Well-posedness in \(C^\infty\) for some weakly hyperbolic equations
The authors study the well-posedness of the Cauchy problem for the third-order homogeneous equation \[ D^3_tu+ a(t)D^2_t D_xu+b(t) D^2_xD_t u+c(t)D^3_x u=f. \] The coefficients are assumed to vanish at \(t=0\), but \(b(t)\) and \(c(t)\) do not vanish simultaneously to infinite order; moreover, the equation is supposed to be weakly hyperbolic. A necessary and sufficient condition is then given for well-posedness in \(H^\infty\). As an introduction to their result, the authors give a detailed survey on weakly hyperbolic equations with variable multiplicity. An extension of the result to higher-order equations is also proposed at the end of the paper.
necessary and sufficient condition for well-posedness
scientific article; zbMATH DE number 1530988