# Project Materials

## Evolution Equations and Applications

Epigraph ii
Preface iii
Acknowledgement iv
Dedication v
1 PRELIMINARIES 1
1.1 Basic notions of Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Differentiability in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Riemann Integration of functions with values in Banach spaces . . . . . . . . 12
1.1.5 Gronwall Lemma, Differential Inequality . . . . . . . . . . . . . . . . . . . . . 15
1.1.6 Function Spaces with Values in a Banach Space . . . . . . . . . . . . . . . . . 16
1.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1 Spectral Theory of linear Operators . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.2 Semigroups of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.3 Examples of Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.4 Infinitesimal Generator of a C0-semigroup . . . . . . . . . . . . . . . . . . . . 26
1.2.5 Lumer-Phillips Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 ABSTRACT LINEAR EVOLUTION EQUATIONS 36
2.1 Linear Evolution Equations in nite dimensional spaces: Well-Posedness . . . . . . . 36
2.2 Linear Evolution Equations in infinite Dimensional Spaces: Abstract Cauchy Problem 39
3 SEMI-LINEAR EVOLUTION EQUATIONS 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Theory for Lipschitz-Type Forcing terms . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.2 Existence of Local Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.3 Continuous Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.4 Extendability of Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.5 Global Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.6 Long-Term Behavior of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Theory for Non-Lipschitz-Type Forcing Terms . . . . . . . . . . . . . . . . . . . . . . 64
3.3.1 Existence and Uniqueness Result . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Theory under compactness assumption . . . . . . . . . . . . . . . . . . . . . . 71
4 APPLICATIONS 75
4.1 The Homogeneous Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 The Classical Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 The Nonlinear Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Nonlinear Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Bibliography 84

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CHAPTER ONE

PRELIMINARIES

1.1 Basic notions of Functional Analysis

In this section we recall some definitions and results from linear functional analysis Definition 1.1.1 Let X be a linear space over a eld K; where K holds either for R or C.

A mapping k:k: X 􀀀! R is called a norm provided that the following conditions hold:

i) kxk 0 for all x 2 X, and kxk= 0 , x = 0

ii) kxk= jjkxk, for all 2 K; x 2 X

iii) kx + yk kxk+kyk, for arbitrary x; y 2 X.

If X is a linear space and k:k is a norm on X, then the pair (X; k:k) is called a normed linear space over K.
Should no ambiguity arise about the norm, we simply abbreviate this pair by saying that X is a normed linear space over K.

Example . Let X = C([0; 1]) be the space of all real-valued continuous functions on [0; 1]. Each of the following expressions denes on the vector space C([0; 1]) a norm which is in common use.

kfkp=
R 1
0 (jf(t)j)pdt
1
p , for every f 2 C([0; 1]), and any p 2 [1;1)
kfk1 = ess sup jfj = inffM 0 : jf(x)j M a:eg

Definition 1.1.2 (Equivalent norms)

Two norms k:k1 and k:k2 dened on a linear space X are said to be equivalent if there exists > 0 and > 0 constants such that kxk1 kxk2 kxk1; 8x 2 X:

Theorem 1.1.1 In a nite dimensional linear space, all the norms are equivalent.

Definition 1.1.3 Every normed linear space E is canonically endowed with a metric d dened on E E by
d(x; y) = jjx 􀀀 yjj 8 x; y 2 E:

Definition 1.1.4 (Cauchy sequence)

A sequence (xn)n1 of elements of a normed vector space X is a Cauchy sequence if lim
n;m!1

kxn 􀀀 xmk= 0:

That is, for any > 0 there is an integer N = N() such that kxn 􀀀 xmklinear space has a completion. The notion of completeness is also dened for metric spaces which need not have any linear structure.

Example (Banach spaces). The normed linear space
􀀀
C([0; 1]); k k1

is a Banach space. Also the space of all bounded linear maps from R to R denoted by B(R) is a Banach space.

The completion of the normed linear space
􀀀
C([0; 1]); k k2

where k k2 is dened by
kfk2 =
Z 1
0
jf(t)j2dt
1
2
is L2(0; 1) (see Definition 1.1.7).

1.1.1 Linear operators

In this section X and Y are normed linear spaces over a eld K.

Definition 1.1.7 A K-linear operator T from X into Y is a map T : X 􀀀! Y satisfying the following property
T(x + y) = Tx + Ty

for all ; 2 K and all x; y 2 X:

When Y = K, such a map is called a linear functional or a linear form.

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