The Klein-Gordon equation in anti-de Sitter spacetime - Seminario ...

278 K. Yagdjian and A. Galstianv tt −△v=0, v(x,0;b)= f(x,b), v t (x,0;b)=0.The next theorem represents the solutions of the equation with the initial dataprescribed at t = 0.THEOREM 6. The solution u=u(x,t) to the Cauchy problem(14) u tt − e 2t △ u+M 2 u=0, u(x,0)=ϕ 0 (x), u t (x,0)=ϕ 1 (x),with ϕ 0 , ϕ 1 ∈ C ∞ 0 (Rn ), n≥2, can be represented as follows:(15)∫ 1u(x,t) = e − 2 t v ϕ0 (x,φ(t))+ 2 v ϕ0 (x,φ(t)s)K 0 (φ(t)s,t)φ(t)ds0∫ 1+2 v ϕ1 (x,φ(t)s)K 1 (φ(t)s,t)φ(t)ds, x∈R n , t > 0,0φ(t) := e t − 1, and where the kernels K 0 and K 1 have been defined in Theorem 4. Herefor ϕ∈ C0 ∞(Rn ) and for x∈R n , n=2m+1, m∈N,())n−3∂ 1 ∂ 2 rv ϕ (x,φ(t)s) :=∂r( n−2 ∫r ∂r ω n−1 c (n) ϕ(x+ry)dS S0n−1 yr=φ(t)swhile for x∈R n , n=2m, m∈N,( ( ))n−2∂ 1 ∂ 2 2r n−1 ∫1v ϕ (x,φ(t)s) :=√∂r r ∂r ω n−1 c (n)ϕ(x+ry)dV B n 0 1 (0) y1−|y| 2r=sφ(t)The function v ϕ (x,φ(t)s) coincides with the value v(x,φ(t)s) of the solution v(x,t) ofthe Cauchy problemv tt −△v=0, v(x,0)=ϕ(x), v t (x,0)=0.As a consequence of the above theorems, we obtain in a forthcoming paper thefollowing L p − L q decay estimate for the particles with “large” mass m, m≥n/2, thatis, with nonnegative curved mass M ≥ 0..(16)‖(−△) −s u(x,t)‖ L q (R n )( )∫≤ Ce t 2s−n( 1 p − q 1 t)0‖ f(x,b)‖ L p (R n )(1+ t− b) 1−sgnM db+C(1+ t) 1−sgnM (e t − 1) 2s−n( 1 p − 1 q ){ ‖ϕ 0 (x)‖ L p (R n )+(1−e −t )‖ϕ 1 ‖ L p (R n )}provided that 1< p≤2, 1 p + 1 q = 1, 1 2 (n+1)( 1p − 1 q)≤ 2s≤n( 1p − 1 q)< 2s+1.We emphasize that the estimate (16) implies exponential decay for large time. Itis essentially different from the decay estimate obtained in [16] for the wave equation