带时变质量的有效阻尼波的不存在性
Nonexistence for effectively damped waves with time-dependent mass
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中文总结 AI 辅助
本文针对带有效时变阻尼和时变质量的半线性波动方程，在特定条件下证明了亚临界及临界情况整体弱解不存在，并给出了条件寿命上界与显式容许系数族。
中文摘要 AI 辅助
本文研究半线性波动方程 $u_{tt}-\Delta u+b(t)u_t+m^2(t)u=|u|^p$（$t\geq0, x\in\mathbb{R}^n$），该方程具有受阻尼主导的有效时变阻尼与时变质量。D'Abbicco、Girardi和Reissig针对初始数据属于 $(L^\eta(\mathbb{R}^n)\cap H^1(\mathbb{R}^n))\times(L^\eta(\mathbb{R}^n)\cap L^2(\mathbb{R}^n))$（$1\leq\eta<2$）的情况，在超临界范围建立了小初值整体存在性，并确定了尺度 $p_{\beta,\eta}(n)=1+\frac{2\eta}{n+2\eta\beta}$，其中 $\beta$ 是与阻尼-质量对相关的下质量指数。为验证该尺度的尖锐性，他们还对对应扩散方程在 $L^\eta(\mathbb{R}^n)$ 中非负初始数据的情况建立了类似的亚临界不存在性结果，留下了带有效阻尼和时变质量的波动方程这一对应问题待解决。本文针对 $\eta=1$，在固有累积质量平衡和Liouville非振荡条件下解决该问题；通过构造正慢伴随模式，证明了当 $1<p<p_{\beta,1}(n)=1+\frac{2}{n+2\beta}$ 时整体弱解不存在，还在Osgood发散条件下处理了临界情况 $p=p_{\beta,1}(n)$，同时给出了条件寿命上界和显式容许系数族。
英文摘要
In this paper, we study the semilinear wave equations $$ u_{tt}-Δu+b(t)u_t+m^2(t)u=|u|^p, \quad t \geq 0, \quad x\in\mathbb{R}^n $$ with effective time-dependent damping and a time-dependent mass dominated by the damping. D'Abbicco, Girardi and Reissig established global small-data existence in supercritical ranges and identified the scale $$ p_{β,η}(n)=1+\frac{2η}{n+2ηβ} $$ for initial data in $(L^η(\mathbb R^n)\cap H^1(\mathbb R^n))\times(L^η(\mathbb R^n)\cap L^2(\mathbb R^n))$ with $1\leqη<2$, where $β$ is the lower mass index associated with the damping-mass pair. To support the expected sharpness of this scale, they also established an analogous subcritical nonexistence result for the corresponding diffusion equation with nonnegative initial data in $L^η(\mathbb{R}^n)$, leaving the wave-equation counterpart with effective damping and time-dependent mass open. We address this problem for $η=1$ under an intrinsic accumulated-mass balance and a Liouville nonoscillation condition. By constructing a positive slow adjoint mode, we prove nonexistence of global weak solutions for $$ 1<p<p_{β,1}(n)=1+\frac{2}{n+2β}, $$ and also treat the critical case $p=p_{β,1}(n)$ under an Osgood divergence condition. Conditional lifespan upper bounds and explicit admissible coefficient families are also given.
