AI 中文总结
针对带分数阶结构阻尼的半线性波动方程，研究其爆破时间的显式上下界，明确了阻尼参数对非线性载荷容许范围的影响，端点情形可恢复已知结果。
AI 中文摘要
我们考虑有界域中带有分数阶结构阻尼的半线性波动方程的初边值问题：$$ u_{tt}+(-\boldsymbol{\triangle})^{\theta}u_{t}-\boldsymbol{\triangle}u=|u|^{p-1}u $$，其中指数$\theta \boldsymbol{\text{∈}}[0,1]$在外部摩擦阻尼（$\theta=0$）和内部Kelvin–Voigt黏弹性阻尼（$\theta=1$）之间插值，因此参数化了耗散机制的频率依赖性。对于能量低于势阱深度且Nehari泛函为负的初值，我们证明了有限时间爆破，并给出了爆破时间的显式上界。该上界来自一个单一的凹性泛函，其中分数阶耗散完全抵消；因此它对每个$\theta \boldsymbol{\text{∈}}[0,1]$都具有相同形式，且存在一个对$\theta$一致的变体。反之，当$n\boldsymbol{\text{≥}}3$且$1<p\boldsymbol{\text{≤}}\frac{n+2\theta}{n-2}$时，我们建立了爆破时间的显式下界。从力学角度看，这两个界限定了该模型的保证存在区间和保证失效时间。下界中指数的容许范围随$\theta$线性扩大，这量化了内部阻尼的强度如何扩大可计算此类保证的非线性载荷类别；我们不主张界的数值关于$\theta$单调。两个定理均针对满足一小组要求的一类能量解证明，因此它们独立于任何特定的局部存在定理，且可原封不动地推广到结构模型中使用的弹性和板算子。两个端点情形恢复了摩擦阻尼和强阻尼的已知结果。
英文摘要
We consider the initial--boundary value problem for the semilinear wave equation with fractional structural damping $$ u_{tt}+(-Δ)^θu_{t}-Δu=|u|^{p-1}u $$ in a bounded domain, where the exponent $θ\in[0,1]$ interpolates between external frictional damping ($θ=0$) and internal Kelvin--Voigt viscoelastic damping ($θ=1$), and therefore parametrises the frequency dependence of the dissipation mechanism. For initial data with energy below the depth of the potential well and negative Nehari functional we prove finite-time blow-up together with an explicit upper bound for the blow-up time. The bound comes from a single concavity functional in which the fractional dissipation cancels identically; it therefore has the same form for every $θ\in[0,1]$, and it admits a variant that is uniform in $θ$. Conversely, for $1<p\le\frac{n+2θ}{n-2}$ when $n\ge3$, we establish an explicit lower bound for the blow-up time. Mechanically, the two bounds delimit a guaranteed interval of existence and a guaranteed failure time for the model. The admissible range of exponents in the lower bound widens linearly with $θ$, which quantifies how the strength of the internal damping enlarges the class of nonlinear loads for which such a guarantee can be computed; we do not claim monotonicity in $θ$ of the numerical value of the bound. Both theorems are proved for a class of energy solutions specified by a short list of requirements, so that they are independent of any particular local existence theorem, and they carry over unchanged to the elasticity and plate operators used in structural models. The two endpoint cases recover known results for frictional and strong damping.
Comments16 pages, no figures
