الفهرس 
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Abstract
In science, a fundamental problem is that of obtaining a mathematical model of physical system, electromagnetic and a problem of two dimensional in the theory of elasticity. Scientific theory, usually, develops by the theoretician postulating a mathematical model, based perhaps on some physical laws; and this model is checked against experimental evidence until the model reasonably represents the process under consideration.
Our study is concerning with the abstract differential equations, which are known as the theory equations of evolution. These equations are based mainly on the results obtained from the semigroup theory of linear operators, and we study the nonlinear case of certain types of those equations. The main theory that solves the initial and mixed problems of hyperbolic and parabolic equations with coefficients that depend on a time variable is known as HiRe-Yosida-Phillps. Around 1959, the application of the theory to parabolic equation was initialed. Since then, the research in this field became quite active and an extensive literature has appeared.
The thesis consists of (75) pages including (44)references, it contains an introduction, basic concepts in chapter 1 and follows by three chapters.
In chapter 1, we shed the light on some fundamental facts from functional analysis, and present the theories behind semigroup, closed operators, and fractional power of operators.
In chapter 2, Cauchy problem is introduced where the general form of the evolution equation is given by:
dt
(1)
u(Q) = go
= 9i
(2)
au
in Banach space X.
Some conditions on the equation are stated, and the correct formulation of the solution is carried out. In addition, an application of that equation has been discussed.
In chapter 3, under certain conditions the following kind of evolution equation
du
— = Au(t] + Aau(t] + /(*, u(t),A^u(t}) (3)
u(0) = uq (4)
is considered in Banach space. The stability of the solution is also discussed here. Furthermore, two bounded operators, H\(t] and H^(t} are inserted in equation (3) to have the new form:
(5)
(6)
(ill
— = Au(t) at
u(G) = uq
equation (5) is also studied by the same method as equation (3).
Finally, in chapter 4 and under certain conditions, we study the following second order nonlinear evolution equation of the form
dt u(0) = 0
(7)
dt2
cfo(O) dt
_ n
(8)
is studied and, we present Zigmond Caldron transformations for applying it in many applications related to equation (7).