Heisenberg群上带结构阻尼的分数阶半线性发展方程的衰减指标与临界指数的新定义
New definitions of decay indicators and critical exponent for fractional semi-linear structurally damped evolution equations on the Heisenberg group
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中文总结 AI 辅助
本文在Heisenberg群上定义衰减指标与特征，推导线性分数阶扩散方程及结构阻尼分数阶半线性发展方程的衰减估计，确定半线性问题的临界指数，拓展了相关研究结果。
中文摘要 AI 辅助
本文在Heisenberg群上引入了上下衰减指标及相关的衰减特征，这些概念用于推导线性分数阶扩散方程的衰减估计，并刻画若干函数空间中初始数据的衰减特性。随后研究柯西问题：$$\partial_t^2u+\left(-\Delta_{\mathrm H}\right)^{\delta_1}u+\left(-\Delta_{\mathrm H}\right)^{\delta_2}\partial_tu=0, \quad \delta_1\in\left[0,\frac{\delta_2}{2}\right], $$并依据初始数据的衰减特征，建立齐次分数阶Sobolev空间中解及其导数的衰减估计，这些结果既包含已知估计又将其扩展至新的数据类。还研究带非线性项$|u|^p$的对应半线性问题，证明当$$ p>1+\frac{2\omega\delta_1}{Q-2\omega\delta_2}, \quad \omega=\frac{Q}{Q+\min\{r_{\mathrm H}(u_0),r_{\mathrm H}(u_1)-2\delta_2\}+2\delta_2} $$时的整体存在性与衰减性，以及对应的临界情形。最后通过构造适配非局部分数阶次拉普拉斯算子的检验函数，建立爆破结果，并确定$\dot H_m^{-\gamma}(\mathbf H_n)$中初始数据的临界指数$$ p=1+\frac{2m\delta_1}{Q+m\gamma-2m\delta_2} $$，其中$m\in(1,2]$，$\gamma\in\left[0,Q-\frac{Q}{m}\right]$。
英文摘要
In this paper, we introduce the lower and upper decay indicators and the associated decay character on the Heisenberg group. These notions are used to derive decay estimates for the linear fractional diffusion equation and to characterize the decay of initial data in several function spaces. We then study the Cauchy problem $$ \partial_t^2u+\left(-Δ{\mathrm H}\right)^{δ_1}u+\left(-Δ_{\mathrm H}\right)^{δ_2}\partial_tu=0, \quad δ_1\in\left[0,\frac{δ_2}{2}\right], $$ and establish decay estimates for solutions and their derivatives in homogeneous fractional Sobolev spaces in terms of the decay characters of the initial data. These results recover known estimates and extend them to new classes of data. We also investigate the corresponding semilinear problem with nonlinearity $|u|^p$. Global existence and decay are proved for $$ p>1+\frac{2ωδ_1}{Q-2ωδ_2}, \quad ω=\frac{Q}{Q+\min{r_{\mathrm H}(u_0),r_{\mathrm H}(u_1)-2δ_2}+2δ_2}, $$ together with the corresponding critical case. Finally, by constructing test functions adapted to the nonlocal fractional sub-Laplacians, we establish blow-up results and identify the critical exponent $$ p=1+\frac{2mδ_1}{Q+mγ-2mδ_2} $$ for initial data in $\dot H_m^{-γ}(\mathbf H_n)$, where $m\in(1,2]$ and $γ\in\left[0,Q-\frac{Q}{m}\right)$.
