Monotonicity in Time and Stationary Solutions for a Quasilinear Heat Equation with Source
We consider the Cauchy problem for the quasilinear parabolic heat equation with source in , , are fixed constants, with nonnegative bounded symmetric initial function. Two properties of monotone behaviors of the solution for are investigated. 1. Monotonicity of large solutions: there exists a constant such that if for some , then for all . 2. does not decrease in . It is shown that sufficient conditions for these properties are quite different for five cases: i) . ii) , iii) , iv) , v) , where for ( for ) is the critical Sobolev exponent.
1991 Mathematics Subject Classification: 35K55, 35K60.