Remarks on the blow-up rate of classical solutions to quasilinear multidimensional hyperbolic systems
Consider the following symmetric hyperbolic system: \[ L(u)=A_0(x,t,u)\partial _tu+\sum_{j=1}^nA_j(x,t,u)\partial_iu+B(x,t,u)=0, \quad(x,t)\in \mathbb{R}^{n+1}, \tag{1} \] where \(u\in \mathbb{R}^{N}\), \(A_j\) and \(B\) are real and smooth, \(A_j\), \(j=0,\dots,n\) are symmetric, \(A_0\) is positive defined. By \(V\) denote a neighborhood of the point \(M_0(x_0,T)\) in \(\{t\leq T\}\). Theorem: Let (1) be a symmetric hyperbolic system. There exists a function \(C(x,t,u)>0\) such that, if \(u\in C(V)\) is a smooth solution of \(L(u)=0\) which is in \(C^{\infty}\) for \(t<T\), and \[ \lim_{(x,t)\to M_0}(T-t)\|\nabla u(x,t)\|\leq C(x_0,T,M_0), \tag{2} \] then \(u\in C^{\infty}\) in some neighborhood of \(M_0\) in \(\{t\leq T\}\). The infimum of all such numbers \(C(x_0,T,u_0)\) \((u_0=u(M_0))\) from (2) corresponding to all possible \(u\) is called the best Minimal Growth (MG) constant for (1) in \(M_0\). If \(B=0\) and \(A_j\) are independent of \((x,t)\) the author proves that \[ C(x_0,T,u_0)\leq \inf_{(j,\xi)\in E} \|r_j(u_0,\xi)\|, \] where \(r_j(v,\xi)\) are the eigenvectors of the matrix \(A^{-1}(v)\sum A_i(v)\xi\), \(|\xi|=1\) and \[ E=\{(j,\xi)\mid (r_j(\nabla_v\lambda_j), (u_0,\xi))\neq 0,\;(r_j(v,\xi),\nabla_{v}\lambda _j (v,\xi))=1\}. \] As a consequence of this theorem for the scalar equation \(\partial _t u+\sum _{j=1}^n A_j(u) \partial _j u =0\) the author obtains \(C(u_0)=\|A'(u_0)\|^{-1}\), \(A(u)=(A_1(u),\dots, A_n(u))\). Also the author computes the (MG) for the \(2\times 2\) diagonal systems of the form \(\partial _t u+\Lambda(u)\partial _x u=0\).
scientific article; zbMATH DE number 1546940