Some well-posed Cauchy problem for second order hyperbolic equations with two independent variables
The authors consider the \(C^\infty\) well-posedness for the second-order hyperbolic equation \(Pu=\partial^2_t u- a(t,x)\partial^2_x u= f\) with two independent variables \((t, x)\). They prove that if the \(C^\infty\) function \(a(t, x)\geq 0\) satisfies \(\partial^p_t a(0,0)\neq 0\) with some \(p\) and if the discriminant of \(a(t, x)\) vanishes of finite order at \(x=0\), the Cauchy problem for \(P\) is \(C^\infty\) well-posed in a neibourhood of the origin. As a generalization of the above result they prove that if \(a(t, x)= \prod^r_{j=1} b_j(t,x)^{m_j}\) with positive integers \(m_j\), where each \(b_j(t, x)\) satisfies \(\partial^{p_j}_t b_j(0,0)\neq 0\), \(b_j(t,x)^{m_j}\geq 0\) and the discriminant of \(b_j\) vanishes of finite order, then the Cauchy problem for \(P\) is \(C^\infty\) well-posed in a neigborhood of the origin.
scientific article; zbMATH DE number 5969043