Project Materials

MATHEMATICS PROJECT TOPICS

Evolution Equations and Applications – Premium Researchers

Do You Have New or Fresh Topic? Send Us Your Topic

Equations and Applications

 

TABLE OF CONTENTS

Epigraph ii
Preface iii
Acknowledgement iv
Dedication v
1 PRELIMINARIES 1
1.1 Basic notions of Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 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 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- 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
84

 

 

Do You Have New or Fresh Topic? Send Us Your Topic 

 

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 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.

 

Evolution Equations and Applications - Premium Researchers

 

Evolution Equations and Applications – Premium Researchers

Not What You Were Looking For? Send Us Your Topic

INSTRUCTIONS AFTER PAYMENT

After making payment, kindly send the following:
  • 1.Your Full name
  • 2. Your Email Address
  • 3. Your Phone Number
  • 4. Amount Paid
  • 5. Project Topic
  • 6. Location you made payment from

» Send the above details to our email; [email protected] or to our support phone number; (+234) 0813 2546 417 . As soon as details are sent and payment is confirmed, your project will be delivered to you within minutes.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Advertisements