具有非线性阻尼的半线性波动方程的尖锐寿命估计
Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping
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中文总结 AI 辅助
该论文通过双曲尺度变换和局部存在性理论，证明了带非线性阻尼的半线性波动方程在大振幅下解的最大存在时间具有与上界同阶的尖锐下界，确定了指数 $\vartheta(p,q)=\min\{(p-1)/2,(p-q)/q\}$。
中文摘要 AI 辅助
我们研究有界域中半线性波动方程 $u_{tt}-\Delta u=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ 在Dirichlet边界条件和初始数据 $(\varrho f,\varrho g)$ 下的解的最大存在时间 $T^{*}(\varrho)$，其中 $1<q<p$ 且振幅 $\varrho$ 很大。对于非平凡的 $f$ 和足够大的 $\varrho$，凹性方法给出：当 $q\leq2p/(p+1)$ 时 $T^{*}(\varrho)\leq C\varrho^{-(p-1)/2}$，当 $q>2p/(p+1)$ 时 $T^{*}(\varrho)\leq C\varrho^{-(p-q)/q}$；而能量方法仅给出阶为 $\varrho^{1-p}$ 的下界。我们证明了与上界具有相同指数的下界，从而得到 $T^{*}(\varrho)\asymp\varrho^{-\vartheta(p,q)}$，其中 $\vartheta(p,q)=\min\{(p-1)/2,\\,(p-q)/q\}$。证明依赖于双曲尺度变换，该变换将大振幅转化为区域的膨胀以及阻尼项前的系数 $\varrho^{(q(p+1)-2p)/2}$；依赖于一致局部能量范数中的局部存在性理论，其存在时间不依赖于该系数；并依赖于当系数较大时对耗散的定量利用。阈值 $2p/(p+1)$ 是阻尼项在尺度变换下不变的 $q$ 值。我们未尝试优化常数：尖锐性始终指指数层面。
英文摘要
We study the maximal existence time $T^{*}(\varrho)$ of the solution of the semilinear wave equation $u_{tt}-Δu=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\varrho f,\varrho g)$, where $1<q<p$ and the amplitude $\varrho$ is large. For nontrivial $f$ and sufficiently large $\varrho$, the concavity method gives $T^{*}(\varrho)\leq C\varrho^{-(p-1)/2}$ for $q\leq2p/(p+1)$ and $T^{*}(\varrho)\leq C\varrho^{-(p-q)/q}$ for $q>2p/(p+1)$, whereas the energy method gives a lower bound of order $\varrho^{1-p}$ only. We prove lower bounds with the same exponents as the upper ones, so that $T^{*}(\varrho)\asymp\varrho^{-\vartheta(p,q)}$ with $\vartheta(p,q)=\min\{(p-1)/2,\,(p-q)/q\}$. The proof rests on a hyperbolic rescaling which converts the large amplitude into a dilation of the domain and a coefficient $\varrho^{(q(p+1)-2p)/2}$ in front of the damping term, on a local existence theory in uniformly local energy norms whose existence time does not depend on that coefficient, and on a quantitative use of the dissipation when the coefficient is large. The threshold $2p/(p+1)$ is the value of $q$ at which the damping term is invariant under the rescaling. No attempt is made to optimize the constants: sharpness is meant throughout at the level of the exponent.
发表机构
- Faculty of Sciences of Tunis（突尼斯科学学院）
- University of Tunis El Manar（突尼斯埃尔曼纳尔大学）
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