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Convergence to a single steady state is shown for non-negative and radiallysymmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneousDirichlet 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 afirst step, the radially symmetric and non-increasing stationary solutions arecharacterized.

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