We study a generalized vanishing discount problem for Hamilton–Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against all Mather measures. Specifically, we consider λa(x)u(x)+H(x,Du(x))-Aλ=c_0, with a suitably chosen constant A>0. By appropriately changing the signs of the function a(x) on different static classes associated with H, we show that the maximal viscosity solution converges uniformly as λ→ 0^+ and that all elementary solutions of the stationary equation H(x,Du(x))=c_0 can be selected as limits. This provides the first result for selecting multiple viscosity solutions in vanishing discount problems beyond the usual monotonicity and integral assumptions, as long as a(x) is positive on one static class. Our results highlight the crucial role of static classes in controlling the asymptotic behavior of viscosity solutions. Previously, under usual monotonicity assumptions, only a single solution could be selected (as discussed in ), whereas our approach allows controlled selection of multiple solutions via static class-guided discount coefficients.

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} λ(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $λ(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $cc_0$. In this case, we determine the basin of the stable solution and investigate the long time behavior of the solution semigroup associated to \eqref{hjs1}. In particular, we relate the lowest convergence rate to the integral of $λ$ over Mather measures, which leads to an asymptotic behavior of Mather measures when $c$ goes to infinity. Assume $c\geqslant c_0$ and the equation admits a solution, we classify ergodic Mather measures and locate their distribution in the phase space.

We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume C^3 Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions.

For contact Hamiltonian systems without monotonicity assumption, there is a family of invariant sets {(N) over tildeu} naturally stratified by the solutions u to the corresponding Hamilton-Jacobi equation. Under convergence assumptions of the solution semigroup, we establish the existence of semi-infinite orbits asymptotic to some (N) over tildeu and heteroclinic orbits between (N) over tildeu and (N) over tildev for different solutions u and v by variational methods. We also give verifiable criteria to ensure the convergence assumptions. As a corollary, we give a description of action minimizing orbits of the model system studied in [26]. (c) 2026 Published by Elsevier Inc.

Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider the evolutionary Hamilton-Jacobi equation \begin{equation*} \left\{ \begin{aligned}&\partial_t u(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in M\times(0,+\infty), \\&u(x,0)=\varphi(x), \end{aligned} \right. \end{equation*} where $\varphi\in C(M)$ and the stationary one \begin{equation*} H(x,u(x),\partial_x u(x))=0, \end{equation*} where $H(x,u,p)$ is continuous, convex and coercive in $p$, uniformly Lipschitz in $u$. By introducing a solution semigroup, we provide a representation formula of the viscosity solution of the evolutionary equation. As its applications, we obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Moreover, we prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially, which is different from the one for the Hamilton-Jacobi equation independent of $u$.

In this paper, we study the asymptotic behavior of globally minimizing orbits of contact Hamiltonian systems. Under some assumptions, we prove that the omega-limit set of globally minimizing orbits is contained in the set of semi-static orbits.

We consider the evolutionary Hamilton–Jacobi equation w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, (x,t)∈ M× [0,+∞ ), where M is a compact manifold, H:T^*M×R→R , H=H(x,p,u) satisfies Tonelli conditions in p and the Lipschitz condition in u. This work mainly concerns with the Lyapunov stability (including asymptotic stability, and instability) and uniqueness of stationary viscosity solutions of the equation. A criterion for stability and a criterion for instability are given. We do not utilize auxiliary functions and thus our method is different from the classical Lyapunov’s direct method. We also prove several uniqueness results for stationary viscosity solutions. The Hamiltonian H has no concrete form and it may be non-monotonic in the argument u, where the situation is more complicated than the monotonic case. Several simple but nontrivial examples are provided, including the following equation on the unit circle w_t(x,t)+1/2w^2_x(x,t)-a· w_x(x,t)+(sin x+b)· w(x,t)=0, x∈S, where a, b∈R are parameters. We analyze the stability, and instability of the stationary solution w=0 when parameters vary, and show that w=0 is the unique stationary solution when a=0 , b>1 and a 0 , b⩾ 1 . The sign of the integral of ∂ H/∂ u with respect to the Mather measure of the contact Hamiltonian system generated by H plays an essential role in the proofs of aforementioned results. For this reason, we first develop the Mather and weak KAM theories for contact Hamiltonian systems in this non-monotonic setting. A decomposition theorem of the Mañé set is the main result of this part.

Combing the weak KAM method for contact Hamiltonian systems and the theory of viscosity solutions for Hamilton-Jacobi equations, we study the Lyapunov stability and instability of viscosity solutions for evolutionary contact Hamilton-Jacobi equation in the first part. In the second part, we study the existence and multiplicity of time-periodic solutions.

This paper concerns with the time periodic viscosity solution problem for a class of evolutionary contact Hamilton-Jacobi equations with time independent Hamiltonians on the torus $\mathbb{T}^n$. Under certain suitable assumptions we show that the equation has a non-trivial $T$-periodic viscosity solution if and only if $T\in D$, where $D$ is a dense subset of $[0,+\infty)$. Moreover, we clarify the structure of $D$. As a consequence, we also study the existence of Bohr almost periodic viscosity solutions.

This paper is a continuation of our study of the dynamics of contact Hamiltonian systems in , but without monotonicity assumption. Due to the complexity of general cases, we focus on the behavior of action minimizing orbits. We pick out certain action minimizing invariant sets {𝒩_u} in the phase space naturally stratified by solutions u to the corresponding Hamilton-Jacobi equation. Using an extension of characteristic method, we establish the existence of semi-infinite orbits that is asymptotic to some 𝒩_u and heteroclinic orbits between 𝒩_u and 𝒩_v for two different solutions u and v.

We study the asymptotic behavior of solutions of an equation of the form * G(x, D_x u,λ u(x)) = c_0 in M on a closed Riemannian manifold M, where G∈ C(T^*M×ℝ) is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and c_0 is the critical constant associated with the Hamiltonian H:=G(·,·,0). By assuming that ∂_u G(·,·,0) satisfies a positivity condition of integral type on the Mather set of H, we prove that any equi-bounded family of solutions of () uniformly converges to a distinguished critical solution u_0 as λ→ 0^+. We furthermore show that any other possible family of solutions uniformly diverges to +∞ or -∞. We then look into the linear case G(x,p,u):=a(x)u + H(x,p) and prove that the family (u_λ)_λ∈ (0,λ_0) of maximal solutions to () is well defined and equi-bounded for λ_0>0 small enough. When a changes sign and enjoys a stronger localized positivity assumption, we show that equation () does admit other solutions too, and that they all uniformly diverge to -∞ as λ→ 0^+. This is the first time that converging and diverging families of solutions are shown to coexist in such a generality.

We are concerned with the stability of viscosity solutions to contact Hamilton-Jacobi equation H ( x , ∂ x u ( x ) , u ( x ) ) = 0 , x ∈ M , \begin{align*} H(x,\partial _x u(x),u(x))=0, \quad x\in M, \end{align*} where H = H ( x , p , u ) H=H(x,p,u) satisfies Tonelli conditions. We study the relationship between Lyapunov stability of viscosity solutions and the structure of the set of weak KAM solutions to the contact Hamilton-Jacobi equation.

We discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation h(x, d(x)u) + ?(x)u = c, x ? M, where M is a closed manifold and ? : M-+ R changes signs on M, via nonlinear semigroup method. It turns out that a bifurcation phenomenon occurs when the parameter c strides over some critical value. As an application of the main result, we analyse the structure of the set of viscosity solutions of an one-dimensional example in detail.

Assume H = H(x, p, u) with (x, p) is an element of T*M and u is an element of S, is smooth and satisfies Tonelli conditions in p, Lipschitz continuity condition in u, where M is a compact connected smooth manifold without boundary. We find a compact interval C such that equationH(x, partial differential xu(x), u(x)) = chas solutions if and only if c is an element of C. We also study the long-time behavior of the unique viscosity solution uc of partial differential tu(x, t) + H(x, partial differential xu(x, t), u(x, t)) = c, u(x, 0) = phi(x) is an element of C(M, R).If c is an element of C, uc is bounded by a constant independent of c and Lipschitz with respect to the argument x with a Lipschitz constant independent of c and phi. If c is an element of/ C, then the long-time average of uc can be characterized by a function c 7 -> rho(c) which admits a modulus of continuity. We obtain these results by analyzing properties of a kind of one-parameter semigroups of operators. All the aforementioned results show the fundamental difference between Hamilton Jacobi equations with Hamiltonians H(x, p, u) and over line H(x, p).

In this article we develop an analogue of Aubry-Mather theory for time periodic dissipative equation {ẋ =∂ _p H(x,p,t), ṗ =-∂ _x H(x,p,t)-f(t)p . with (x,p,t)∈ T^*M×𝕋 (compact manifold M without boundary). We discuss the asymptotic behaviors of viscosity solutions for the associated Hamilton-Jacobi equation ∂ _t u+f(t)u+H(x,∂ _x u,t)=0, (x,t)∈ M×𝕋 w.r.t. certain parameters, and analyze the meanings in controlling the global dynamics. We also discuss the prospect of applying our conclusions to many physical models.

We are concerned with the existence and multiplicity of nontrivial time periodic viscosity solutions to partial differential tw(x, t) + H(x, partial differential xw(x, t), w(x, t)) = 0, (x, t) E S x [0, +oo), 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 partial differential tw(x, t) + H(x, partial differential xw(x, t), lambda w(x, t)) = 0 where lambda is a parameter.(c) 2023 Elsevier Masson SAS. All rights reserved.

This paper aims at providing a dynamic perspective of viscosity subsolutions to contact Hamiltonian–Jacobi equations H( x, ∂ x u, u) = 0 on connected, closed manifold M. Based on implicit variational principles, we focus on the connection between positive invariant sets under Hamiltonian flow and viscosity subsolutions. We apply the connection to give a new necessary and sufficient condition for the existence of viscosity solutions of the stationary equation from the dynamical view. Finally, we discuss the multiplicity of viscosity solutions and give several illustrative examples.

Suppose that $H(x,u,p)$ is strictly decreasing in $u$ and satisfies Tonelli conditions in $p$. We show that each viscosity solution of $H(x,u,u_x)=0$ can be reached by many viscosity solutions of $$ w_t+H(x,w,w_x)=0, $$ in a finite time.

Abstract We study the Hamilton–Jacobi equations in M and in where the Hamiltonian 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.