Acta Mathematica Vietnamica最新文献

In this paper, we investigate the Robin-Dirichlet problem for a nonlinear pseudoparabolic equation of Kirchhoff-Carrier type with viscoelastic term. Under suitable assumptions on the initial data and the relaxation function included in the viscoelastic term, we obtain sufficient conditions for the existence, uniqueness, blow-up, and decay of a weak solution. The results obtained here extend the ones in a previous paper of the authors (Ngoc et al., Math. Meth. Appl. Sci. 44(11), 8697–8725, 26).

This paper especially focuses on a general estimate, called ((f-mathcal M)^{k}), for the (bar partial )-Neumann problem

({(f-mathcal M)^{k}} qquad | f({varLambda })mathcal M u|^{2}le c(|bar partial u|^{2}+|bar partial ^{*}u|^{2}+|u|^{2})+C_{mathcal M}|u|^{2}_{-1})

for any (uin C^{infty }_{c}(Ucap bar {Omega })^{k}cap text {Dom}(bar {partial }^{*})), where f(Λ) is the tangential pseudodifferential operator with symbol f((1 + |ξ|²)^1/2), (mathcal M) is a multiplier, and U is a neighborhood of a given boundary point z₀. Here the domain Ω is q-pseudoconvex or q-pseudoconcave at z₀. We want to point out that under a suitable choice of f and (mathcal M), ((f{-}mathcal M)^{k}) is the subelliptic, superlogarithmic, compactness and so on. Generalizing the Property (P) by Catlin (1984), we define Property ((f-mathcal M-P)^{k}). The result we obtain in here is: Property ((f-mathcal M-P)^{k}) yields the ((f-mathcal M)^{k}) estimate. The paper also aims at exhibiting some relevant classes of domains which enjoy Property ((f-mathcal M-P)^{k}).

In this paper, we investigate local derivations and local generalized derivations on path algebras associated with finite acyclic quivers. We show that every local derivation on a path algebra is a derivation, and every local generalized derivation on a path algebra is a generalized derivation. Also, we apply main results on several related maps to local derivations. The established results generalize several ones on some known algebras such as incidence algebras.

We study families of porous medium equations with nonlocal pressure. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the L^p norms.

We derive an expression for the collision-induced amplitude dynamics in a fast collision between two (N + 1) −dimensional spatiotemporal solitons in saturable nonlinear media with weak perturbations in a spatial dimension of N, where N ≥ 1. The perturbed spatiotemporal soliton evolution is under a framework of the coupled saturable (N + 1 + 1) −dimensional nonlinear Schrödinger equations in the presence of weakly nonlinear loss and delayed Raman response. The perturbation approach is based on an extended perturbation technique for analyzing the collision-induced dynamics of one-dimensional temporal solitons and two-dimensional solitons. The accuracy of our theoretical calculations is validated by numerical simulations of the interaction of two 3D spatiotemporal solitons, also known as two light bullets, of the coupled nonlinear Schrödinger equations in the presence of delayed Raman response and cubic loss.

We establish several identities and inequalities of Hardy type and Sobolev type with Dunkl weights. We also investigate the sharp constants and optimal functions of the Hardy-Sobolev inequality with Dunkl weights.

Let R be a commutative noetherian ring. The n-semidualizing modules of R are generalizations of its semidualizing modules. We will prove some basic properties of n-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about n-semidualizing modules.

We consider gradient estimates for H¹ solutions of linear elliptic systems in divergence form (partial _{alpha }(A_{ij}^{alpha beta } partial _{beta } u^{j}) = 0). It is known that the Dini continuity of coefficient matrix (A = (A_{ij}^{alpha beta }) ) is essential for the differentiability of solutions. We prove the following results:

(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L² mean oscillation ω_A,2 of A satisfies

where C_∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

(b) If X_A,2 = 0, then ∇u ∈ V MO.

(c) Finally, examples satisfying X_A,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when (nabla u in L^{infty } cap VMO).

We give a proof of the Lieb–Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering lemma.