For the discounted Hamilton-Jacobi equation,$$\lambda u+H(x,d_x u)=0, \ x \in M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on projected Aubry set. The smoothness of such can be improved under additional hyperbolicity assumptions. As applications, we use to identify maximal global attractor associated conformally symplectic flow and control convergent speed Lax-Oleinik semigroups

This paper is devoted to study the vanishing contact structure problem which a generalization of discount problem. Let $H^\lambda(x,p,u)$ be family Hamiltonians type with parameter $\lambda>0$ and converges $G(x,p)$. For Hamilton-Jacobi equation respect $H^\lambda$, we prove that, under mild assumptions, associated viscosity solution $u^{\lambda}$ specific $u^0$ vanished equation. As applications, give some convergence results for nonlinear

We study the representation formulae for fundamental solutions and viscosity of Hamilton-Jacobi equations contact type. also obtain a vanishing structure result relevant Cauchy problems which can be regarded as an extension to discount problem.

For the discounted Hamilton-Jacobi equation,$$\lambda u+H(x,d_x u)=0, \ x \in M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on projected Aubry set. The smoothness of such can be improved under additional hyperbolicity assumptions. As applications, we use to identify maximal global attractor associated conformally symplectic flow and control convergent speed Lax-Oleinik semigroups

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

We study the representation formulae for fundamental solutions and viscosity of Hamilton-Jacobi equations contact type. also obtain a vanishing structure result relevant Cauchy problems which can be regarded as an extension to discount problem.

We study a multi-objective variational problem of Herglotz' type with cooperative linear coupling. established the associated Euler-Lagrange equations and characteristic system for weakly coupled systems Hamilton-Jacobi equations. also relation value functions this viscosity solutions Comparing to previous work in stochastic frame, approach affords pure deterministic explanation under more general conditions. showed is valid linearly coupling matrix short time.

This paper deals with the long-time behavior of viscosity solutions evolutionary contact Hamilton-Jacobi equations <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w Subscript t Baseline plus upper H left-parenthesis x comma w right-parenthesis equals 0 comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo>...

We are concerned with the stability of viscosity solutions to contact Hamilton-Jacobi equation <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper H left-parenthesis x comma partial-differential Subscript Baseline u right-parenthesis equals 0 element-of M EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 0em" side="left" displaystyle="true">...

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

Vardy has investigated the basic property of sectionalization in conventional trellis his paper. His proofs are based on structure which fails to be generalized linear tail-biting trellis. We already have normal our paper and now we will give another method get an optimal a given could better than one. All properties preserved

We discuss various kinds of representation formulas for the viscosity solutions contact type Hamilton-Jacobi equations by using Herglotz' variational principle.

For the conformally symplectic system \[ \left\{ \begin{aligned} \dot{q}&=H_p(q,p),\quad(q,p)\in T^*\mathbb{T}^n\\ \dot p&=-H_q(q,p)-\lambda p, \quad \lambda>0 \end{aligned} \right. \] with a positive definite Hamiltonian, we discuss variational significance of invariant Lagrangian graphs and explain how KAM torus impacts $W^{1,\infty}-$convergence speed Lax-Oleinik semigroup.

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

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

The main purpose of this paper is to study the global propagation singularities viscosity solution discounted Hamilton-Jacobi equation \begin{equation}\label{eq:discount 1}\tag{HJ$_\lambda$} \lambda v(x)+H( x, Dv(x) )=0 , \quad x\in \mathbb{R}^n. \end{equation} We reduce problem for \eqref{eq:discount 1} into that a time-dependent evolutionary equation. proved propagate along locally Lipschitz singular characteristics which can extend $+\infty$. also obtained homotopy equivalence between set...

&lt;p style='text-indent:20px;'&gt;The main purpose of this paper is to study the global propagation singularities viscosity solution discounted Hamilton-Jacobi equation&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE333"&gt; \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p...