Search

This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in ℝn. Firstly, the global existence and uniqueness of classical solutions for small initial data are established. Then, we obtain the Lp, 2 ⩽ p ⩽ +℞ decay rate of solutions. The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.

In this paper, we propose a new combined scalarization method of multi-objective optimization problems by using the surplus variables and the generalized Tchebycheff norm and then use it to obtain some equivalent scalarization characterizations of (weakly, strictly, properly) efficient solutions by adjusting the range of parameters. These scalarization results do not need any convexity assumption conditions of objective functions. Furthermore, we establish some scalarization results of approximate solutions by means of the method. Moreover, we also present some examples to illustrate the main results.

In this paper, we first give some characterizations of improvement sets via quasi interior. Furthermore, as applications of these characterizations, we establish an alternative theorem via improvement sets and quasi interior, and then obtain a scalarization result of weak $$E$$ -efficient solutions defined by improvement sets and quasi interior for vector optimization problems with set-valued maps. Moreover, we also present some examples to illustrate the main conditions and results.

In this paper, two kinds of nonlinear scalarization functions are applied to characterize $E$-efficient solutions and weak $E$-efficient solutions of vector optimization problems and some nonlinear scalarization characterizations are obtained. Some examples also are given to illustrate the main results.

A class of vector optimization problems is considered and a characterization ofE-Benson proper efficiency is obtained by using a nonlinear scalarization function proposed by Göpfert et al. Some examples are given to illustrate the main results.

Based on the ideas of the classical Benson proper efficiency, a new kind of unified proper efficiency namedS-Benson proper efficiency is introduced by using Assumption (B) proposed by Flores-Bazán and Hernández, which unifies some known exact and approximate proper efficiency including(C,ε)-proper efficiency andE-Benson proper efficiency in vector optimization. Furthermore, a characterization ofS-Benson proper efficiency is established via a kind of nonlinear scalarization functions introduced by Göpfert et al.