  We consider the heat flow of Yang-Mills-Higgs functional where the base
manifold is a Riemannian surface and the fiber is a compact symplectic
manifold. We show that the corresponding Cauchy problem admits a global weak
solution for any $H^1$-initial data. Moreover, the solution is smooth except
finitely many singularities. We prove an energy identity at finite time
singularities and give a description of the asymptotic behavior at time
infinity.
