The asymptotic behavior of solutions of a semilinear parabolic equation

Minkyu Kwak; Kyong Yu

doi:10.3934/dcds.1996.2.483

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1996, Volume 2, Issue 4: 483-496. Doi: 10.3934/dcds.1996.2.483

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The asymptotic behavior of solutions of a semilinear parabolic equation

Minkyu Kwak^1, and
Kyong Yu¹

1.
Department of Mathematics, Chonnam National University, Kwangju, 500-757

Received: September 30, 1995

Revised: February 29, 1996

Published: September 1996

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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.

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Mathematics Subject Classification: 35B30, 35B40, 35K15.

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