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

• Date: 2012
• Main Authors: Martynenko, A. V., Tedeev, A. F., Shramenko, V. N., Мартыненко, А. В., Тедеев, А. Ф., Шраменко, В. Н.
• Format: Article
• Language: Russian; English
• Published: Institute of Mathematics, NAS of Ukraine 2012
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/2676
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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 ex...