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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 ut−Δu=up+M|∇ 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 ut−Δu=up+M|∇ u|2, which corresponds to the supercritical case.

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Cite

BibTeXRIS
BibTeX
@misc{li2026liouville,
  title = {{Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term}},
  author = {Xue Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28481},
  eprint = {2610.28481},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28481},
}
RIS
TY  - UNPB
TI  - Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
AU  - Xue Li
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28481
UR  - https://arxiv.org/abs/2610.28481
AB  - In this paper, we prove the Liouville-type theorem for equation u_t−Δu=u^p+M|∇ 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|∇ u|^2, which corresponds to the supercritical case.
ER  -