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.

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.

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