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Large time behavior of solutions of Hamilton-Jacobi equations with periodic boundary data. (English) Zbl 1184.35061
The author investigates the large time behavior of viscosity solutions of the Cauchy-Dirichlet problem with periodic boundary data for Hamilton-Jacobi equations \[ \begin{aligned} u_t(x, t)+ H(x, Du(x,t))= 0\quad &\text{in }\Omega\times(0,\infty),\\ u(x, t)= g(x, t)\quad &\text{on }\partial\Omega\times(0,\infty),\\ u(x, 0)= f(x)\quad &\text{in }\Omega,\end{aligned} \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^N\), \(H= H(x, p)\) is a real-valued function (Hamiltonian) on \(\overline\Omega\times\mathbb{R}^N\) which is coercive and convex in the variable \(p,u: \overline\Omega\times [0,\infty)\to\mathbb{R}\) is the unknown function, \(f:\overline\Omega\to\mathbb{R}\) is a given function, \(g: \partial\Omega\times (0,\infty)\to\mathbb{R}\) is a given function which is assumed to be asymptotically periodic with respect to the \(t\) variable. More precisely, the author deals with the time-dependent Dirichlet boundary condition given by \(g\in C(\partial\Omega\times [0,\infty))\) which satisfies \[ g(x,t)- g_1(x,t)\to 0\text{ uniformly on }\partial\Omega\text{ as }t\to\infty \] for some time-periodic function \(g_1\in C(\partial\Omega\times\mathbb{R})\) with period \(T> 0\). Representations for a state constraint asymptotic solution and a periodic asymptotic solution are given.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic 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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