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The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations. (English) Zbl 1171.35332
The large-time behavior of solutions of the Cauchy-Dirichlet problem for the Hamilton-Jacobi equation \[ \begin{cases} u_t(x,t)+ H(x, Du(x,t)= 0\quad &\text{in }Q,\\ u(x,t)= f(x)\quad &\text{in }\Omega\times \{0\},\\ u(x,t)= g(x)\quad &\text{on }\partial\Omega\times (0,\infty)\end{cases}\tag{1} \] is investigated. Here \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(Q= \Omega\times (0,\infty)\), the Hamiltonian \(H= H(x,p)\) is a real-valued function on \(\overline\Omega\times \mathbb{R}^n\) which is coercive and convex in the variable \(p\) and \(f:\overline\Omega\to \mathbb{R}\), \(g: \partial\Omega\to\mathbb{R}\) are given functions. This work deals only with the viscosity solutions of the Hamilton-Jacobi equations. In recent years, many researchers have investigated the large-time behavior of the solution \(u(x,t)\) of (1) as \(t\to\infty\).
In this paper, the author does not assume the compatibility condition on the initial and boundary data \(f\), \(g\). This gives a viewpoint which unifies the state constraint and Dirichlet boundary conditions. In the last part of the paper, general convergence results for viscosity solutions of (1) by using the Aubry-Mather theory are established and representation formulas for asymptotic solutions are given.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35F25 Initial value problems for nonlinear first-order PDEs
35F30 Boundary value problems for nonlinear first-order PDEs
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