The Research Of Stability For Nonlinear Evolution Equations With State-dependent Delays

The evolution equation with state-dependent delays(SDD)is a class of important functional equation.State-dependent delays have been incorporated into a variety of partial differential equation models,since they are more realistic in the real world problems.In recent years,the theory and applications of evolution equation with SDD are emerging as an important area of investigation.Meanwhile,the complexity of SDD makes the analysis of nonlinear evolution equation with SDD more difficult.And SDD make many of the standard tools of dynamical systems theory inapplicable for partial functional equations at first sight.Then we have to establish the basic theory of such equation as soon as possible.In this dissertation,we mainly investigates the stability problem of nonlinear evolution equation with SDD.By establishing the principles of linearized stability for two kinds of nonlinear evolution equation with SDD and choosing some novel Lyapunov functionals for for virus infection model,we can perfectly work out the problem.More specifically,the main work is as follows:Firstly,we establish the principle of linearized stability for the following equation with SDD(?)where A:D(A)(?)X?X is the generator of a semigroup(T(t))t?0 of bounded linear operators on X and(X,?·?x)is a Banach space.Such equations contain more general delay terms which cover the discrete delay and distributed delay as special cases.We first obtain the existence of mild solution for initial value problem(IVP)by using Arzela-Ascoli theorem,Schauder fixed point theorem,B anach fixed point theorem and semigroup theory.And according to sectorial operator theory,we get the existence and uniqueness of classical solution for IVP.Moreover,in order to obtain the main results,we need some error estimation between nonlinear function and linear function.After that,we can prove the result of linearized stability by applying semigroup theory,variation of constant formula,Gronwall-Bellman s inequality.Next,we gives a constructive proof of linearized instability for such equations in frame of sectorial operator theory.Finally,we present an example of Hematopoiesis model to show the effectiveness of the proposed results.Furthermore,we can predict the density of blood circulation mature cells in the circulatory system.Secondly,we establish the principle of linearized stability for the following equation with infinite state-dependent delays(ISDD)(?)where the abstract phase space B is a Banach space consisting of mappings from(-?,0]into X that satisfy some axioms.It should be pointed out that such equation contain more general delay terms which not only cover the discrete delay and distributed delay as special cases,but also extend the SDD to infinite delay.Moreover,the unboundedness of the delay stirs up that the choice of phase space is a unconventional job linked with the specific research items.And the solution semigroup of such equation is not compact.This implies that there are numerous technical difficulties in dealing with such equation.We first obtain the existence of mild solution for IVP by using Banach fixed point theorem and semigroup theory.And by employing the same argument as in the proof of the above chapter,we get the existence and uniqueness of classical solution for IVP.Next,we provide an error estimation which will be useful in the sequel.Then we can prove the result of linearized stability by applying semigroup theory,variation of constant formula.On the other hand,we mainly give the theory of linearized instability for such equations by employing sectorial operator theory.Finally,we apply the obtained theoretical results to the predator-prey model with spatial diffusion and infinite state-dependent memory effects.Thirdly,the stability of the interior equilibrium for the following virus infection dynamical model with diffusion and state-dependent delays is established by using dynamical systems theory and Lyapunov functional.At first,by applying the same line,the existence and uniqueness of classical solution for IVP can be easily proved.Next,according to constructing a dynamical system on a nonlinear space endowed with it topology of uniform convergence,we get that the system is described a dynamical system on such nonlinear space.Then we prove the stability of the interior equilibrium of the system by choosing a novel Lyapunov functional and LaSalle invariance principle.Further,the proposed algorithm can be extended to the models with Logistic growth rate and strong Allee effect growth rate,respectively.