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<p>We obtain some new Li-Yau inequalities for semilinear parabolic equation on complete Riemannian manifolds with Ricci curvature bounded below. Compared to previous research, we do not add additional conditions for the positive solutions and our inequalities can be applied to more semilinear equations. Concretely, we first derive Li-Yau inequalities for the general semilinear equations with a weak growth condition. Second, for parabolic Lichnerowicz equation, we derive its Li-Yau inequalities and the sharp bounds for the ancient solutions. Consequently, sharp Liouville theorems for ancient solutions and eternal solutions are obtained. Last, we revisit the Li-Yau inequalities for parabolic Lane-Emden equation. We get an improved inequality on Ricci nonnegative manifolds. Especially, for Eucliden case, we derive its Li-Yau inequality with all subcritical index. Naturally, the corresponding Harnack inequalities are obtained from these Li-Yau inequalities.</p>

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Some New Li-Yau Inequalities for Semilinear Parabolic Equations and Its Applications

  • 期刊论文

  • 发表日期：

    2025年5月30日

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  • Zhihao Lu,
  • Lili Zhu

摘要

We obtain some new Li-Yau inequalities for semilinear parabolic equation on complete Riemannian manifolds with Ricci curvature bounded below. Compared to previous research, we do not add additional conditions for the positive solutions and our inequalities can be applied to more semilinear equations. Concretely, we first derive Li-Yau inequalities for the general semilinear equations with a weak growth condition. Second, for parabolic Lichnerowicz equation, we derive its Li-Yau inequalities and the sharp bounds for the ancient solutions. Consequently, sharp Liouville theorems for ancient solutions and eternal solutions are obtained. Last, we revisit the Li-Yau inequalities for parabolic Lane-Emden equation. We get an improved inequality on Ricci nonnegative manifolds. Especially, for Eucliden case, we derive its Li-Yau inequality with all subcritical index. Naturally, the corresponding Harnack inequalities are obtained from these Li-Yau inequalities.