Global existence of the solutions to nonlinear hyperbolic equations in exterior domains
The authors prove global existence and uniqueness results concerning smooth solutions of initial boundary value problems for the nonlinear wave equation: \[ \partial_ t^ 2u- \sum^{n}_{i,j=1}a_{ij}(u,Du)\partial_{x_ i}\partial_{x_ j}u=b(u,Du);\quad t>0,\quad x\in \Omega \] \[ u(0,x)=u_ 0(x),\quad \partial_ tu(0,x)=u_ 1(x);\quad x\in \Omega;\quad u(t,x)=0;\quad t>0,\quad x\in \partial \Omega \] in an exterior domain \(\Omega\) of a compact set in \({\mathbb{R}}^ n\) and \(Du:=(\partial_ tu,\partial_{x_ 1}u,...,\partial_{x_ n}u)\). Under that assumption that all \(a_{ij}\) and b are infinitely differentiable functions of their \(n+2\) arguments and if there is a \(k\in {\mathbb{N}}\) such that \(| a_{ij}(y)- \delta_{ij}| ={\mathcal O}(| y|^ k)\) and \(| b(y)| ={\mathcal O}(| y|^{k+1})\) where \(\delta_{ij}\) denotes the Kronecker symbol, the existence and uniqueness of a global smooth solution of the afore-mentioned problem is shown for the two cases \(n\geq 7\), \(k=1\) and \(n\geq 4\), \(k\geq 2\). Furthermore, decay estimates for the problem \(\partial_ t^ 2u-\nabla u=f\) with initial and boundary data as in the nonlinear case are given.
scientific article; zbMATH DE number 4198624