On a local existence theorem for quasilinear hyperbolic mixed problems with Neumann type boundary conditions
Let S be a sufficiently smooth and compact hypersurface in \(R^ n\) and \(\Omega\) be its interior or exterior domain. Consider the problem for the system of the equations \[ \partial^ 2_ tu_ a- \sum^{n}_{i=1}\partial_{x_ i}(A_{ia}\quad ((t,x,u,\nabla u))+\phi_ a(t,x,u,\partial_ tu,\nabla u)=f_ a(t,x) \] in [0,T]\(\times \Omega\) with initial conditions \[ u_ a(0,x)=u^ 0_ a(x),\quad \partial_ tu_ a(0,x)=u^ 1_ a(x),\quad (x\in \Omega) \] and boundary conditions \[ \sum^{n}_{i=1}\nu_ i(x)A_{ia}(t,x,u,\nabla u)+\psi_ a(t,x,u)=g_ a(t,x),\quad ((t,x)\in [0,T]\times S), \] for \(a=1,...,m\), where \(u=(u_ 1,...,u_ m)\), \(\nabla =(\partial x_ 1,...,\partial x_ n)\), \(\partial_ t=\partial /\partial t\), \(\partial_{x_ i}=\partial /\partial x_ i\) and \((\nu_ 1(x),...,\nu_ n(x))\) is the outer unit normal of S. Under certain assumptions on \(A_{ia}\), \(\phi_ a\) and \(\psi_ a\), the authors obtain the local existence in time of classical solutions for the problem.
homogeneous, isotropic, hyperelastic material
scientific article; zbMATH DE number 3969197