AI 中文总结
研究与一类多项式非调和振荡器相关的分数阶热半群，利用Weyl - Hörmander演算证明其分数幂是伪微分算子，给出在Lebesgue和调制空间的衰减估计，还应用于相关非线性方程，扩展了热半群理论。
AI 中文摘要
我们研究了在\(\mathbb R^n\)上由一类非调和振荡器生成的分数阶热半群\(\mathcal H_{P,Q}=Q(D)+P(x)\)，其中\(P\in\mathcal P_{2k}\)且\(Q\in\mathcal P_{2\ell}\)是具有各向异性增长的实值多项式。利用与由\((P,Q)\)确定的自然度量相关的Weyl - Hörmander演算，我们表明分数幂\(\mathcal H_{P,Q}^s\)，\(s>0\)，是具有适应类\(\Sigma_{P,Q}^{2s}\)中符号的伪微分算子。我们证明了分数阶非调和热半群\(e^{-t\mathcal H_{P,Q}^s}\)在Lebesgue空间和调制空间上的固定时间衰减估计。在Lebesgue空间设置中，我们为整个范围\(1\le p,q\le\infty\)建立了精确的\(L^p - L^q\)估计。对于大时间，衰减是指数的且由\(\mathcal H_{P,Q}\)的最小特征值\(\lambda_0\)控制，即通过因子\(e^{-t\lambda_0^s}\)，而对于小时间，估计揭示了与\(P\)和\(Q\)的强制增长相关的两个不同的相空间尺度，导致各向异性的\(L^p - L^q\)平滑。作为应用，我们研究了与\(\mathcal H_{P,Q}^s\)相关的非线性分数阶热方程。我们证明了在超临界Lebesgue范围\(p>\frac{n(\beta - 1)}{2\ell s}\)中的局部适定性，推导了有限时间爆破解的较低爆破率，并获得了临界小数据全局存在性。我们进一步证明了调制空间中小初始数据的全局适定性和指数衰减。这些结果将调和和模型非调和振荡器的热半群理论扩展到了一类广泛的各向异性多项式哈密顿量。
英文摘要
We investigate fractional heat semigroups generated by a class of anharmonic oscillators on $\mathbb R^n$ of the form $\mathcal H_{P,Q}=Q(D)+P(x),$ where $P\in\mathcal P_{2k}$ and $Q\in\mathcal P_{2\ell}$ are real-valued polynomials with anisotropic growth. Using the Weyl--Hörmander calculus associated with the natural metric determined by $(P,Q)$, we show that the fractional powers $\mathcal H_{P,Q}^s$, $s>0$, are pseudo-differential operators with symbols in adapted classes $Σ_{P,Q}^{2s}$. We prove fixed-time decay estimates for the fractional anharmonic heat semigroup $e^{-t\mathcal H_{P,Q}^s}$ on both Lebesgue and modulation spaces. In the Lebesgue setting, we establish sharp $L^p$--$L^q$ estimates for the full range $1\le p,q\le\infty$. For large time, the decay is exponential and governed by the smallest eigenvalue $λ_0$ of $\mathcal H_{P,Q}$, namely through the factor $e^{-tλ_0^s}$, while for small time the estimates reveal two distinct phase-space scales associated with the coercive growth of $P$ and $Q$, leading to anisotropic $L^p$--$L^q$ smoothing. As applications, we study nonlinear fractional heat equations associated with $\mathcal H_{P,Q}^s$. We prove local well-posedness in the supercritical Lebesgue range $ p>\frac{n(β-1)}{2\ell s},$ derive a lower blow-up rate for finite-time blow-up solutions, and obtain critical small-data global existence. We further prove global well-posedness and exponential decay for small initial data in modulation spaces. These results extend the heat semigroup theory for harmonic and model anharmonic oscillators to a broad class of anisotropic polynomial Hamiltonians.
Comments35 pages
