On second order hyperbolic equations with coefficients degenerating at infinity and the loss of derivatives and decays

In this paper, we study well-posedness issues in the weighted L2L2 space for the Cauchy problem on [0,T]×Rx[0,T]×Rx for wave equations of the form ∂t2u−a(t,x)∂x2u=0. We shall give the condition a(t,x)>0a(t,x)>0 for all (t,x)∈[0,T]×Rx(t,x)∈[0,T]×Rx which is between the strictly hyperbolic condition and weakly hyperbolic one, and allows the decaying coefficient a(t,x)a(t,x) such that lim|x|→∞⁡a(t,x)=0lim|x|→∞⁡a(t,x)=0 for all t∈[0,T]t∈[0,T]. Our concerns are the loss of derivatives and decays of the solutions.