RMC - Vol. 4, Num. 2, 3 - V. A. Galaktionov/S. A. Posashkov

Full paper in PDF:
$% V. A. Galaktionov and S. A. Posashkov, Monotonicity in time and stationary solutions for a quasilinear heat equation with source, Rev. Mat. Univ. Complut. Madrid 4 (1991), no. 2, 3, 251–264.%$

Monotonicity in Time and Stationary Solutions for a Quasilinear Heat Equation with Source

Victor A. GALAKTIONOV and Sergey A. POSASHKOV

Keldysh Institute of Applied Mathematics USSR
Miusskaya Sq. 4
125047 Moscow — USSR

Received: September 3, 1990

ABSTRACT

We consider the Cauchy problem for the quasilinear parabolic heat equation with source $ut = Da+1+ ub$ in $RN × (0,T)$ , $s > 0$ , $b > 1$ are fixed constants, with nonnegative bounded symmetric initial function. Two properties of monotone behaviors of the solution $u(|x|,t)$ for $x= 0$ are investigated. 1. Monotonicity of large solutions: there exists a constant $Mc >0$ such that if $u(0,t0)= Mc$ for some $t0 (- [0,T)$ , then $ut(0,t)> 0$ for all $t (- [t0,T)$ . 2. $u(0,t)$ does not decrease in $(0,T)$ . It is shown that sufficient conditions for these properties are quite different for five cases: i) $1< b < a+ 1$ . ii) $b = a+ 1$ , iii) $s+ 1< b < b*$ , iv) $b = b*$ , v) $b >b*$ , where $b = (a+ 1)(N + 2)/(N - 2) *$ for $N > 2$ ( $b = 8 *$ for $N = 1,2$ ) is the critical Sobolev exponent.

1991 Mathematics Subject Classification: 35K55, 35K60.