We study the evolution of Tsallis entropy along the heat flow and establish concavity results in arbitrary dimensions. Extending earlier one-dimensional results, we prove that Tsallis entropy is concave along the heat flow for $q\in(0,3]$ in dimension one and for $q\in[1,3]$ in higher dimensions. The upper endpoint $q=3$ is sharp in every dimension. The proof is based on a nonlinear transformation of the heat equation, a sharp dimension-free functional inequality with constant $C_u=3$, and a rigorous justification of the integration-by-parts identities used in the argument. The sharp inequality is proved by an explicit integration-by-parts sum-of-squares identity, rather than by a computer-assisted semidefinite-programming search. As consequences, we recover a generalized de Bruijn identity, prove monotonicity of the associated $q$-Fisher information along the heat flow, and establish concavity results for Tsallis entropy power, including the Shannon entropy-power case and Costa's EPI as an endpoint. We also obtain an asymptotic entropy-power concavity statement for general initial data and a sharp auxiliary functional inequality which may be of independent analytic interest.

翻译：我们研究了Tsallis熵沿热流的演化过程，并在任意维度下建立了凹性结果。通过将先前一维结果进行推广，我们证明：在一维情形下，当$q\in(0,3]$时Tsallis熵沿热流是凹的；在高维情形下，当$q\in[1,3]$时该凹性成立。在所有维度中，上界端点$q=3$均为极优的。证明基于热方程的非线性变换、常数为$C_u=3的锐利无维泛函不等式，以及对论证中所用分部积分恒等式的严格证明。该锐利不等式通过显式的分部积分平方和恒等式得以证明，而非借助计算机辅助的半正定规划搜索。作为推论，我们恢复了一类广义de Bruijn恒等式，证明了相关$q$-Fisher信息沿热流的单调性，并建立了Tsallis熵幂的凹性结果，其中包括香农熵幂情形以及作为端点的Costa EPI。我们还获得了针对一般初始数据的渐近熵幂凹性陈述，以及一个可能具有独立分析价值的锐利辅助泛函不等式。