Regularizing estimates for Schrödinger and wave equations

Alberto Ruiz

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We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space ${\dot{B}}_{2}^{\frac{n}{2} - \frac{2}{p}, \infty} (𝐑^{n})$ , 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.