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Journal of the London Mathematical Society 2005 72(3):663-688; doi:10.1112/S0024610705006782
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© The London Mathematical Society

Asymptotics for Fractional Nonlinear Heat Equations

Nakao Hayashi, Elena I. Kaikina and Pavel I. Naumkin

Department of Mathematics, Graduate School of Science, Osaka University Osaka, Toyonaka 560-0043, Japan nhayashi{at}math.wani.osaka-u.ac.jp
Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia CP 58120, Morelia, Michoacán, Mexico
Instituto de Matemáticas UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico pavelni{at}matmor.unam.mx

Received 16 March 2004. Revision received 14 October 2004.

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian

Formula

where {alpha} isin (0,2), with critical {sigma} = {alpha}/n and sub-critical {sigma} isin (0,{alpha}/n) powers of the nonlinearity. Let u0isin L1,a{cap} L{infty} {cap} C, u0(x) ≥ 0 in Rn, {theta} = Formula. The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution u isin C([0,{infty}); L{infty} {cap} L1,a{cap} C) and the large time asymptotics are obtained.


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