具有变指数对数源的强阻尼粘弹性波动方程在任意能量水平下的爆破
Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source
- Hohai University（河海大学）
- Wuhan University（武汉大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究强阻尼粘弹性波动方程在对数源项下的有限时间爆破，通过移位能量方法证明任意能量水平均可爆破，并给出显式寿命界与示例。
AI中文摘要：
我们研究了一类具有对数源的强阻尼粘弹性波动方程，其可测指数满足 $2<p_-\le p(x) \le p_+<2^*$。对源-势不等式的最优点态修正定义了一个“移位能量”。在适当的核条件下，我们证明了负移位能量以及一类无固定上能量阈值的非负移位能量的有限时间爆破。通过达到负移位能量的估计和凹性论证，我们得到了显式的上寿命界。我们还构造了在每个给定的实数能量水平下的光滑爆破数据，并表明我们的寿命估计涵盖了常数指数情形下的额外数据。给出了显式例子以说明爆破结果。
英文摘要：
We study a strongly damped viscoelastic wave equation with a logarithmic source whose measurable exponent satisfies $2<p_-\le p(x) \le p_+<2^*$. An optimal pointwise correction to the source--potential inequality defines a \emph{shifted energy}. Under suitable kernel conditions, we prove finite-time blow-up for negative shifted energy and for a class of nonnegative shifted energies with no fixed upper energy threshold. Explicit upper lifespan bounds follow from an estimate for reaching negative shifted energy and a concavity argument. We also construct smooth blow-up data at every prescribed real energy level and show that our lifespan estimate covers additional data in the constant-exponent case. Explicit examples are given to illustrate the blow-up results.
