Spherical harmonics and maximal estimates for the Schrödinger equation.

Sjölin, Per

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We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian ${(- Δ)}^{κ / 2}$ with $1 \leq κ \leq 2$ . Consequently the classical Hamiltonian corresponding to the previous Schrödinger type equations is studied. As applications, a series of new boundedness results for...

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We consider the Cahn-Hilliard equation in $H^{1} (ℝ^{N})$ with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as $| u | \to \infty$ and logistic type nonlinearities. In both situations we prove the $H^{2} (ℝ^{N})$ -bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012).