The heat flow and harmonic maps on a class of manifolds (Q1366656)

We study the heat flow for harmonic maps from a class of complete non-compact domain manifolds \(M\) which satisfy: (a) there exists a constant \(A>1\) such that for any \(x\in M\) and for all \(R>0\), \(V_x(2R)\leq AV_x(R)\); (b) there exist constants \(N>1\), \(a>0\) such that for any function \(f\in C^\infty(B_x(NR))\), \(\frac{a}{R^2} \inf_{\alpha\in R}\int_{B_x(R)} (f-\alpha)^2\leq \inf_{B_x(NR)}| \nabla f|^2\). We show that if the target manifold \(N\) is complete, the \(C^2\) initial map \(h\) has bounded image in \(M\) and has bounded energy density and bounded tension field, then, for heat equations for harmonic maps \(u_t=\text{tension} (u)\), \(u(x,0)=h(x)\), short-time existence holds and the solution is unique. Additionally, if the sectional curvature of the target manifold is bounded from above, either long-time existence holds or the energy density of the heat flow blows up at a finite time. Moreover, if the target manifold has nonpositive sectional curvature and long-time existence holds, whose energy density increases logarithmically in time, then the long-time solution converges to a harmonic map up to a subsequence. On this class of (domain) manifolds, we also get an \(L^p\) \((p>0)\) mean value inequality for subsolutions of heat equations and gradient estimates for solutions of heat equations.