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Viscosity Solutions of Cauchy Problems for Hamilton-Jacobi Equations

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Abstract

The viscosity solutions of the Cauchy problem u_t+H(x, u, Du)=0,u(x, 0)=u_0(x) in R^N, where H : R_N×R×R^N→R is a continuous function, are considered. We prove an existence and uniqueness theorem under a condition which is more general than the usual one with respect to the u dependence of the Hamiltonian H(x, u, p). This generalized condition would not necessarily guarantee that the stationary problem u+H(x, u, Du)=ƒ in R^N has a continuous viscosity solution. Our main method is based on the technique from nonlinear semigroup theory.The viscosity solutions of the Cauchy problem u_t+H(x, u, Du)=0,u(x, 0)=u_0(x) in R^N, where H : R_N×R×R^N→R is a continuous function, are considered. We prove an existence and uniqueness theorem under a condition which is more general than the usual one with respect to the u dependence of the Hamiltonian H(x, u, p). This generalized condition would not necessarily guarantee that the stationary problem u+H(x, u, Du)=ƒ in R^N has a continuous viscosity solution. Our main method is based on the technique from nonlinear semigroup theory.

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