Nonlinear Evolution Equations In Non-reflexive Banach Spaces

In this dissertation, we use the contraction mapping argument and the tool of harmonic analysis, especially, Littlewood-Paley decomposition and time-space estimates. A class of self-similar solutions are obtained by studying the global well-posedness in non-reflexive Banach spaces for the Cauchy problem of the nonlinear evolution equations of higher order, the system of nonlinear wave equations in three space dimensions and the coupled system of nonlinear Schrodinger equations. we use some non-reflexive Banach spaces as basic work spaces which include the initial data with self-similar structure. Also, we show the existence and uniqueness of finite energy solutions to the Cauchy problem for nonlinear wave equations of higher order and the system of nonlinear wave equations in three space dimensions.This thesis is divided into the following five parts.Part â… : We consider the Cauchy problem of nonlinear wave equations for higher order with nonlinear term |u|^αu. The existence and uniqueness of global solutions and the finite energy solutions in space L^α+2(Rⁿ) are proved for α₀ < α < 4m/(n-2m) if n > 2m and α₀ < α < +∞ if n ≤ 2m. where α₀ is the positive root of equation (n-m)x² + (n- 4m)x — 4m = 0. Afterwards, we show the existence of global solutions and self-similar solutions in space H_p^s(Rⁿ).Part â…¡: By establishing some L^p-L^q estimates of solutions to the initial value problem of homogeneous linear parabolic equations of higher order, we study the global solutions and self-similar solutions of some nonlinear parabolic equations of higher order by means of the contraction mapping argument. Meanwhile, we also get asymptoticallyself-similar solutions.Part III: We establish the global well-posedness of the Cauchy problem of nonlinear Schrodinger equations of higher order in some non-standard function spaces which contain many homogeneous functions by building some nonlinear estimates in the Lorentz spaces and Besov spaces and using the generalized Strichartz estimates. Furthermore, we derive the self-similar solutions in the case of small data.Part IV: We prove the global well-posedness of some nonlinear wave equation system with a class of data which admits homogeneous data f3 and g-j of degree —rn-j and —rrij — 1 ,j = 1,2; respectively. Hence the existence of self-similar solutions is shown, and the solutions are not necessarily radially symmetric. Moreover, we show the existence of asymptotically self-similar solutions.Part V: On the basis of the study of the global well-posedness to the Cauchy problem of single Schrodinger equations, we prove the existence of global solutions and self-similar solutions to the initial value problem of a coupled system of Schrodinger equations. Meantime, we get the existence of asymptotically self-similar solutions.