Breakdown of \(C^1\) solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity
The author of this interesting paper studies the first order quasilinear hyperbolic system \({\partial u\over \partial t}+A(u){\partial u\over \partial x}=0\), where \(u=(u_1,\dots ,u_n)^T\) is the unknown vector function of \((t,x)\) and \(A(u)\) is an \(n\times n\) matrix with suitably smooth elements \(a_{ij}\) (\(i,j=1,2,\dots ,n\)). It is given the initial data at \(t=0\), \(u(0,x)=\varepsilon \psi (x)\), where \(\varepsilon >0\) is a small parameter and \(\psi \in C^1\) is a vector function satisfying \(\sup_{x\in {\mathbb R}}\{(1+|x|)^{1+\mu }|\psi '(x)|\}<\infty \) (\(\mu >0\)). It is assumed that in a neighbourhood of \(u=0\) the eigenvalues satisfy the relations \(\lambda (u)\triangleq \lambda _1(u)\equiv \cdots \equiv \lambda _p(u)<\lambda _{p+1}(u)<\cdots \lambda _n(u)\), where \(1\leq p\leq n\). In the case \(p=1\), the system is strictly hyperbolic, and when \(p>1\) it is non-strictly hyperbolic system with characteristics with constant multiplicity \(p>1\). It is supposed that the system under consideration possesses linearly degenerate (only at \(u=0\)) characteristics with constant multiplicity. The existence of blowing-up \(C^1\) solution to the Cauchy problem for the considered system with small \(C^1\) initial data with certain decaying properties is discussed. The author introduces a nonempty set of indices \(G\) such that when \(i\in G\) then \(\lambda_i(u)\) is genuinely nonlinear simple characteristic. The main result is that if there exists at least an \(m\in G\) and a real number \(x_0\) such that \((\nabla \lambda _m(0)r_m(0))l_m(0)\psi '(x_0)< 0\) (\(l_m(u)\) is the left eigenvector, and \(r_m(u)\) is the right eigenvector corresponding to \(\lambda_m(u)\)) then there exists \(\varepsilon_0>0\) so small that for any fixed \(\varepsilon \in (0,\varepsilon_0]\), the first order partial derivative \(u_x\) of the \(C^1\) solution \(u=u(t,x)\) to Cauchy problem must blow up in a finite time and the life-span \(\widetilde T (\varepsilon)\) of \(u=u(t,x)\) satisfies \(\lim_{\varepsilon \to 0}(\varepsilon \widetilde T (\varepsilon))=M_0\), where \(M_0\) is certain number determined by author.
scientific article; zbMATH DE number 1990021