Grow up and slow decay in the critical Sobolev case

Marek Fila; John R. King; Marek Fila; John R. King

doi:10.3934/nhm.2012.7.661

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 661-671. doi: 10.3934/nhm.2012.7.661

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Grow up and slow decay in the critical Sobolev case

Marek Fila ^{1
,},
John R. King ^{2
,}

1.
Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava
2.
Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD

Received: 01 January 2012 Revised: 01 May 2012
Primary: 35K57; Secondary: 35B40, 35B33.

We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as $t→∞$. The rates of grow up or decay are computed formally using matched asymptotics.
- Semilinear heat equation,
- asymptotic behaviour,
- Sobolev exponent
Citation: Marek Fila, John R. King. Grow up and slow decay in the critical Sobolev case[J]. Networks and Heterogeneous Media, 2012, 7(4): 661-671. doi: 10.3934/nhm.2012.7.661

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Abstract

We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as $t→∞$. The rates of grow up or decay are computed formally using matched asymptotics.

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