We study the evolution of Tsallis entropy along the heat flow and establish its concavity in arbitrary dimensions. Extending prior results that were restricted to the one-dimensional setting, we prove that the Tsallis entropy is concave in time for a nontrivial range of the entropic index $q$ in both the one-dimensional and higher-dimensional settings. The analysis is based on a nonlinear transformation, together with a novel estimate for the second-order time derivative of the entropy and a rigorous justification of the integration-by-parts identities required in the argument. Our approach is fully analytic and avoids the use of computer-assisted methods that have limited previous works in higher dimensions. As consequences, we recover a generalized de Bruijn identity, establish the monotonicity of the associated $q$-Fisher information along the heat flow, and derive concavity properties for the Tsallis entropy power, including asymptotic results under general initial conditions. In addition, our method yields a new functional inequality that may be of independent interest. These results contribute to the broader program of extending classical information-theoretic inequalities beyond the Shannon framework to non-additive entropy settings.

翻译：我们研究了Tsallis熵沿热流的演化过程，并建立了其在任意维度下的凹性。我们克服了先前仅适用于一维情形的局限，证明在非平凡参数范围$q$内，Tsallis熵在时间维度上具有凹性，且该结论同时适用于一维与高维情形。分析基于非线性变换、熵的二阶时间导数的新型估计以及论证中所需分部积分恒等式的严格证明。我们的方法完全基于解析推导，避免了限制先前高维研究的计算机辅助方法。作为推论，我们恢复了广义de Bruijn恒等式，证明了关联的$q\)-Fisher信息沿热流的单调性，并推导了Tsallis熵功率的凹性性质（包括一般初始条件下的渐近结果）。此外，本方法还得到了一个可能具有独立价值的新泛函不等式。这些结果拓展了将经典信息论不等式从香农框架推广至非可加熵体系的广泛研究方向。