Study On Convergence In Viscosity Solution Of Contact Hamilton-Jacobi Equation

The systems without energy dissipation are generally called Hamiltonian systems,the corresponding function is H(x,Du(x))=0.However,many physical and dynamical systems with energy dissipation are contact Hamiltonian systems whose function is H(x,u(x),Du(x))=0.The difference between them is that the latter system contains u(x).The relationship between them is similar to which between contact geometry and symplectic geometry.In recent years,contact Hamiltonian systems have been widely used in non-conservative mechanical systems,dissipative mechanical systems,microdynamics and balanced statistical mechanics systems.The study of the long time asymptotic behavior of the viscosity solution of the Hamilton-Jacobi equation is an important research direction of the viscosity solution theory.There are weak KAM theory based on variational method and PDE method for its research.The article mainly studies the long time asymptotic behavior of the viscosity solution of the discounted Hamilton-Jacobi equation,the discounted Hamilton-Jacobi(H-J)equation is a special form of contact H-J equation.Because its solution can be expressed explicitly,we can obtain some more intuitive and profound conclusions.The article mainly contains the following two parts.In the first part,we study an expression of the viscosity solution u?(x,t)for the evolutionary discounted H-J equation in non-compact space under certain conditions first.Then we discuss the relationship between the convergence of the viscosity solution u?(x,t)of a specific discounted H-J equation with ?>0 in non-compact space and its initial value in different cases as t?+?.In the second part,we provide a counterexample about the convergence of the viscosity solution of the evolutionary discounted H-J equation in non-compact space as t?+?.We will give the counterexample in the Tonelli framework,in contrast to what happens in the compact space,where the viscosity solutions of the evolutionary H-J equation and evolutionary contact H-J equation are convergent as t?+?.The construction of the counterexample is based on a nonautonomous H-J equation.For the case of autonomous system,the discussion in the first part shows that u?(x,t)is convergent as t?+? if u?(x,t)is finite.