Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity partial derivative(t)(2)u+partial derivative(t)u-Delta u+lambda u(1+2/n)=0, x is an element of R-n, t>0, u(0,x)=epsilon u(0) (x), partial derivative(t)u(0,x)=epsilon u(1)(x),x is an element of R-n, where epsilon>0, and space dimensions n=1, 2, 3. Assume that the initial data u(0) is an element of H-delta,H-0 boolean AND H-0,H-delta, u(1) is an element of H-delta-1,H-0 boolean AND H--1,H-delta where delta>n/2, weighted Sobolev spaces are H-l,H-m={phi is an element of L-2; parallel to < x >(m) < i partial derivative(x)>(l) phi (x) parallel to(L 2)=root 1+x(2.) Also we suppose that lambda theta(2/n)>0, integral u(0) (x) dx>0, where theta=integral(u(0) (x)+u(1) (x)) dx. Then we prove that there exists a positive epsilon(0) such that the Cauchy problem above has a unique global solution u is an element of C ([0,infinity); H-delta,H-0) satisfying the time decay property parallel to u(t)-epsilon theta G(t,x)e(-pi(t))parallel to(L p)<= C epsilon(1+2/n) g(-1-n/2)(t)< t >(-n/2(1-1/p)) for all t>0, 1 <= p <=infinity, where epsilon is an element of (0, e(0)].