Journal of Pseudo-Differential Operators and Applications最新文献

The main goal of this paper is to show that if (uin W^{m,p}(mathbb R^n)) is a weak solution of (Qu = f) where (f in X^{r,q}_{p,k}(mathbb R^n)), then (u in X^{m+r,q}_{p,k}(mathbb R^n)) with (1< p,q < infty ), (0< r < 1), k is a temperate weight function in the Hörmander sense, (Q = sum _{|beta | le m} c_{beta }partial ^{beta }) is a linear partial differential operator of order (m ge 0) with non-zero constant coefficients (c_{beta }), and where (X^{r,q}_{p,k}(mathbb R^n)) is either the weighted Triebel-Lizorkin or the weighted Besov space. The way to prove this result is based on the boundedness of the continuous wavelet transform with rotations.

In this article, we introduce infinite-order integro-differential operator related to Lebedev–Skalskaya transform. Some characteristics of this operator are obtained. Furthermore, we establish the necessary and sufficient conditions for a class of infinite-order integro-differential operators to be unitary on ( L^2({mathbb {R}}_{+}; , dx)). Some classes of related integro-differential equations are also studied at the end.

Abstract

This paper is devoted to a simpler derivation of energy estimates and a proof of the well-posedness, compared to previously existing ones, for effectively hyperbolic Cauchy problem. One difference is that instead of using the general Fourier integral operator, we only use a change of local coordinates x (of the configuration space) leaving the time variable invariant. Another difference is an efficient application of the Weyl-Hörmander calculus of pseudodifferential operators associated with several different metrics.

We consider wave equations with a nonlocal polynomial type of damping depending on a small parameter (theta in (0,1)). This research is a trial to consider a new type of dissipation mechanisms produced by a bounded linear operator for wave equations. These researches were initiated in a series of our previous works with various dissipations modeled by a logarithmic function published in (Charão et al. in Math Methods Appl Sci 44:14003-14024, 2021; Charão and Ikehata in Angew Math Phys 71:26, 2020; Piske et al. in J Diff Eqns 311:188-228, 2022). The model of dissipation considered in this work is probably the first defined by more than one sentence and it opens field to consider other more general. We obtain an asymptotic profile and optimal estimates in time of solutions as (t rightarrow infty ) in (L^{2})-sense, particularly, to the case (0<theta <1/ 2).

We study the Cauchy problem for the higher-order KdV–BBM type equation

where (varvec{Lambda }) (=mathcal {F}^{-1}Lambda mathcal {F}) and (Theta ) (=mathcal {F}^{-1}Theta mathcal {F}) are the pseudodifferential operators, defined by their symbols (Lambda left( xi right) ) and ( Theta left( xi right) ), respectively. The aim of the present paper is to develop a general approach through the Factorization Techniques of evolution operators which can be applied for finding the large time asymptotics of small solutions to a wide class of nonlinear dispersive KdV- type equations including the KdV or the improved version of the KdV with higher order dispersion terms.

Abstract

In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction–diffusion systems posed in a bounded domain of (mathbb {R}^N) . The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide nonnegativity of the solutions and uniform control of the total mass. The diffusion operators are of type (u_imapsto d_i(-Delta )^s u_i) where (0<s<1) . Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type (u_imapsto -d_iDelta u_i) . On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case (s=1) .

Abstract

Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier (text {Op}(p)) of order (d>0) , real-analytic near (p^{-1}(0)) , has a fundamental solution (mu _0) . We give an integral representation of (mu _0) in terms of the resolutions supplied by Hironaka’s theorem. This (mu _0) is weakly approximated in (H^t_{text {loc}}({mathbb {R}}^n)) for (t<d-frac{n}{2}) by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

In this paper, we consider the semilinear pseudo-parabolic equation with cone degenerate viscoelastic term

with initial and boundary conditions, where (f(u)=|u|^{p-2}u-frac{1}{|mathbb B|}displaystyle int _{mathbb B}|u|^{p-2}ufrac{dx_1}{x_1}dx'). We construct several conditions for initial data which leads to global existence of the solutions or the solutions blowing up in finite time. Moreover, the asymptotic behavior and the bounds of blow-up time for the solutions are given.

In this work we consider Toeplitz operators and composition operators on the q-Bergman space.We give some spectral properties of Toeplitz operators in general and a sufficient condition for hyponormality of Toeplitz operators in the case of a symbol where the analytic part is a monomial. We also give a necessary condition for hyponormality in the general case of a harmonic symbol as well as a necessary and sufficient condition for such operators to commute. For composition operators we give necessary conditions and sufficient conditions for their compactness and normality, as well as necessary conditions for cohyponormality in the case of a linear fractional map and we finally compute the adjoint in the case of a linear map.

Abstract

In this paper, we investigate the blow-up mechanism to the bipolar compressible Navier–Stokes–Poisson system in three dimensions. It is essentially shown that the mass of the model will concentrate in some spatial points, even if the initial density contains vacuum states, provided that the smooth solution develops singularity in finite time.