Blow-up mechanism of classical solutions to quasilinear hyperbolic systems
The Cauchy problem for the first order hyperbolic system \[ \frac{\partial u} {\partial t} + A(u)\frac{\partial u} {\partial x} = 0, \qquad u(x,0)= \varphi(x), \tag{1} \] where \( u =(u_1 ,u_2,\dots,u_n)^T \) is unknown vector function of \((t,x)\), \(A(u)\) is an \( n \times n \) matrix with suitably smooth elements \( a(u) \) \((i,j=1,\dots,n)\) and \( \varphi (x)=(\varphi_1(x),\dots,\varphi _n(x))^T \) is a \( C^1 \) vector function of \( x \) is discussed. It is supposed that in a neighborhood of \( u = 0 \) system is strictly hyperbolic: \( A(u)\) has \(n\) distinct real eigenvalues \( \lambda _1(u)< \lambda _2(u)<\cdots<\lambda_n(u)\). The blowup phenomenon of the classical solutions to the problem (1), particularly, the geometric blowup mechanism in the noncritical case for general initial data satisfying \[ \sup_{x \in\mathbb R}\{(1+| x|^{1+\mu}(|\varphi(x)|+ |\varphi'(x)|))\}\ll1 \] is studied. The estimate of lifespan and under the additional hypothesis sharp estimate of the \( C^1 \) solution is obtained. It is proved that the envelope of the same family of characteristics which yields the blowup of classical solutions to the problem (1).
scientific article; zbMATH DE number 5161908
scientific article