Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term

Xue Li

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arXiv · 2610.28481

Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term

Xue Li

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Abstract

In this paper, we prove the Liouville-type theorem for equation $u_t-Δu=u^p+M|\nabla u|^q$ via the integral Bernstein method, where $q=2p/(p+1)$ is critical and $M<0$. As an application, we can establish a universal priori estimate for this equation by using the doubling lemma. These results generalize the previous work of Liang and Zhang. We can also prove the Liouville-type theorem for equation $u_t-Δu=u^p+M|\nabla u|^2$, which corresponds to the supercritical case.

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BibTeX RIS

Xue Li. 2026-10-20. Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term. https://arxiv.org/abs/2610.28481

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