The Existence And Uniqueness Of The Solution To Cauchy Problem Of Heat Equation

The existence and uniqueness of the global classical solution to cauchy problem of n-D nonlinear heat equation has been firstly studied by S.klainerman in 1982 . the following result has been obtained: if n,the dimension of the space ,satisfies the following condition the initial value is small,which means the norm of (x) in certain sobolev space is very small,then the cauchy problem (0.1) has a unique global classical solution when t≥0. In 1985, ,Zheng Song-mu ,Chen Yun-mei and G.Ponce almost simultaneously obtained the same result for n >α2. In this paper ,the method that Zheng and Chen used in problem (0.1) is adopted to deal with the following Cauchy problem Where (t ,x)∈R+×Rn, f ( x)∈Hs (Rn)∩L∞(Rn),∈W s ,1 ( Rn)∩Hs+1(Rn),and F (u ,Dx u,Dx2u) satisfies some suitable conditions and n satisfies n >1+ 42α,Make use of the property of the solution to cauchy problem (0.2) in certain sobolev space and Banach fixed point principle to get the conclusion that Cauchy problem (0.2) has and only has a unique global solution.