Weakly hyperbolic equations with time degeneracy in Sobolev spaces
In this paper the author proves existence and uniqueness theorems in Sobolev spaces of time local solutions to the Canchy problem for weakly hyperbolic second order equation as follows \[ u_{tt}- \lambda(t)^2\Delta u=f\bigl(t,x,u_t, \mu_i(t) \partial_{x_i} u\bigr). \] In this equation the regularity loss of derivatives solutions occurs as in the case of linear equations. Under some assumptions on the functions \(\lambda(t),\mu(t)\) and \(f(t,x,u, p,q)\) the author determines precisely the loss of Sobolev regularity of solutions of the above nonlinear Cauchy problem.
scientific article; zbMATH DE number 1370653