An extension of Glaeser inequality and its applications (Q1431086)

The authors study \(C^\infty\) well-posedness for the pseudosymmetric hyperbolic system \[ (D_t+ A(x) D_x+ B(x)) u= f(t,x),\quad u(0,x)= 0 \] (the notion pseudosymmetric hyperbolic system has been introduced by D'Ancona/Spagnolo). They describe Levi conditions to \(B\). Moreover, they discuss the case \(B\equiv 0\) under low regularity of \(A(x)\). In special situations such systems can be transferred to a second-order weakly hyperbolic Cauchy problem. To guarantee the Levi condition an extension of Glaeser's classical inequality is discussed. The proof of well-posedness results (a priori estimates) is based on an application of the energy method.