Volume 20, Issue 1

Abstract. We consider the nonlinear wave equation in a bounded domain with a delay term in the internal feedback and prove the global existence of its solutions in Sobolev spaces by means of the energy method combined with the Faedo–Galerkin procedure under a certain condition between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study the asymptotic behavior of solutions using the multiplier method and general weighted integral inequalities.

Abstract. We consider a problem associated to a linear elasticity system in the dynamic regime in the three-dimensional thin domain . We first discuss the existence of weak solutions of this problem. Then we study the asymptotic behavior when one dimension of the domain tends to zero. The uniqueness result for the limit problem is also proved.

Abstract. We introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy–Littlewood maximal operator as well as the boundedness of Calderón–Zygmund operators. The boundedness of Riesz type potential operators is also obtained in the framework of homogeneous and also in the nonhomogeneous cases in generalized grand Morrey spaces.

Abstract. New oscillation criteria are proved for second-order two-term linear delay differential equations with locally integrable coefficients and measurable delays. In particular, a suitable a priori lower bound of non-oscillatory solutions is found, which plays a crucial role in the proofs of the results obtained.

Abstract. The paper discusses the solvability of the singular Dirichlet boundary value problem Here and f satisfies the local Carathéodory conditions on , where . It is shown that the cardinality of the set of all positive solutions to the problem is a continuum. In addition, the structure and properties of the set are described. Applications and numerical simulations of the results are presented.

Abstract. In this paper, the B -separability properties of anisotropic convolution-operators are investigated. In particular, the sharp estimates for the resolvent are established. Thus these operators are positive and also are generators of analytic semigroups. Moreover, we find sufficient conditions that guarantee the maximal B -regularity of the Cauchy problem for a parabolic convolution equation. Finally, the convolution equation is applied to establish maximal regularity properties for an anisotropic integro-differential equations and their infinite systems.

Abstract. The weighted Cauchy problem is considered for nonlinear n -th order delayed differential equations having nonintegrable singularities with respect to a time variable. The conditions are found under which the set of solutions is compact and connected in the topology of the space C n -1 .

Abstract. The main aim of this paper is to find the estimation for Vilenkin–Fourier coefficients.

Abstract. The nonlinear evolution backward time problem where A is a positive self-adjoint unbounded operator and f is a given Lipschitz function, is regularized by a new quasi-reversibility method. A new error estimate of the regularized solution is obtained. This paper extends the work by M. Denche and K. Bessila [J. Math. Anal. Appl. 301 (2005), 419–426] to nonlinear ill-posed problems. A numerical example for application to heat equations is given to show the effectiveness of our method.

Abstract. In this paper, by means of the theory of singular integrals and linear commutators, we establish the regularity in grand Morrey spaces of the solution to an elliptic equation in nondivergence form with VMO coefficients.