Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations

This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These spaces are local, thus adapted to finite speed of propagation, and related to a conservation law of the linear wave equation. We also construct the adapted profile decomposition

An improved maximal inequality for 2D fractional order Schr\"{o}dinger operators

The local maximal inequality for the Schr\"{o}dinger operators of order \a>1 is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to fractional order Schr\"{o}dinger equations. Our method is inspired by Bourgain's argument in case of \a=2. The extension from \a=2 to general \a>1 confronts three essential obstacles: the lack of Lee's reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma at our disposal and analyzing all the possibilities in using the separateness of the segments to obtain the analogous bilinear $L^2-$estimates. To compensate the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality in \cite{ref Bourgain Guth} is also rebuilt to dominate the solution of fractional order Schr\"{o}dinger equations.Comment: Pages47, 3figures. To appear in Studia Mathematic