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This book is devoted to prove a nonexictence theorems for solutions of some nonlinear evolution equations and systems of the same type on the Heisenberg group. The proofs are based on the test function method developed by Mitidieri and Pohozaev, Pohozaev and Tesei, Kirane and L. Ragoub. The method relies on a suitable choice of the test function in the weak formulation of the sought solutions.