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We are concerned with the existence and multiplicity of nontrivial time periodic viscosity solutions to ∂ t w (x , t) + H (x , ∂ x w (x , t) , w (x , t)) = 0 , (x , t) ∈ S × [ 0 , + ∞) , where S is the unit circle and H = H (x , p , u) satisfies Tonelli conditions with respect to the argument p and is strictly decreasing in the argument u. We also study the long time behavior of viscosity solutions of the Cauchy problem for the above contact Hamilton-Jacobi equation. It is shown that for a class of initial data the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. As an application of the existence result we analyze a bifurcation phenomenon for ∂ t w (x , t) + H (x , ∂ x w (x , t) , λ w (x , t)) = 0 where λ is a parameter. [ABSTRACT FROM AUTHOR]

We study the Hamilton–Jacobi equations H (x , D u , u) = 0 in M and ∂ u / ∂ t + H (x , D x u , u) = 0 in M × (0 , ∞) , where the Hamiltonian H = H (x , p , u) depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem. [ABSTRACT FROM AUTHOR]

First, we provide a necessary and sufficient condition of the existence of viscosity solutions of the nonlinear first order PDE F (x , u , D u) = 0 , x ∈ M , under which we prove the compactness of the set of all viscosity solutions. Here, F (x , u , p) satisfies Tonelli conditions with respect to the argument p and − λ ≤ ∂ F ∂ u < 0 for some λ > 0 , and M is a compact manifold without boundary. Second, we study the long time behavior of viscosity solutions of the Cauchy problem for w t + F (x , w , w x) = 0 , (x , t) ∈ M × (0 , + ∞) , from the weak KAM point of view. The dynamical methods developed in [13–15] play an essential role in this paper. [ABSTRACT FROM AUTHOR]

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity. [ABSTRACT FROM AUTHOR]

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in the paper (P. Cannarsa, W. Cheng, K. Wang, J. Yan, 2019 [17]) in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation D t u (t , x) + H (t , x , D x u (t , x) , u (t , x)) = 0 and study the related Lax-Oleinik evolution. [ABSTRACT FROM AUTHOR]