Arkiv för Matematik

  • Ark. Mat.
  • Volume 28, Number 1-2 (1990), 371-381.

Long time small solutions to nonlinear parabolic equations

Chen Zhimin

Full-text: Open access


A sharp result on global small solutions to the Cauchy problem $u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $

In Rn is obtained under the the assumption that f is C1+r for r>2/n and ‖u0‖C2(Rn) +‖u0‖W ${}_{1}^{2}$ (Rn) is small. This implies that the assumption that f is smooth and ‖u0 ‖W ${}_{1}^{k}$ (Rn)+‖u0‖W ${}_{2}^{k}$ (Rn) is small for k large enough, made in earlier work, is unnecessary.

Article information

Ark. Mat. Volume 28, Number 1-2 (1990), 371-381.

Received: 16 May 1989
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

1990 © Institut Mittag-Leffler


Zhimin, Chen. Long time small solutions to nonlinear parabolic equations. Ark. Mat. 28 (1990), no. 1-2, 371--381. doi:10.1007/BF02387387.

Export citation


  • Fujita, H., On the blowing up of solution of the Cauchy problem ut=Δu+u1+r, J. Fac. Sci. Univ. Tokyo Sect.I13 (1966), 109–124.
  • Klainerman, S., Long time behavior of solutions to nonlinear evolution equations, Arch. Rat. Mechs. Anal. 78 (1982), 73–98.
  • Li Tatsien and Chen Yunmei, Nonlinear Evolution Equations, to appear.
  • Ponce, G., Global existence of small solutions to a class of nonlinear evolution equations, Nonl. Anal. T. M. A. 9 (1985), 399–418.
  • Weissler, F. B., Existence and non-existence of global solutions for a semilinear heat equation, Israel. J. Math. 38 (1981), 29–40.
  • Zheng Songmu and Chen Yunmei, Global existence for nonlinear parabolic equations, Chin. Ann of Math. 7B (1986), 57–73.