Higher-Order Heat Estimates and Semilinear Heat Equations on the Noncommutative Torus

We establish sharp higher-order heat estimates with complete bound on the noncommutative tori \(\mathbb{T}_{\theta}^{n}\) and show the optimality in the small-time order. As an application in polynomial semilinear heat equations on \(\mathbb{T}_{\theta}^{n}\), we give local well-posedness, the blow-up alternative, persistence of higher regularity, and instantaneous smoothing in the Sobolev algebra scale \(H^k_\theta\), \(k>n/2\).