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Formation of singularities of Hamilton-Jacobi equation. II. (English) Zbl 0655.35009
[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 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 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.

MSC:
35F25 Initial value problems for nonlinear first-order PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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