Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations
Marco Cappiello Fabio Nicola
Discrete & Continuous Dynamical Systems - A 2016, 36(4): 1869-1880 doi: 10.3934/dcds.2016.36.1869
We consider semilinear equations of the form $p(D)u=F(u)$, with a locally bounded nonlinearity $F(u)$, and a linear part $p(D)$ given by a Fourier multiplier. The multiplier $p(\xi)$ is the sum of positively homogeneous terms, with at least one of them non smooth. This general class of equations includes most physical models for traveling waves in hydrodynamics, the Benjamin-Ono equation being a basic example.
    We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in $p(\xi)$. When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well as holomorphic extension to a strip, for analytic nonlinearity.
keywords: Benjamin-Ono equation. Nonlocal semilinear equations solitary waves algebraic decay holomorphic extension
Remarks on dispersive estimates and curvature
Fabio Nicola
Communications on Pure & Applied Analysis 2007, 6(1): 203-212 doi: 10.3934/cpaa.2007.6.203
We investigate connections between certain dispersive estimates of a (pseudo) differential operator of real principal type and the number of non-vanishing curvatures of its characteristic manifold. More precisely, we obtain sharp thresholds for the range of Lebesgue exponents depending on the specific geometry.
keywords: Strichartz estimates Dispersive estimates Gaussian curvature.
Time-frequency analysis of fourier integral operators
Elena Cordero Fabio Nicola Luigi Rodino
Communications on Pure & Applied Analysis 2010, 9(1): 1-21 doi: 10.3934/cpaa.2010.9.1
Time-frequency methods are used to study a class of Fourier Integral Operators (FIOs) whose representation using Gabor frames is proved to be very efficient. Indeed, similarly to the case of shearlets and curvelets frames [10, 35], the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\infty, 1}$ [33], for some Fourier multipliers [6] and metaplectic operators [14, 31]. Moreover, this paper provides the mathematical tools to numerically solving the Cauchy problem for Schr¨odinger equations using Gabor frames [17]. Finally, similar arguments can be employed to study other classes of FIOs [16].
keywords: Fourier integral operators modulation spaces Gabor frames. short-time Fourier transform
Schrödinger equations with rough Hamiltonians
Elena Cordero Fabio Nicola Luigi Rodino
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 4805-4821 doi: 10.3934/dcds.2015.35.4805
We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.
    Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.
keywords: pseudodifferential operators Schrödinger equation semilinear equations. Sjöstrand class modulation spaces

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