Abstract
The classical theory of differential inequalities generated by Hamilton-Jacobi equations or systems was developed in monographs [138, 140, 195] widely. In particular, uniqueness results for initial problems on the Haar pyramid with nonlinear a priori estimates, were obtained as consequences of suitable comparison theorems for differential inequalities. The authors deal with solutions which admit first order partial derivatives and are totally differentiable on a subset of the boundary of the Haar pyramid. For the Cauchy problem on an unbounded domain the assumption of total differentiability can be omitted provided the comparison function is linear ([21]). In this case the right-hand side of the differential equation satisfies the Lipschitz condition with respect to the unknown function and the uniqueness result is based on the fact that a first-order linear differential inequality can be solved. An interesting result for the global uniqueness of the Cauchy problem, when the right-hand side of the equation satisfies the Hőlder condition can be found in [23]. Mixed inequalities between solutions of first order partial differential equations were considered in [173 – 175].
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© 1999 Springer Science+Business Media Dordrecht
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Kamont, Z. (1999). Initial Problems on the Haar Pyramid. In: Hyperbolic Functional Differential Inequalities and Applications. Mathematics and Its Applications, vol 486. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4635-7_1
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DOI: https://doi.org/10.1007/978-94-011-4635-7_1
Publisher Name: Springer, Dordrecht
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