New Yamabe-type flow in a compact Riemannian manifold

In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold (M, g) of dimension n & GE; 3. Let & psi;(x) be any smooth function on M. Let p = n+2 n-2 and cn = 4(n-1) n-2 . We study the Yamabe-type flow u = u(t) satisfyingut = u1-p(cn & UDelta;u -& psi;(x)u) + r(t)u, in M x (0 , T) , T > 0withr(t) = M (cn| backward difference u|2 + & psi;(x)u2)dv/ M up+1 ,which preserves the Lp+1(M )-norm and we can show that for any initial metric u0 > 0, the flow exists globally. We also show that in some cases, the global solution converges to a smooth solution to the equationcn & UDelta;u - & psi;(x)u + r(& INFIN;)up = 0 , on M