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In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schrödinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L p -L q estimates on the Schrödinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).

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This paper investigates higher order wave-type equations of the form ∂ttu+P(Dx)u=0, where the symbol P(ξ) is a real, non-degenerate elliptic polynomial of the order m≥4 on boldRn. Using methods from harmonic analysis, we first establish global pointwise time–space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) Lp−Lq estimates on the wave solution in terms of the initial conditions.

In this paper, we consider the Cauchy problem for the H s -critical inhomogeneous nonlinear Schrödinger (INLS) equation2 and f (u) is a nonlinear function that behaves like λ|u| σ u with λ ∈ C and σ = 4−2b n−2s . First, we establish the local well-posedness as well as the small data global well-posedness in H s (R n ) for the H s -critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, we obtain some standard continuous dependence results for the H s -critical INLS equation. Our results about the well-posedness and standard continuous dependence for the H s -critical INLS equation improve the ones of Aloui-Tayachi [Discrete Contin. Dyn. Syst. 41 (11) (2021), 5409-5437] by extending the validity of s and b. Based on the local well-posedness in H 1 (R n ), we finally establish the blow-up criteria for H 1 -solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.