In this work, we present some applications of the \(L^{p}\) - \(L^{q}\) boundedness of Fourier multipliers to PDEs on the noncommutative (or quantum) Euclidean space. More precisely, we establish \(L^{p}\) - \(L^{q}\) norm estimates for solutions of heat, wave, and Schrödinger type equations with Caputo fractional derivative in the case \(1< p \le 2 \le q < \infty \) . Moreover, we obtain well-posedness of nonlinear heat and wave equations on the noncommutative Euclidean space.
