Local existence theorems and blow-up of solutions for quasi-linear hyperbolic Cauchy problems in a domain with characteristic boundary generated by initial data
The quasilinear Cauchy problem: \[ (1)\quad A(t,x;\phi^{m- 1}u;\partial)u\equiv \partial^ m_ tu+\sum_{j+| v| \leq m,j\leq m-1}a_{jv}(t,x;\phi^{m-1}u)\partial^ j_ t\partial^ v_ xu=f(t,x;\partial^{m-1}u) \] in (0,T)\(\times \Omega\), \[ \partial^ j_ tu=\phi_ j(x),\quad j=0,1,...,m-1\text{ on } \{t=0\}\times \Omega, \] where the simple notations: \[ \partial^ ku=\{\partial^ j_ t\partial^ v_ xu| \quad j:| v| =k\},\quad \phi^ ku=\{\partial^ j_ t\partial^ v_ xu| \quad j+| v| \leq k\}=(u,\partial u,...,\partial^ ku) \] are used, is studied. It is proved that if initial data are ``degenerate-structure generating'', there is a unique classical solution in a small time of (1). By proving existence of the solution a standard iterative method is used. Some examples of equations with solutions blowing-up on the boundary are given.
scientific article; zbMATH DE number 4173513