Global well-posedness of a class of singular hyperbolic Cauchy problems (Q2684474)

In the present paper the authors study the following linear Cauchy problem forward in time: \[ u_{tt} + b_0(t,x)u_t + a(t,x,D_x) u +b(t,x,D_x)=f(t,x),\quad u(0,x)=u_0(x), \,\,\,u_t(0,x)=u_1(x),\tag{1} \] where \[ a(t,x,\xi)=\sum_{k,l=1}^n a_{kl}(t,x)\xi_k\xi_l,\,\,\,b(t,x,\xi)=i\sum_{k=1}^n b_k(t,x)\xi_k + b_{n+1}(t,x). \] Under reasonable assumptions for the coefficients, data and right-hand side one expects well-posedness results in suitable evolution spaces. But, the authors are interested in non-standard assumptions for the coefficients. On the one hand they allow some unbounded behavior in time if \(t \to +0\), on the other hand they allow some unbounded behavior for \(|x| \to \infty\). Simple examples verify that the unbounded behaviors should be controlled somehow, otherwise there will be no (even distributional) solution. The authors introduce a scale of weighted (with respect to \(x\) and \(D_x\)) spaces of infinite order. The scale allows to describe the loss of regularity (of positive order or infinite order) which is a typical phenomenon for linear hyperbolic Cauchy problems with unbounded coefficients. The approach is standard. One needs scales of symbol classes and related pseudo-differential operators. The extended phase space will be divided into zones. Then one applies factorization in the principal part which allows to transform to first order pseudo-differential systems. A change of variables and a conjugation result transform the original problem to an auxiliary problem. Finally, application of sharp Gaarding inequality allows transformation to a \(L^2\) well-posed Cauchy problem, to derive a standard energy inequality as well. The change of variables contains the loss of regularity. To complete the well-posedness the authors verify the property of existence of cone of dependence.