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.

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