Existence of a maximal solution of singular parabolic equations with absorptions: quenching phenomenon and the instantaneous shrinking phenomenon

TL;DR

This paper proves the existence of maximal solutions for certain singular degenerate parabolic equations, analyzing phenomena like quenching, finite speed of propagation, and instantaneous shrinking of support.

Contribution

It establishes sharp gradient estimates and extends results from Dirichlet to Cauchy problems for these equations, highlighting new behaviors such as support shrinking.

Findings

Existence of maximal solutions under specified conditions

Demonstration of quenching and finite speed of propagation

Identification of instantaneous support shrinking phenomenon

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 gradient estimate of |u_x|. Besides, we investigate the behaviors of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem will be extended to the associated Cauchy problem. In addition, we show that the phenomenon of the instantaneous shrinking of compact support of the nonnegative solutions occurs if f satisfies some growth condition.