Formation of singularities of Hamilton-Jacobi equation. II
[For part 1 see Proc. Jap. Acad., Ser. A 59, 55-58 (1983; Zbl 0531.35015).] Consider the Cauchy problem for Hamilton-Jacobi equation in two space dimensions: \[ (1)\quad \partial u/\partial t+f(\partial u/\partial x)=0\quad in\quad \{t>0,\quad x\in R^ 2\};\quad (2)\quad u(0,x)=\phi (x)\in {\mathcal S}(R^ 2). \] We assume that f is \(C^{\infty}\) and uniformly convex. It is well known that, even for smooth initial data, the Cauchy problem (1) and (2) does not admit a smooth solution for all t. Therefore we consider a generalized solution. The existence of global generalized solution for (1) and (2) is already established [for example see \textit{S. N. Kruzkov}, Math. USSR Sb. 10, 217-243 (1970); translation from Mat. Sb., Nov. Ser. 81(123), 228-255 (1970; Zbl 0202.112)]. For detailed bibliography, refer to \textit{Stanley H. Benton} [``The Hamilton-Jacobi equations. A global approach'' (1977; Zbl 0418.49001)]. This paper is concerned with the singularities of global generalized solutions.
scientific article; zbMATH DE number 4069398