Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations

Planchon, Fabrice 

Journées équations aux dérivées partielles (1999), article no. 9, 11 p.

Résumé

We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space ${\dot{B}}_2^{\frac{n}{2}-\frac{2}{p},\infty }\left({𝐑}^n\right)$, when the nonlinearity is of type $u^p$, for $p\in 𝐍$. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.

Bibliographie

[1] J. Bergh and J. Löfstrom. Interpolation Spaces, An Introduction. Springer-Verlag, 1976. | Zbl

[2] J.-M. Bony. Calcul symbolique et propagation des singularités dans les équations aux dérivées partielles non linéaires. Ann. Sci. Ecole Norm. Sup., 14:209-246, 1981. | Zbl | MR | mathdoc-id

[3] J. Bourgain. Refinements of Strichartz inequality and applications to 2-D NLS with critical nonlinearity. I.M.R.N., 5:253-283, 1998. | Zbl | MR

[4] M. Cannone. Ondelettes, Paraproduits et Navier-Stokes. Diderot Editeurs, Paris, 1995. | Zbl | MR

[5] T. Cazenave and F. Weissler. The Cauchy problem for the critical nonlinear Schrödinger equation in Hs. Nonlinear Anal. T.M.A., 14:807-836, 1990. | Zbl | MR

[6] T. Cazenave and F. Weissler. Asymptotically self-similar global solutions of the non linear Schrödinger and heat equations. Math. Zeit., 228:83-120, 1998. | Zbl | MR

[7] T. Cazenave and F. Weissler. More self-similar solutions of the nonlinear Schrödinger equation. No D.E.A., 5:355-365, 1998. | Zbl | MR

[8] J. Ginibre and Velo. The global Cauchy problem for the NLS equation revisited. Ann. IHP, An. non-linéaire, 2:309-327, 1985. | Zbl | MR | mathdoc-id

[9] M. Keel and T. Tao. Enpoint Strichartz estimates. American Journal of Mathematics, 120(5):955-980, 1998. | Zbl | MR

[10] H. Lindbladt and C. D. Sogge. On existence and scattering with minimal regularity for semilinear wave equations. J. funct. Anal., 130:357-426, 1995. | Zbl | MR

[11] R. O'Neil. Convolution operators and L(p,q) spaces. Duke Mathematical Journal, 30:129-142, 1963. | Zbl | MR

[12] F. Oru. Rôle des oscillations dans quelques problèmes d'analyse non-linéaire. PhD thesis, ENS Cachan, 1998.

[13] H. Pecher. Self-similar and asymptotically self-similar solutions of nonlinear wave equations. preprint. | Zbl

[14] J. Peetre. New thoughts on Besov Spaces. Duke Univ. Math. Series. Duke University, Durham, 1976. | Zbl | MR

[15] F. Planchon. On the Cauchy problem in Besov spaces for a non-linear Schrödinger equation. preprint. | Zbl

[16] F. Planchon. Self-similar solutions and semi-linear wave equations in Besov spaces. preprint. | Zbl

[17] F. Planchon. Asymptotic Behavior of Global Solutions to the Navier-Stokes Equations. Rev. Mat. Iberoamericana, 14(1), 1998. | Zbl | MR

[18] F. Planchon. Solutions autosimilaires et espaces de données initiales pour une équation de Schrödinger non-linéaire. C. R. Acad. Sci. Paris, 328, 1999. | Zbl | MR

[19] F. Ribaud and A. Youssfi. Self-similar solutions of nonlinear wave equation. preprint. | Zbl

[20] F. Ribaud and A. Youssfi. Regular and self-similar solutions of nonlinear Schrödinger equations. J. Math. Pures Appl., 9(10):1065-1079, 1998. | Zbl | MR

[21] F. Ribaud and A. Youssfi. Solutions globales et solutions auto-similaires de l'équation des ondes non linéaire. C. R. Acad. Sci. Paris, 328, 1999. | Zbl | MR

[22] R. Strichartz. Restriction of Fourier transform to quadratic surfaces and decay of solutions of the wave equations. Duke Mathematical Journal, 44:705-714, 1977. | Zbl | MR

[23] T. Tao. Low regularity semi-linear wave equations. Comm. in Partial Diff. Equations, 24(3&4):599-629, 1999. | Zbl | MR