Energy estimates for a class of degenerate hyperbolic equations
Considered is the Cauchy problem for a linear degenerate inhomogeneous hyperbolic equation in one dimension. The leading coefficient which is allowed to vanish is decomposed in \(a(x ,t)= q(x ,t)K(x ,t)\), where \(K(x ,t)\), is a nonnegative smooth function and \(q(x ,t)\) is a positive smooth function. In case all coefficients are smooth functions on \(\mathbb R\times[0,T]\), \(K(x ,t)\) has real roots only and some additional conditions are satisfied, a priori estimates for smooth solutions are obtained. At the end of the paper, there is a discussion on implications of stated results on the well-posedness.
scientific article; zbMATH DE number 5700557
scientific article