On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with source in the case where the initial function slowly vanishes

  • A. V. Martynenko Луган. нац. ун-т им. Т. Шевченко
  • A. F. Tedeev Ин-т прикл. математики и механики НАН Украины, Донецк
  • V. N. Shramenko Нац. техн. ун-т Украины „КПИ”, Киев

Abstract

We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form $$u_t = \text{div}(\rho(x)u^{m-1}|Du|^{\lambda-1}Du) + u ^p $$ in the case where initial function decreases slowly to zero as $|x| \rightarrow \infty$. We establish conditions for the existence and nonexistence of a global-in-time solution, which substantially depend on the behavior of the initial data as $|x| \rightarrow \infty$. In the case of global solvability, we obtain an exact estimate of a solution for large times.
Published
25.11.2012
How to Cite
MartynenkoA. V., TedeevA. F., and ShramenkoV. N. “On the Behavior of Solutions of the Cauchy Problem for a Degenerate Parabolic Equation With Source in the Case Where the Initial Function Slowly Vanishes”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 11, Nov. 2012, pp. 1500-15, http://umj.imath.kiev.ua/index.php/umj/article/view/2676.
Section
Research articles