DEPELOPMENTS IN OPERATOR THEORY TOWARDS EVOLUTION EQUATIONS

我们考虑了(1)非负算子的超几何函数，(2)复Ginzburg-Landau方程初边值问题的完备性。(1)高斯超几何函数F（α,β,γ; -z）首先由复平面单位圆盘上的幂级数定义。如果0 < Re α < Re γ，则F在单位圆盘外的解析延拓由C\（-∞,1）上有意义的积分表示给出。用一类闭线性算子代替复变量，得到了相应的算子值函数公式。注意到幂函数和对数函数~log z是用F（α,β,γ; -z）和F（α',β',γ'; -z^<-1>）在C\（-∞,0）上表示的，我们可以用算子值超几何函数统一定义非负算子（带逆）的分式幂函数和对数。(2)建立了具有L^2初始数据的复Ginzburg-Landau方程初边值问题（对初始数据有平滑效应）全局强解的存在唯一性。此外，我们还证明了解算子在L^2上形成一个局部Lipschitz连续算子的非线性半群。这部分改进和推广了在严格限制非线性项复系数的条件下，解算子在L^2上形成拟压缩的非线性半群的结论。

We have considered(1)hypergeometric functions of non-negative operators, and(2)the weliposedness of initial-boundary value problems for the complex Ginzburg-Landau equation.(1) The Gauss hypergeometric function F(α,β,γ ; -z) is first defined by the power series in the unit disk of the complex plane. If 0 < Re α < Re γ, then an analytic continuation of F outside the unit disk is given by the integral representation which makes sense on C＼ (-∞, 1].Replacing the complex variable with a class of closed linear operators, we obtain the corresponding formula for the operator-valued functions. Noting that the power and logarithmic functions ~log z are written down in terms of F(α,β,γ ; -z)and F(α',β',γ' ; -z^<-1>) on C＼(-∞, 0], we can define in a unified way the fractional powers and logarithm of a non-negative operator (with inverse) in terms of operator-valued hypergeometric functions.(2)The existence and uniqueness of global strong solutions to the initial-boundary value problem for the complex Ginzburg-Landau equation with L^2-initial data(smoothing effect on the initial data)is established under a condition on the power of the nonlinear term without any restriction on the complex coefficients.Moreover, we have shown that the solution operator forms a nonlinear semigroup of locally Lipschitz continuous operators on L^2.This improves and extends partially the previous result which asserts that the solution operator forms a nonlinear semigroup of quasi-contractions on L^2 under strict restriction on the complex coefficient of the nonlinear term.