a(x_1)f(u(x, t)), \ \ \mbox{in}\ { \mathbb R^N}\times\mathbb R, \end{equation*}$

    where $0<s<1$ and the mass $m>0$. We first prove the monotonicity of positive bounded solutions by using the method of moving planes. Moreover, the nonexistence of positive bounded solutions is established.

    MSC: 35R11, 35B53, 35K58
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