Existence time for solutions of semilinear different speed Klein-Gordon system with weak decay data

A 2−2| log 2|−α (α = 2 if d ≥ 3, α = 3 if d = 2) result is obtained for the existence time of solutions of semilinear different speed Klein-Gordon system with weakly decaying Cauchy data, of size 2, in certain circumstances of nonlinearity. keywords: Weakly decaying Cauchy data, Existence time, Klein-Gordon equations with different speeds. Subject class: 35L70 1 Statement of the Main Result We know that there are a lot of results on the lower bound problem of the life-spain for solutions to the following semilinear Klein-Gordon equation { ¤u + u = F (u, ∂tu, ∂xu), u|t=0 = 2u0, ∂tu|t=0 = 2u1 (1) with small, weak decay Cauchy data. In [1] Delort studied the problem (1) with periodic Cauchy data , and got a lower bound for the time of existence, of magnitude c2−2, for a general nonlinearity, and there are examples showing the optimality of that result for specific nonlinearities. Another type of Cauchy data with weak decay properties at infinity has been considered in [2], in dimension d = 1: it was shown that if u0, u1 are in Sobolev spaces H (R), HN−1(R), without any other assumption at infinity, and if the right hand side satisfies a “null condition” of the type of those considered by Kosecki [4], the solution to (1) exists over an interval of length c2−4| log 2|−6. Very recently, Delort and Fang in [3] investigated the higher dimensional case. We show that if the nonlinearity F is a combination of a polynomial, depending only on u, vanishing at order at least 2 at 0, and of [(∂tu) − (∂xu)]G(u), where G is also a polynomial, the solution to (1) with u0 ∈ H (R), u1 ∈ HN−1(Rd), N > (d + 3)/2, exists on an interval of time of length exp[c2−μ] with μ = 2/3 if d = 2, μ = 1 if d ≥ 3. A natural problem is: can we get a lower bound for the time of existence for the different speed Klein-Gordon system with weak decay data? In one space dimension this problem has been ∗The author was supported by NNSF of China 19671072 and partially by 19971077