Xingquan Li, Yuli Feng, Han Yang
Abstract:
The purpose of this article is to study the Cauchy problem for the viscoelastic
damped wave equation on the Heisenberg group.
We first prove the global existence of small data solutions for
\(p\in [2,Q/(Q-4)]\) if \(n=2,3\), \(p>2\) if \(n=1 \) using
the contraction principle. Then, a blow-up result is obtained by using the
test function method under certain integral sign assumptions for the Cauchy
data when \(1< p\leq1+2/(Q-1)\), where \(Q=2n+2\) is the homogeneous dimension of
the Heisenberg group. Moreover, we obtain the upper bound for the lifespan of the
solution by employing a revisited test function method.
Submitted March 24, 2025. Published July 14, 2025.
Math Subject Classifications: 35A01, 35B44, 35R03.
Key Words: Viscoelastic damped wave equation; Heisenberg group; test function method; global existence; blow up.
DOI: 10.58997/ejde.2025.71
Show me the PDF file (380 KB), TEX file for this article.
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Xingquan Li School of Mathematics Southwest Jiaotong University Chengdu 611756, China email: lixingquan@my.swjtu.edu.cn |
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Yuli Feng School of Mathematics Southwest Jiaotong University Chengdu 611756, China email: fyl@my.swjtu.edu.cn |
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Han Yang School of Mathematics Southwest Jiaotong University University Chengdu 611756, China email: hanyang95@263.net |
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