Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation

Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the $p$-Laplacian operator, $p\ge 2$, and the source term a power of the norm of the gradient of $u$. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.

[1]  P. Laurençot.  Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions , 2007 .

[2]  Philippe Souplet,et al.  Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions , 2002, Differential and Integral Equations.

[3]  Philippe Souplet,et al.  Global solutions of inhomogeneous Hamilton-Jacobi equations , 2006 .

[4]  G. Barles,et al.  EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM* , 2001 .

[5]  Robert Kersner,et al.  The Cauchy problem for u t = ?u+|? u| q , 2003 .

[6]  H. Ishii,et al.  Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains , 1991 .

[7]  Philippe Laurençot,et al.  Decay estimates for a viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions , 2007, Asymptot. Anal..

[8]  C. Stinner,et al.  Convergence to steady states in a viscous Hamilton–Jacobi equation with degenerate diffusion , 2010 .

[9]  G. Barles,et al.  Exit Time Problems in Optimal Control and Vanishing Viscosity Method , 1988 .

[10]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[11]  P. Souplet,et al.  Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena , 2004 .

[12]  G. Barles,et al.  Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non lipschitz nonlinearity , 1992 .

[13]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[14]  J. Roquejoffre,et al.  Remarks on the long time behaviour of the solutions of hamilton-jacobi equations , 1999 .

[15]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[16]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[17]  B. Gilding.  The Cauchy problem for ut=?u+|?u|qut=?u+|?u|q, large-time behaviour , 2003 .

[18]  Guy Barles,et al.  On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations , 2004 .

[19]  Jean-Michel Roquejoffre,et al.  Convergence to steady states or periodic solutions in a class of Hamilton–Jacobi equations , 2001 .

[20]  Thierry Tabet Tchamba.  Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian , 2009, Asymptot. Anal..