Complete quenching phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption
Nguyen Anh Dao, Jesús Ildefonso Díaz Díaz, Huynh Van Kha · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2019
Abstract This paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.