Existence、Uniqueness And Stability Of Mild Solutions For Stochastic Evolution Equations

For more than half a century,stochastic evolution equations have played a major role in analyzing the dynamic phenomena in the biological and physical sciences,as well as engineering.Many researches have shown that the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena.Furthermore,the effect of nonlocal diffusion is better than that of classical diffusion for describing some real phenomena.Thus people gradually focus on non-local problems.On the other hand,retarded differential equations have attracted considerable interest in the literature due to physical reasons with non-instant transmission phenomena such as high velocity fields in wind tunnel experiments,or other memory processes,or biological motivations like species growth or incubating time in disease models among many others.Therefore,this dissertation investigate stochastic nonlocal or delay evolution equations.In the first chapter,we summarize the development and the current research situation of theories related to stochastic evolution equations,and show the main contents,research methods and innovation points of this dissertation.Then we briefly recall some relevant preliminaries from stochastic analysis theory.The results of this dissertation are divided into the following six parts.In the second chapter,we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions.We introduce a lemma to overcome the difficulty of the continuity of the Lévy semigroup.Based on this result and the technique of the measure of noncompactness,we establish the local existence of mild solutions in C(O,T;Lq(Ω;Lq(RN))under some weaker growth conditions.Moreover,we obtain the existence of mild solutions on any finite interval by using the general growth conditions on nonlinear terms.Finally,the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear term.In the third chapter,we investigate large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise.However,as far as we know,the polynomial stability problem of stochastic delay differential equations has been studied in recent years.In contrast,there are relatively few works on stochastic partial differential equations with pantograph delay.We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate(including both polynomial and logarithmic rates)in the pth moment and almost sure senses.The major difference in proof of pth moment stability is that the analysis is based on the Banach fixed point theorem and various estimates involving the gamma function.Moreover,by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis,we construct the nontrivial equilibrium solution,defined for t∈R,for stochastic pantograph delay evolution equations with nonlinear multiplicative noise.In particular,the uniqueness,H?lder regularity in time and general stability,in pth moment and almost sure senses,of the nontrivial equilibrium solution are established.In the fourth chapter,we consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order p>2 driven by a tempered fractional Brownian motion(tfBm)Bσ,λ(t)with-1/2<σ<0 and λ>0.First,the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to tfBm in the sense of pth moment.Moreover,based on the relations between the stochastic integrals with respect to tfBm and fractional Brownian motion(fBm),we show the continuity of mild solutions in the case of λ→0,σ∈(-1/2,0)or A>0,σ→σ0∈(-1/2,0).In particular,we obtain pth moment H?lder regularity in time and pth polynomial stability of mild solutions.This paper can be regarded as a first step to study the challenging model:stochastic 2D-Navier-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise.In the fifth chapter,we investigate the asymptotic behaviour of stochastic pantograph delay evolution equations driven by a tfBm with Hurst parameter H>1/2.First of all,the global existence,uniqueness and mean square stability with general decay rate of mild solutions are established.In particular,we would like to point out that our analysis is not necessary to construct Lyapunov functions,but we deal directly with stability via the Banach fixed point theorem,the fractional power of operators and the semigroup theory.It is worth emphasizing that a novel estimate of stochastic integrals with respect to tfBm is presented,which greatly contributes to the stability analyses.Then after extending the factorization formula to the tfBm case,we construct the nontrivial equilibrium solution,defined for t∈R,by means of an approximation technique and a convergence analysis.It is worth mentioning here that this is the first work on the construction of the nontrivial equilibrium solution to stochastic evolution equations with pantograph delay and tempered fractional noise.Moreover,we analyze the H?lder regularity in time and general stability(including both polynomial and logarithmic stability)of the nontrivial equilibrium solution in the sense of mean square.As an example of application,the reaction diffusion neural network system with pantograph delay is considered,and the nontrivial equilibrium solution and general stability of the system are proved under the Lipschitz assumption.In the sixth chapter,a theoretical framework for general stability and global convergence analyses is established for stochastic pantograph delay evolution equations driven by tfBm with Hurst index H∈(0,1/2)U(1/2,1)and tempered parameterρ>0.A novel estimate of the stochastic integral with respect to tfBm is the key of our analyses,being valid for both H<1/2 and H>1/2.We first prove the global existence,mean square stability with general decay rate and global H?lder regularity in time of the exact solution by utilizing the convolution method,the Young convolution inequality,the semigroup theory and the Banach fixed point theorem.These stabilities may be specialized as the exponential,polynomial,and logarithmic one.The result of H?lder regularity we present here is available for t∈[0,∞).Then we use the spectral Galerkin method to discretize the fractional Laplacian,and further show the general stability for the mild solution of the corresponding full-discrete system in the sense of mean square.Based on these stability and regularity results,we establish a convergence theorem for the global solution of stochastic pantograph delay evolution equations with the infinite dimensional tempered fractional noise given in a trace class.Finally,the numerical experiments are performed to confirm the theoretical results.In particular,the influence of Hurst index H and tempered parameter p on the sample paths of the stochastic pantograph delay evolution system is investigated from both theory and numerical simulation.To the best of our knowledge,this is the first work on theoretical research and numerical analysis for the stochastic PDE driven by tempered fractional Gaussian noise with H∈(0,1/2)U(1/2,1)and ρ>0.In the seventh chapter,we are interested in stochastic delay evolution equations driven by tfBm BQσ,λ(t)with time fractional operator of order α∈(1/2+σ,1),whereσ∈(-1/2,0)and λ>0.First,we establish the global existence and uniqueness of mild solutions by using the new established estimation of stochastic integrals with respect to tfBm.Moreover,based on the relations between the stochastic integrals with respect to tfBm and fBm,we show the continuity of mild solutions for stochastic delay evolution equations when tempered fractional noise is reduced to fractional noise.We also analyze the stability with general decay rate(including exponential,polynomial and logarithmic stability)of mild solutions for stochastic delay evolution equations with tfBm and time tempered fractional operator.