Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity

Jean Bricmont Antti Kupiainen Jack Xin

Analysis of PDEs mathscidoc:1912.43846

Journal of differential equations, 130, (1), 9-35, 1996.9
We study the initial value problem of the thermal-diffusive combustion systemu 1, t= u 1, x, x u 1 u 2 2, u 2, t= du 2, xx+ u 1 u 2 2, x R 1, for non-negative spatially decaying initial data of arbitrary size and for any positive constantd. We show that if the initial data decay to zero sufficiently fast at infinity, then the solution (u 1, u 2) converges to a self-similar solution of the reduced systemu 1, t= u 1, xx u 1 u 2 2, u 2, t= du 2, xx, in the large time limit. In particular, u 1 decays to zero like O (t 1/2 ), where> 0 is an anomalous exponent depending on the initial data, andu 2 decays to zero with normal rate O (t 1/2). The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self-similarity of the solutions in the large time limit.
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@inproceedings{jean1996global,
  title={Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity},
  author={Jean Bricmont, Antti Kupiainen, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210149227538410},
  booktitle={Journal of differential equations},
  volume={130},
  number={1},
  pages={9-35},
  year={1996},
}
Jean Bricmont, Antti Kupiainen, and Jack Xin. Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity. 1996. Vol. 130. In Journal of differential equations. pp.9-35. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210149227538410.
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