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Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations. (English) Zbl 1008.35076
Journées “Équations aux dérivées partielles”, Saint-Jean-de-Monts, France, 31 mai au 4 juin 1999. Exposés Nos. I–XIX. Nantes: Université de Nantes. Exp. No. IX, 11 p. (1999).
Summary: We prove that the initial value problem for the semilinear Schrödinger and wave equations is well-posed in the Besov space \(\dot B_2^{{n\over 2}-{2\over p},\infty} (\mathbb R^n)\), when the nonlinearity is of type \(u^p\), for \(p\in\mathbb N\). 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.
For the entire collection see [Zbl 0990.00047].

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35L70 Second-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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