Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian

Details Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian

2009-03-26

arxiv.org/abs/0903.4679v1

Mathematics

Analysis of PDEs

Scientific paper

We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a bounded domain of $\mathbb{R}^N$. We mainly focus on the superquadratic case ($m>2$) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on $f,$ the initial and boundary data, we connect the problem studied to its associated stationary generalized Dirichlet problem on one hand and to a stationary problem with a state constraint boundary condition on the other hand.

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