`a`
Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

The asymptotic behavior of solutions of a semilinear parabolic equation

Pages: 483 - 496, Volume 2, Issue 4, October 1996

doi:10.3934/dcds.1996.2.483       Abstract        Full Text (197.8K)       Related Articles

Minkyu Kwak - Department of Mathematics, Chonnam National University, Kwangju, 500-757, South Korea (email)
Kyong Yu - Department of Mathematics, Chonnam National University, Kwangju, 500-757, South Korea (email)

Abstract: We study the long-time behavior of solutions of the Cauchy problem

$u_t=\Delta u - (u^q)_y- u^p, \quad p, q >1,$

defined in the domain $Q=\{ (\x, t): \x=(x, y) \in \mathbf{R}^{N-1} \times \mathbf{R}, t >0 \}$ with nonnegative initial data in $L^1( \mathbf{R}^N)$. We completely classify the asymptotic profiles of solutions as $t \to \infty$ according to the parameters $p$ and $q$. We use rescaling transformations and a priori estimates.

Keywords:  Asymptotic behaviour, a semilinear heat equation, self-similar solution, singular solution.
Mathematics Subject Classification:  35B30, 35B40, 35K15.

Received: October 1995;      Revised: March 1996;      Published: July 1996.