Dynamic Study For The Solution Of Time-Periodic Hamilton-Jacobi Equation

Critical value is a very important concept in PDE and dynamic system.Mather used the Calculus of Variations to verify the existence of critical value under the framework of Tonelli(H(x,t,p)is strictly convex on p,superliner growth on p,C2 continuous on(x,t,p)).We intend to prove the existence of critical value under weaker conditions(time 1-periodic Hamiltonian H(x,t,p)is continuous on(x,t,p),linear on t,convex on p and coercive on p).There are two parts in this paper.In the first part,we mainly use the PDE method to prove that there exists a unique 1-periodic viscosity solution of discounted Hamilton-Jacobi equation ?u(x,t)+ut(x,t)+H(x,t,Dxu(x,t)= 0,where the 1-periodic Hamiltonian H(x,t,p)is continuous on(x,t,p),linear on t and coercive on p.We can use sup-convolutions and test-function to prove comparison theorem,and then we can prove final conclusion by Perron method and comparison theorem.In the second part,we constructed a value function.Using the properties of the value function and the properties of the unique time 1-periodic solution obtained in the first part,we can get uniform boundedness of time 1-periodic solution u?(x,t)of discounted Hamilton-Jacobi equation.Then we can prove that there exists a constant c such that u(x,t)-ct is bounded with ergodic approximation method of Partial Differential Method,where u(x,t)is the viscosity solution of Hamilton-Jacobi equation ut+H(x,t,Dxu)=0 and H(x,t,Dxu)is a dual function obtained by the Lengendra transformation of Lagrangian function L(x,t,x).