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Uniqueness for discontinuous ODE and conservation laws. (English) Zbl 0948.34006
The paper is concerned with the Cauchy problem $x'= f(t,x),\quad x(0)=\overline x,\tag{1}$ where $$f:[0,T]\times \mathbb{R}\to\mathbb{R}$$ is a measurable function and the solutions are understood in Carathéodory sense. The main result ensures the existence of a unique solution to (1) under the following assumptions:
(i) For every point $$(\overline t,\overline x)$$ there exists a slope $$\lambda(\overline t,\overline x)$$ such that the function $$f$$ is constant along the segment $$s(\overline t,\overline x)$$, with $$s(\overline t,\overline x)= \{(t,x): t\in(0,\overline t)$$, $$x=\overline x+(t-\overline t)\lambda(\overline t,\overline x)\}$$.
(ii) There exist disjoint intervals $$[a,b]$$ and $$[c,d]$$ such that $$f(t,x)\in [a,b]$$ and $$\lambda(\overline t,\overline x)\in [c,d]$$ for all $$(\overline t,\overline x)\in[0, T]\times\mathbb{R}$$.

MSC:
 34A36 Discontinuous ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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References:
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