Nagoya Mathematical Journal

Solutions in Morrey spaces of some semilinear heat equations with time-dependent external forces

Xiaofang Zhou

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Abstract

In this paper, we consider the Cauchy problem for some semilinear heat equations with time-dependent external forces. Both the external force and the initial data are assumed to be small in some Morrey spaces. We first prove the unique existence of a small time-global solution. We next show the stability of that solution by proving the time-global sovability of perturbation problems.

Article information

Source
Nagoya Math. J., Volume 174 (2004), 127-163.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114632069

Mathematical Reviews number (MathSciNet)
MR2066106

Zentralblatt MATH identifier
1053.35062

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B35: Stability 35K15: Initial value problems for second-order parabolic equations

Citation

Zhou, Xiaofang. Solutions in Morrey spaces of some semilinear heat equations with time-dependent external forces. Nagoya Math. J. 174 (2004), 127--163. https://projecteuclid.org/euclid.nmj/1114632069


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References

  • P. Baras, M. Pierre, Problèmes paraboliques semi-linéaires avec données measures , Applicable Anal., 18 (1984), 111–149.
  • H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions , J. Math. Pure. Appl. (9), 62 (1983), 73–97.
  • H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^1 + \alpha$ , J. Fac. Sci. Univ. Tokyo, I, 13 (1966), 109–124.
  • A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem , Indiana Univ. Math. J., 31(1982), 167-189.
  • K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations , Proc. Japan Acad. A, 49(1973), 503–505.
  • K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations , J. Math. Soc. Japan, 29(1977), 407–424.
  • H. Kozono,and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions as initial data , C. R. Acad. Sci. Paris, Sér. I, 317(1993), 1127–1132.
  • ––––, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data , Comm. in P.D.E., 19(1994), 959–1014.
  • ––––, The stability of small stationary solutions in Morrey spaces of the Navier-Stokes equation , Indiana Univ. Math. J., 44, No.3 (1995), 1307–1336.
  • ––––, Small stable stationary solutions in Morrey spaces of the Navier-Stokes equation , Proc. Japan Acad. Ser.A, 71(1995), 199–201.
  • T.-Y. Lee, Some limit theorems for super-Brownian motion and semilinear differential equations , Annals of Probability, 21 (1993), 979–995.
  • Y. Niwa, Semilinear heat equations with measures as initial data , Thesis, Univ. of Tokyo, 1986.
  • C.V. Pao, Periodic solutions of systems of parabolic equations in unbounded domains , Nonlinear Analysis, 40 (2000), 523–535.
  • J. Peetre, On convolution operators leaving $L^p, \lambda$ spaces invariant , Ann. Mat. Pura Appl., 72 (1966), 295–304.
  • M.E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations , Comm. in P.D.E., 17 (1992), 1407–1456.
  • F.B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation , Israel J. Math., 38(1981), 29–40.
  • J. Wu, Well-posedness of a semilinear heat equation with weak initial data , J. Fourier Anal. Appl., 4(1998), 629–642.
  • M. Yamazaki, Solutions in the Morrey spaces of the Navier-Stokes equation with time-dependent external force , Funkcial. Ekvac., 43 (2000), No.3, 419–460.
  • X.F. Zhou, The stability of small stationary solutions in Morrey spaces of the semilinear heat equations , J. Math. Sci. Univ. Tokyo, 6 (1999), 793–822.