Analysis and Mathematical Physics最新文献

In this paper, we consider blow-up solutions of a nonlinear hyperbolic fractional equation with variable exponents of nonlinearities in the fractional space (mathcal {H}_{p(xi )}^{alpha }(Omega )). To achieve this, we introduce a control function and use energy inequalities to discuss various estimates. In this sense, we address the problem of non-existence of solutions and derive an estimate for the upper bound of the blow-up time. Finally, we provide classical theoretical insights into possible special cases of the results obtained in this study.

In this article, we introduce variable Lorentz–Karamata spaces ({mathcal {L}}_{p(cdot ),q,b}(R)) defined by rearrangement functions and develop the martingale theory in this framework. The real interpolation theory for variable Lorentz–Karamata spaces is presented. Based on this and the new atomic decomposition, we study the real interpolation theory for variable martingale Hardy–Lorentz–Karamata spaces. We also characterize the real interpolation spaces between variable martingale Hardy spaces and (BMO_2) spaces. The results obtained here generalize the previous results for variable Lorentz spaces as well as for variable martingale Hardy–Lorentz spaces. Moreover, we remove the condition (theta +p_->1) in [Banach J. Math. Anal. 2023, 17(3): 47].

The inverse scattering problem for the Dirac equation on the real line are considered. It is shown that the potential on the real line is uniquely determined in terms of the mixed scattering data which consists of the knowledge of the potential on the right (left) half line of the real axis and the reflection coefficient from the right (left). In particular, neither the bound states or the bound state norming constants are needed. The method is based on a factorization of a scattering matrix.

Let ({f_n}) be a sequence of meromorphic functions defined in a domain D, and let ({psi _n}) be a sequence of holomorphic functions on D, whose zeros are multiple, such that (psi _nrightarrow psi ) converges locally uniformly in D, where (psi (not equiv 0)) is holomorphic in D. If, (1) (f_nne 0) and (f_n^{(k)}ne 0); (2) all zeros of (f_n^{(k)}-psi _n) have multiplicities at least ((k+2)/k), then ({f_n}) is normal in D.

We show that the complex conjugate function (zmapsto overline{z}) cannot be pointwise approximated by holomorphic polynomials on the Alice Roth’s Swiss cheese (Q_Rsubset mathbb {C}). Moreover, under some extra hypotheses, we also show that the complex conjugate cannot be pointwise approximated either by functions holomorphic on (Q_R).

We study the multiplicity of normalized solutions of the following (p, q)-Laplacian equation

where (1<p<q<N), a, (epsilon >0), (Delta _lu:=hbox {div}(|nabla u|^{l-2}nabla u)) with (lin {p,q}), stands for the l-Laplacian operator. (lambda in mathbb {R}) is an unknown parameter that appears as a Lagrange multiplier. (V:mathbb {R}^Nrightarrow mathbb {R}) is a continuous function with some proper assumptions. f is a continuous function with (L^p)-mass subcritical growth. By using variational methods, we prove that the equation has multiple normalized solutions, as (epsilon ) is small enough. Precisely, the number of normalized solutions is at least twice that of the global maximum points of V.

The set of partial isometries in a (W^*)-algebra possesses a structure of Banach Lie groupoid. In this paper, the differential structure on the set of partial isometries over the restricted Grassmannian is constructed, which makes it into a Banach Lie groupoid.

We prove that Lipschitz perturbations of nonautonomous contracting or expanding linear dynamics are Lipschitz shadowable provided that the Lipschitz constants are small on average. This is in sharp contrast with previous results where the Lipschitz constants are assumed to be uniformly small. Moreover, we show by means of an example that a natural extension of these results to the context of linear dynamics admitting an exponential dichotomy does not hold in general.

We give the first example of a non-invariant cycle of Baker domains of infinite connectivity for non-entire meromorphic functions. We also prove the necessary and sufficient condition for a cycle of Baker domains to be infinitely connected in terms of critical points for the family (f(z)=lambda e^z+frac{mu }{z}), where (lambda ) and (mu ) are defined in the paper.

Two classes of fractional type variable weights are established in this paper. The first kind of weights ({A_{vec { p}( cdot ),q( cdot )}}) are variable multiple weights, which are characterized by the weighted variable boundedness of multilinear fractional type operators, called multilinear Hardy–Littlewood–Sobolev theorem on weighted variable Lebesgue spaces. Meanwhile, the weighted variable boundedness for the commutators of multilinear fractional type operators are also obtained. This generalizes some known work, such as Moen (Collect Math 60(2):213–238, 2009), Bernardis et al. (Ann Acad Sci Fenn-M 39:23–50, 2014), and Cruz-Uribe and Guzmán (Publ Mat 64(2):453–498, 2020). Another class of weights ({{mathbb {A}}_{p( cdot ),q(cdot )}}) are variable matrix weights that also characterized by certain fractional type operators. This generalize some previous results on matrix weights ({{mathbb {A}}_{p( cdot )}}).