Properties Of Solutions To Some Classes Of Abstract Nonlinear Evolution Equations With Delay And Nonlocal Initial Conditions

Nonlinear evolution equations,i.e.,pa.rtial differential equations with time t as one of the independent variables arise not only from many fields of mathematics but also from other branches of science such as physics,mechanics and material science.For example,Navier-Stokes and Euler equations from fluid mechanics,nonlinear reaction-diffusion equations from heat transfers and biological sciences,nonlinear Klein-Gorden equa-tions and nonlinear Schrodinger equations from quantum mechanics and Cahn-Hilliard equations from material science,to name just a few,are special examples of nonlinear evolution equations.See the books[1-4].Exact solutions of nonlinear evolution equa-tions play an import ant role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science.Functional differential equations and inclusions arise in a variety of areas of biological,physical,and engineering applications,and such equations have received much attention in recent,years.A good guide to the literature for functional differential equations is the books by Hale[5],Hale and Verduyn Lunel[6],Kolmanovskii and Myshkis[7],and the references therein.During the last decades,exist.ence and uniqueness of mild,strong,classical,almost periodic,almost a.utomorphic solutions of semi-linear functional differential equations and inclusions have been studied extensively by many authors using the semigroup theory,fixed point argument,degree theory,and measures of noncompactness.We mention,for instance,the books by Ahmed[8],Diagana[9],Engel a.nd Nagel[10],Kamenskii et al.[11],Pazy[12],Wu[13],Zheng[14],and the references therein.In recent years,there has been a significant development in evolut ion equations and inclusions;see the monograph of Perestyuk et al.[15],the papers of Baliki and Benchohra[16,17],Benchohra and Medjedj[18,19],Benchohra et al.[20],and the references therein.Furthermore,Some recent contributions on C0-solutions to the evolution equations and inclusions have been established in Vrabie[21-24]Burlica and Rosu[25],Meknani and Zhang[26],Meknani[27],Garcia[28]and Paicu and Vrabie[29]and Burlica[30],Burlica and Rosu[31,32].Diaz and Vrabie[33],Necula and Vrabie 34],Rosu[35,36].However,these researches were motivated by the applications cases of these mathematical models,it is t.hought to covers most of the nonlinear evolution equations with nonlocal retarded initial conditions,arising in physics,described in Deng[37]and McKibben[38,section 10.2,pp.394-398].Delay differential equations are one of the oldest branches of the theory of infinite dimensional dynamical systems theory which describes the qualitative properties of systems changing in time.We refer to the classical monographs on the theory of ordinary delay equations[39,40].However,complicated situations in which the delay depends on the unknown functions have been proposed in modeling in recent years.These equations are frequently called equations with state-dependent delay.Often,it has been assumed that the delay is either a fixed constant or is given as an integral in which case is called distributed delay;see for instance the books by Hale and Verduyn Lunel[6],Kol-Manovskii and Myshkis[7],Smith[41],Abbas and Benchohra[42],and Wu[13],and the references therein.Existence results and among other things were derived recently for functional differential equations when the solution is depending on the delay on a bounded interval for impulsive problems.We refer the reader to the papers by Hernandez et al[43]and Li et al.[44].Very recently,Baghli et al.considered when the solution is depending on the delay for evolution equations in[45].It should be pointed out that,to study the abstract functional differential equations with infinite delay,people usually employ an axiomatic definition of the phase space introduced by Hale and Kato[46],but defined as in the book[47].Reaction-diffusion equations describe distributions of temperature,concentrations of some other variables in space and in time.These equations are characterized by the presence of diffusion and production terms.Originally,diffusion was understood as the random motion of atoms and molecules and described by the Laplace operator.Heat conduction was described by similar differential expressions.This simplest description of heat and mass transport was later completed by other mechanisms.Among them,cross-diffusion,anomalous diffusion,ot.her mechanisms of heat conduction.see[48].For the results concerning reaction-diffusion systems,we mention Burlica[30].Burlica and Rosu[31,32],Diaz and Vrabie[33],Necula and Vrabie[34],Rosu[35,36]and the references therein.For(non-delayed)evolution equations subjected to nonlocal initial conditions see Paicu and Vrabie[29]and the references therein.For delay evolution equations with nonlocal initial data see Burlica and Rosu[25]and Vrabie[22-24].The single-valued system was considered by Burlica and Rosu and Vrabie[49].The theory of almost periodic functions was introduced in the literature around 1924-1926 the pioneering work of the Danish mathematician Bohr[50].A decade later,various significant contributions were then made to that theory mainly by Bochner[51],von Neumann[52],and van Kampen[53].The notion of almost periodicity,which generalizes the concept of periodicity,plays a crucial role in various fields including harmonic analysis,physics,dynamical systems,etc.In the early 1990s.Zhang[54-56] introduced a concept of pseudo-almost periodicity(PAP)as a natural generalization of the notion of almost periodicity.Since its intro-duction in the literature,the notion of pseudo-almost periodicity has generated several developments and extensions,see,e.g.,[57-62].Among other things,it has been uti-lized to study the qualitative behavior to various differential and partial differential equations involving pseudo almost periodic coefficients,see,e.g.,Diagana et al.[57-71]Cuevas et al.[72-74],Ding et al.[75-77],Zhang[54-56]Ji and Zhang[78],Liang et al.[79],Fan et al.[80,81],Agarwal et al.[82,83],Ait et al.[84,85],Pinto[86],Al-Islam et al.[87],Amir and Maniax[88],Bugajewski et al.[89],Boukli-Hacenea and Ezzinbi[90,91],Ezzinbi et al.[92,93],Li et al.[94],Liang et al.[95],and Zhang and Li[96-98].This thesis is devoted to the study the existence of bounded C0-solutions for a class of reaction-diffusion system with a nonlocal retarded initial condition in Chapter2,as well as the existence,uniqueness and global asymptotic stability of C0-solutions nonlinear retarded reaction-diffusion system with a nonlocal retarded initial condition in Chapter3.Fu.rthermore,in Chapter4,we present the conclusion of our study by proving that each C0-solutions is pseudo almost periodic.Except in the last chapter,all problems have been studied with delay and in a Banach space.The proofs are based on the compactness arguments,Kakutani fixed point theorem,Tychonoff fixed point theorem,Banach fixed point theorem and invariance technique to solve our problems.We have arranged this thesis as follows:Chapter1:In this chapter,we give a brief introduction to the theory of C0-semigroups,m-dissipative operators,the nonlinear evolutions governed by them,evolution systems,m-dissipative linear as well as nonlinear partial differential operators.Some basic facts on delay evolution equations subjected to initial conditions,as well as on differential and integral inequalities found their place in this dissertation.Moreover,a general introduction to the almost periodic,asymptotic almost periodic and pseudo almost periodic functions are presented.Chapter2:In this chapter,we study the existence of bounded C0-solutions for a class of reaction-diffusion system with nonlocal retarded initial conditions,under suitable assumptions the main results were obtained.We adopt the compactness method and Kakutani fixed-point theorem for multi-functions in a locally convex space to solve the system(2.1).We reinforced the theoretical study by an illustrated example.All the results in this chapter are new and are the extension of the results Vrabie[22].Chapter3:In this chapter,the existence and uniqueness results for nonlinear nonlocal retarded reaction-diffusion system initial conditions are studied.Based on the com-pactness arguments,Tychonoff fixed point theorem and invariance technique,the proof of our main results is presented.We reinforced the theoretical study by an illustrated example.All the results in this chapter are new and are the extension of the Vrabie 1991.Chapter4:In this chapter,the existence and uniqueness result of pseudo-almost peri-odic C0-solutions to the evolutions equation with nonlocal initial conditions in Banach spaces are presented.Under several suitable assumptions combined with an integral solution definition,the mains results are proved.An example is given in the end to illustrate the theoretical result.All the results in this chapter are new and are the extension of the results Vrabie[21].