Convergence of the Viscosity Solution of Time-periodic Discounted Hamilton-Jacobi Equation
The discounted Hamilton-Jacobi equation is a special form of the contact Hamilton-Jacobi equation.Hence,the study of the discounted Hamilton-Jacobi equation is more intuitionistic.In this paper,we mainly study the convergence of the viscosity solutions of time-periodic discounted Hamilton-Jacobi equation by variational method.For the evolutionary Hamilton-Jacobi equation ut+λu+H(x,Dxu,t)=0,under the assumptions that H(x,p,t)is a Tonelli Hamiltonian,and H(x,p,t)is 1-period in the variable t.we can get the 1-period solution ūλ(x,t)and the 1-periodic solution is unique under the given conditions.Further more,we prove limn→∞ uλ(x,t+n)=ūλ(x,t),where uλ(x,t+n)is the viscosity solution of ut+λu+H(x,Dxu,t)=0.At last,we explain the conclusion with a specific example of Hamilton-Jacobi equation.
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