Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian
This paper concerns the asymptotic behavior as \(t \to \infty\) of viscosity solutions of the degenerate diffusion equation with the infinity-Laplacian \(u_t=\Delta_\infty u\), where \[ \Delta_\infty u= \sum_{i,j}\frac {\partial u}{\partial x_i}\frac {\partial u}{\partial x_j}\frac {\partial^2u}{\partial x_i\partial x_j}. \] Both the Cauchy problem for initial data having a compact support and the Cauchy-Dirichlet problem for bounded domains are dealt with. For the Cauchy problem and the Cauchy-Dirichlet problem under the homogeneous Dirichlet boundary condition, the optimal decay rates for bounded viscosity solutions are obtained and, in particular, it is shown that the rates are independent of the dimension of the Euclidean space. For the Cauchy-Dirichlet problem under the inhomogeneous Dirichlet boundary condition, it is shown that the solution converges to the unique stationary solution as \(t \to \infty\). The authors use the comparison principle and several kinds of barriers. For the comparison principle and the existence theorem for viscosity solutions, see \textit{G. Akagi} and \textit{K. Suzuki} [Calc. Var. Partial Differ. Equ. 31, No. 4, 457--471 (2008; Zbl 1139.35064)].
scientific article; zbMATH DE number 5533131
scientific article