# A note on critical point and blow-up rates for singular and degenerate parabolic equations

Document Type : Research Paper

Authors

1 College of Science‎, ‎China University of Petroleum‎, ‎Qingdao 266580‎, ‎Shandong Province‎, ‎P.R‎. ‎China

2 College of Science, China University of PetroleumCollege of Science‎, ‎China University of Petroleum‎, ‎Qingdao 266580‎, ‎Shandong Province‎, ‎P.R‎. ‎China

Abstract

In this paper, we consider singular and degenerate parabolic equations
$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quad v_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to null
Dirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determined. Additionally, we obtain blow-up rates and sets for the solutions. The singular rates for the derivation of the solutions are given.

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### History

• Receive Date: 29 April 2014
• Revise Date: 21 July 2014
• Accept Date: 21 July 2014