Search

Abstract: In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256]. [Copyright &y& Elsevier]

Abstract: In this paper we study the global regularity for solutions of the -Laplacian in two-dimensional convex domains with Dirichlet boundary conditions. Here with and . [Copyright &y& Elsevier]

Abstract: In this paper, we consider a transmission problem for the Laplace operator when an interface contains angular points and time derivatives of the unknown functions enter in the transmission condition. We prove the existence of a unique solution in the weighted Hölder classes. [Copyright &y& Elsevier]

Abstract: As is the case for the Laplace operator, in Euclidean Clifford analysis also the Helmholtz operator can be factorized, more precisely by using perturbed Dirac operators. In this paper we consider the Helmholtz equation in a circulant matrix form in the context of Hermitian Clifford analysis. The aim is to introduce and study the corresponding inhomogeneous Hermitian Dirac operators, which will constitute a splitting of the traditional perturbed Dirac operators of the Euclidean Clifford analysis context. This will not only lead to special solutions of the Hermitian Helmholtz equation as such, but also to the study of boundary value problems of Riemann type for those solutions, which are, in fact, solutions of the Hermitian perturbed Dirac operators involved. [Copyright &y& Elsevier]

Abstract: Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time–space Caputo–Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals , and , respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchkoʼs Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a discrete version of the following semilinear heat equation with absorption with , which is said to be the ω-heat equation with absorption on a network. Using the discrete Laplacian operator on a weighted graph, we define the ω-heat equations with absorption on networks and give their physical interpretations. The main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish. It is proved that the asymptotic behaviors of the solutions as t tends to +∞ strongly depend on the sign of . [Copyright &y& Elsevier]

Abstract: This paper deals with the heat equation posed in a bounded regular domain Ω of () coupled with a dynamical boundary condition of reactive–diffusive type. In particular we study the problem where , , , , denotes the Laplacian operator with respect to the space variable, while denotes the Laplace–Beltrami operator on Γ, ν is the outward normal to Ω, and k and l are given real constants, . Well-posedness is proved for data such that . We also study higher regularity of the solution. [ABSTRACT FROM AUTHOR]

Abstract: Haïm Brezis and Augusto Ponce introduced and studied several extensions of Kato''s inequality, in particular Kato''s inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a function in a smooth domain [H. Brezis, A.C. Ponce, Kato''s inequality when Δu is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599–604; H. Brezis, A.C. Ponce, Kato''s inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241]. Using potential theoretic methods we answer here some questions raised in [H. Brezis, A.C. Ponce, Kato''s inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241] about the relations between the normal derivative of a function u and the normal derivative of its positive part . The results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given. In the final appendix, H. Brezis solves an old question of J. Serrin about pathological solutions of certain elliptic equations [J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385–387]. This is used in the paper to extend the first version of our main result. [Copyright &y& Elsevier]