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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity


Authors: Jong-Shenq Guo, Hirokazu Ninomiya, Masahiko Shimojo and Eiji Yanagida
Journal: Trans. Amer. Math. Soc. 365 (2013), 2447-2467
MSC (2010): Primary 35K57; Secondary 35K40, 35B44
Published electronically: October 9, 2012
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Abstract: This paper is concerned with the Cauchy problem for a system of parabolic equations which is derived from a complex-valued equation with a quadratic nonlinearity. First we show that if the convex hull of the image of initial data does not intersect the positive real axis, then the solution exists globally in time and converges to the trivial steady state. Next, on the one-dimensional space, we provide some solutions with nontrivial imaginary parts that blow up simultaneously. Finally, we consider the case of asymptotically constant initial data and show that, depending on the limit, the solution blows up nonsimultaneously at space infinity or exists globally in time and converges to the trivial steady state.


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Additional Information

Jong-Shenq Guo
Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137, Taiwan
Email: jsguo@mail.tku.edu.tw

Hirokazu Ninomiya
Affiliation: Department of Mathematics, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan
Email: ninomiya@math.meiji.ac.jp

Masahiko Shimojo
Affiliation: Department of Mathematics, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan
Email: shimojotw@gmail.com

Eiji Yanagida
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
Email: yanagida@math.titech.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05797-7
PII: S 0002-9947(2012)05797-7
Keywords: Complex-valued heat equation, parabolic system, convergence, blow-up
Received by editor(s): August 30, 2011
Published electronically: October 9, 2012
Article copyright: © Copyright 2012 American Mathematical Society