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

Published online by Cambridge University Press:  22 January 2016

Xiaofang Zhou
Xiaofang Zhou*
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, Hubei, P.R.China, xfzhou2003@hotmail.com
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Abstract

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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.

Keywords

35K0535K5535B35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 174 , 2004 , pp. 127 - 163
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Baras, P., Pierre, M., Problèmes paraboliques semi-linéaires avec données measures, Applicable Anal., 18 (1984), 111–149.Google Scholar
[2] Brezis, H., Friedman, A., Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure. Appl. (9), 62 (1983), 73–97.Google Scholar
[3] Fujita, H., On the blowing up of solutions of the Cauchy problem for ut = ?u + u1+a , J. Fac. Sci. Univ. Tokyo, I, 13 (1966), 109–124.Google Scholar
[4] Haraux, A. and Weissler, F. B., Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167–189.CrossRefGoogle Scholar
[5] Hayakawa, K., On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad. A, 49 (1973), 503–505.Google Scholar
[6] Kobayashi, K., Sirao, T. and Tanaka, H., On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407–424.Google Scholar
[7] Kozono, H.,and Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions as initial data, C. R. Acad. Sci. Paris, Sér. I, 317 (1993), 1127–1132.Google Scholar
[8] Kozono, H.,and Yamazaki, M., 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.Google Scholar
[9] Kozono, H.,and Yamazaki, M., The stability of small stationary solutions in Morrey spaces of the Navier-Stokes equation, Indiana Univ. Math. J., 44, No.3 (1995), 1307–1336.Google Scholar
[10] Kozono, H.,and Yamazaki, M., Small stable stationary solutions in Morrey spaces of the Navier-Stokes equation, Proc. Japan Acad. Ser.A, 71 (1995), 199–201.Google Scholar
[11] Lee, T.-Y., Some limit theorems for super-Brownian motion and semilinear differen tial equations, Annals of Probability, 21 (1993), 979–995.Google Scholar
[12] Niwa, Y., Semilinear heat equations with measures as initial data, Thesis, Univ. of Tokyo, 1986.Google Scholar
[13] Pao, C.V., Periodic solutions of systems of parabolic equations in unbounded domains, Nonlinear Analysis, 40 (2000), 523–535.Google Scholar
[14] Peetre, J., On convolution operators leaving Lp’x spaces invariant, Ann. Mat. Pura Appl., 72 (1966), 295–304.Google Scholar
[15] Taylor, M.E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. in P.D.E., 17 (1992), 1407–1456.Google Scholar
[16] Weissler, F.B., Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29–40.Google Scholar
[17] Wu, J., Well-posedness of a semilinear heat equation with weak initial data, J. Fourier Anal. Appl., 4 (1998), 629–642.Google Scholar
[18] Yamazaki, M., Solutions in the Morrey spaces of the Navier-Stokes equation with time-dependent external force, Funkcial. Ekvac, 43 (2000), No.3, 419–460.Google Scholar
[19] Zhou, X.F., The stability of small stationary solutions in Morrey spaces of the semi-linear heat equations, J. Math. Sci. Univ. Tokyo, 6 (1999), 793–822.Google Scholar