The cauchy problem for semilinear parabolic equations in Besov spaces

Houston Journal of Mathematics

Volumn 30, Issue 3, 2004, Pages 829-878

  Miao, Changxing ^a   Zhang, Bo ^b  

^a INSTITUTE OF APPLIED PHYSICS AND COMPUTATIONAL MATHEMATICS   (China)

^b Faculty of Engineering and Computing   (United Kingdom)

Author keywords

Besov spaces; Cauchy problem; Parabolic equations; Scaling; Spacetime estimates

Indexed keywords

EID: 4644283096     PISSN: 03621588     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (38)

1. J. Bergh and J. Löfström, Interpolation Spaces, Springer, New York (1976).
2. J. Bourgain, The Global Solution of Nonlinear Schrödinger Equations, AMS, Providence (1999).
3. M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoramericana 13 (1997), 515-541.
4. s, Nonlinere Anal., TMA 14 (1990), 807-836.
5. D.B. Dix, Nonuniqueness and uniqueness in the initial value problem for the Burger's equation, SIAM J. Math. Anal. 27 (1996), 208-224.
6. n, J. Differ. Equations 100 (1991), 119-161.
7. α+1, J. Fac. Sci. Univ. Tokyo, Sect. I 13 (1966), 109-124.
8. 3) et d'autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoamericana 16 (2000), 605-667.
9. 3, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), 339-342.
10. p and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equations 61 (1986), 186-212.
11. r of Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267-281.
12. J. Ginibre and G. Velo, Generalized Strichartz inequality for the wave equations, J. Funct. Anal. 133 (1995), 50-68.
13. J. Ginibre and G. Velo, The Cauchy problem in local spaces for complex Ginzburg-Landau equation, Comm. Math. Phys. 187 (1997), 45-79.
14. B. Guo and C. Miao, Space-time means and solutions to a class of nonlinear parabolic equations, Science in China 41 (1998), 682-693.
15. A. Haraux and F.B. Weissler, Non-uniquenes for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167-189.
16. n with applications to weak solutions, Math. Z. 187 (1984), 471-480.
17. T. Kato and G. Ponce, The Navier-Stokes equation with weak initial data, Int. Math. Res. Notices No. 10 (1994), 435-444.
18. H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distribution in new functionspace as initial data, Comm. Part. Differ. Equations 19 (1994), 959-1024.
19. C. Miao, Harmonic Analysis and Applications to Partial Differential Equations (in chinese), Science Press, Beijing (1999).
20. C. Miao, Time-space estimates of solutions to general semi-linear parabolic equations, Tokyo J. Math. 24 (2001), 247-278.
21. C. Miao and Y. Gu, Space-time estimates for parabolic type operator and application for nonlinear parabolic equations, J. Part. Differ. Equations 11 (1998), 301-312.
22. M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funt. Anal. 155 (1998), 365-380.
23. T. Ozawa, On critical cases of Sobolev's inequalities, J. Funt. Anal. 127 (1995), 259-269.
24. A. Pazy, Linear Operator Semigroup Theory and Applications to Partial Differential Equations, Springer-Verlag, New York (1983).
25. G. Ponce, On the well posedness of some nonlinear evolution equations, in: Nonlinear Wave, Proc. of Fourth MSJ Int. Research Institute (eds. R. Agemi, Y. Giga and T. Ozawa), 1996, pp. 379-409.
26. F. Ribaud, Semilinear parabolic equation with distributions as initial data, Discrete Cont. Dynam. Sys. 3 (1997), 305-316.
27. n) spaces, Rev. Mat. Iberoramericana 14 (1998), 1-45.
28. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemyskij Operators, and Nonlinear Partial Differential Equations, (De Gruyter Series in Nonlinear Analysis and Applications 3), Walter De Gruyter, Berlin (1996).
29. E.M. Stein, Singular Integral and Differential Property of Functions, Princeton Univ. Press (1970).
30. R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solution of wave equations, Duke Math. J. 44 (1977), 705-714.
31. T. Tao, Lectures on Harmonic Analysis, Department of Mathematics, University of California at Los Angeles, USA (2000).
32. M.E. Taylor, Pseudodifferential Operators, Princeton University Press (1981).
33. E. Terraneo, Non-uniqueness for a critical nonlinear heat equation, Comm. Partial Differential Equations 27 (2002), 185-218.
34. H. Triebel, Theory of Function Spaces, Springer, New York (1983).
35. H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel (1992).
36. p, Indiana Math. J. 29 (1980), 79-102.
37. F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equations, Israel J. Math. 38 (1981), 29-40.
38. F.B. Weissler, Asympototically self-similar solutions of the two-power nonlinear Schroedinger equation, Adv. Differential Equations 6 (2001), 419-440.