Qualitative Study On Dichotomous Solutions For A Class Of Infinite Dimensional Differential Equations

Some functional differential equations or partial differential equations can be rewritten as some abstract ordinary differential equations with a certain operator on the suitable infinite dimensional Banach space,so studying the existence and related properties of the solution for these infinite dimensional differential equations can be transformed into studying the properties of the solution for the corresponding abstract ordinary differential equations.The main work in the doctoral dissertation is to study a class of abstract ordinary differential equations with sectorially dichotomous operator on the infinite dimensional Banach space by utilizing operator and semigroup theory,dynamical system theory and qualitative analysis.Under the condition of the given dichotomous initial value,we obtain some theoretical results for this kind of ab-stract ordinary differential equations,they include that the existence and uniqueness,regularity,continuous dependence on the dichotomous initial value and norm estimate of dichotomous solutions,and the existence and smoothness of invariant manifolds near the equilibrium.More-over,these theoretical results are applied to a class of quasilinear elliptic partial differential equations in infinite cylindrical domain.More specifically,they consist of three parts below.In the first part,we study a densely-defined and hyperbolic bisectorial operator S defined on the Banach space Z.Firstly,we give a sufficient criterion for the hyperbolic bisectorial operator S,and this criterion is easier to verify than the definition of hyperbolic bisectorial operator.Secondly,based on the spectral distribution of S on the complex plane,by obatining the results that the spectral decomposition of S at the infinity and the direct sum decomposition of Z,Z=X?Y,we give a sufficient condition that S being sectorially dichotomous,such that S|_Xand-S|_Yboth are densely defined and sectorial operators on X and Y,respectively.These results generalize the work of Bart et al.[9]for which the operator being exponential dichotomous,and improve the work of Winklmeier and Wyss[94]concerning the spectral decomposition of hyperbolic bisectorial operator.Thirdly,We generalize the definitions of fractional power and intermediate space of the sectorial operator to the sectorially dichotomous operator.We construct the fractional power of the sectorially dichotomous operator S and two intermediate spaces between whole space Z and the domain D?S?,that is,the fractional power space Z_?and the interpolation space D_S??,??,?,???0,1?.Moreover,we obtain the direct sum decomposition of Z_?and D_S??,??,respectively,and give the interpolation estimation for D_S??,???Z_??Z?0<?<?<1?.In this doctoral dissertation,Z_?will be used to study the nonlinear term of the abstract ordinary differential equation,and D_S??,??will be used to study the regularity of dichotomous solutions.?See the third chapter of this doctoral dissertation.?In the second part,based on the given dichotomous initial condition,we study a class of abstract ordinary differential equations with sectorially dichotomous operator on the infinite dimensional Banach space,it consists of a class of linear nonhomogeneous equations and three kinds of semilinear equations with different assumptions for nonlinear terms.Firstly,we study a class of linear inhomogeneous equations,and obtain the existence,uniqueness and regularity of solution.Secondly,we study a class of semilinear equations,and obtain the existence,uniqueness,continuous dependence on the dichotomous initial value,regularity and Z_?-estimate of dichotomous solutions.Thirdly,we study a class of nonautonomous differential equations with local C^k,?nonlinear term with respect to the state variable,and show the existence and C^k,?smoothness of locally stable integral manifold near the equilibrium.Locally unstable integral manifold follows from locally stable integral manifold by reversing time variable directly.Fourthly,we study a class of autonomous differential equations with global C^k,?nonlinear term,and show the existence and C^k,?smoothness of globally stale manifold near the equilibrium.Globally unstable manifold follows from globally stable manifold by reversing time variable directly.?See the fourth and fifth chapter of this doctoral dissertation.?In the third part,we apply the above perturbation results of sectorially dichotomous operator to a class of quasilinear elliptic paritial differential eqautions in infinite cylindrical domain,and show the existence and asymptotic behavior of solutions for elliptic paritial differential eqaution under the specified boundary value conditions.Compared with ElBialy's work[27]that he uses the generator of strongly continuous bisemigroup to study this kind of elliptic equations,we get a higher regularity of solutions of the elliptic equation under the assumption that the nonlinear term has the same Lipschitz property.Then,we consider an abstract ordinary differential equation with non-densely defined and hyperbolic bisectorial operator,and prove that the above theortical results can be applied on ???.?See the sixth chapter of this doctoral dissertation.?...