International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 8, Pages 377-405
Asymptotics for critical nonconvective type equations
1Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
2Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, Morelia CP 58120, Michoacán, Mexico
3Instituto de Matemáticas, Universidad Nacional Autonoma de México (UNAM), Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico
Received 18 March 2003
Copyright © 2004 Nakao Hayashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study large-time asymptotic behavior of solutions to the Cauchy
problem for a model of nonlinear dissipative evolution equation. The linear part is a pseudodifferential operator and the nonlinearity is a cubic pseudodifferential operator defined by means of the inverse Fourier transformation and represented by
bilinear and trilinear forms with respect to the direct Fourier transform of the dependent variable. We consider nonconvective type nonlinearity, that is, we suppose that the total mass of the nonlinear term does not vanish. We consider the initial data,
which have a nonzero total mass and belong to the weighted Sobolev space with a sufficiently small norm. Then we give the main term of the large-time asymptotics of solutions in the critical case. The time decay rate have an additional logarithmic
correction in comparison with the corresponding linear case.