World Scientific
ISBN: 9810235356, Autumn 1998
It can be ordered online from:
World Scientific,
Amazon,
Barnes and Noble
Reviews by American Mathematical Society and Zentralblatt MATH
The book has been used as a textbook all over the world. Any
feedback would be very appreciated.
On the rest of this page you can find
Preface,
Contents,
Text Samples and
Index of the book.
Partial Differential Equations by Avner Friedman
Geometric Theory of Semilinear Parabolic Equations by Dan Henry
Semigroups of Linear Operators and Applications to PDEs by A. Pazy.
The need for such a book is due to the fact that in order to study PDEs one needs to know some functional analysis, which requires a thorough knowledge of real analysis (Lebesgue integral). Therefore, if real analysis is studied in the first year of graduate school, and functional analysis in the second year, the student only begins with PDEs in the third year  and may even have to relearn functional analysis if the prior instructor ignored unbounded operators (which sometimes happens).
The reader is expected to be comfortable with the Lebesgue integral; more specifically, with the material presented in Examples 1.3.4 and 1.5.2. The Cauchy Theorem is also used in a couple of places, with the most difficult version used in (4.44). These are the only real prerequisites for the whole book. Above this level, all theorems used are proved in the text. One may, and perhaps should, skip over some of the proofs. However, they are included in case they are needed.
With regard to the writing style, all formal statements, like Theorems, contain all assumptions except for those declared at the beginning of the section in which the statement appears. This should make it easy, even for a casual reader, to figure out what is actually assumed in a given statement. There is, however, one exception. Throughout Chapter 3 it is assumed, unless otherwise specified, that O is an arbitrary nonempty open set in R^{n}.
In the first two chapters functional analysis tools are developed and differential operators are studied as examples. SturmLiouville operators are nice examples of selfadjoint operators with compact resolvent and are reused in Chapter 4 as generators of strongly continuous semigroups. Hormander's treatment of weak solutions of constant coefficient PDEs is also presented early on as an example. The foundation of elliptic, parabolic and wave equations, as well as of Galerkin approximations, is given in the section on Sectorial Forms. Throughout the text, completeness of a number of orthonormal systems is proven.
The Fourier transform and its applications to constant coefficient PDEs are presented in Chapter 3. We briefly touch upon distributions and fundamental solutions, and prove the MalgrangeEhrenpreis Theorem. Most of Chapter 3 is devoted to study of Sobolev spaces. Many sharp results concerning existence and compactness of imbeddings, as well as interpolation inequalities, are proven. These results are applied to elliptic problems in the last two sections.
The study of evolution equations begins in Chapter 4 where the semigroup theory is introduced. The HilleYosida Theorem for strongly continuous semigroups and Hille's construction of analytic semigroups are presented. The semigroup theory and the results of the previous chapters enable us to discuss linear parabolic and wave equations. In preparation for studies of nonlinear evolution equations, the invariant subspaces associated with the semigroups and the inhomogeneous problem are also examined.
A dynamical systems approach to weakly nonlinear evolution equations is given in Chapter 5 with a nonlinear heat equation studied as an example. Trotter's approximation theory is adapted to such equations giving convergence of Galerkin and finite difference type approximations.
The chapter on semilinear parabolic equations begins with a very technical section on fractional powers of operators. Our main results contain existence, uniqueness, continuous dependence, maximal interval of existence, stability and instability results. These results are applied to the NavierStokes equations, to a stability problem in fluid mechanics, to showing how a classical solution can be obtained, and to the ChafeeInfante problem as an example of a gradient system.
I wish to thank S. N. Chow and D. R. Dunninger for their
early encouragement and my wife Pam for checking the
grammar.
Preface  v  
1  Linear Operators in Banach Spaces  1 
1.1  Metric Spaces  1 
1.2  Vector Spaces  6 
1.3  Banach Spaces  8 
1.4  Linear Operators  11 
1.5  Duals  16 
1.6  Spectrum  23 
1.7  Compact Linear Operators  29 
1.8  Boundary Value Problems for Linear ODEs  39 
1.9  Exercises  43 
2  Linear Operators in Hilbert Spaces  47 
2.1  Orthonormal Sets  47 
2.2  Adjoints  54 
2.3  Accretive Operators  58 
2.4  Weak Solutions  62 
2.5  Example: Constant Coefficient PDEs  68 
2.6  Selfadjoint Operators  70 
2.7  Example: SturmLiouville Problem  76 
2.8  Sectorial Forms  80 
2.9  Example: Harmonic Oscillator and Hermite Functions  88 
2.10  Example: Completeness of Bessel Functions  92 
2.11  Example: Finite Element Method  96 
2.12  Friedrichs Extension  99 
2.13  Exercises  104 
3  Sobolev Spaces  109 
3.1  Introduction  109 
3.2  Fourier Transform  114 
3.3  Distributions  122 
3.4  Weak Derivatives  126 
3.5  Definition and Basic Properties of Sobolev Spaces  135 
3.6  Imbeddings of W^{m,p } (O)  140 
3.7  Elliptic Problems  153 
3.8  Regularity of Weak Solutions  160 
3.9  Exercises  163 
4  Semigroups of Linear Operators  167 
4.1  Introduction  167 
4.2  Bochner Integral  171 
4.3  Basic Properties of Semigroups  176 
4.4  Example: Wave Equation  187 
4.5  Sectorial Operators and Analytic Semigroups  192 
4.6  Invariant Subspaces  204 
4.7  The Inhomogeneous Problem  Part I  209 
4.8  Exercises  212 
5  Weakly Nonlinear Evolution Equations  215 
5.1  Introduction  215 
5.2  Basic Theory  218 
5.3  Example: Nonlinear Heat Equation  222 
5.4  Approximation for Evolution Equations  224 
5.5  Example: Finite Difference Method  233 
5.6  Example: Galerkin Method for Parabolic Equations  236 
5.7  Example: Galerkin Method for the Wave Equation  239 
5.8  Friedrichs Extension and Galerkin Approximations  243 
5.9  Exercises  245 
6  Semilinear Parabolic Equations  247 
6.1  Fractional Powers of Operators  247 
6.2  The Inhomogeneous Problem  Part II  254 
6.3  Global Version  257 
6.4  Main Results  259 
6.5  Example: NavierStokes Equations  267 
6.6  Example: A Stability Problem  271 
6.7  Example: A Classical Solution  273 
6.8  Dynamical Systems  276 
6.9  Example: The ChafeeInfante Problem  279 
6.10  Exercises  282 
Bibliography  285  
List of Symbols  289  
Index  291 
Friedrichs extension and Galerkin approximations
Excerpt on Amazon shows first 6 pages of the book.
Abel's integral equation, 45 accretive, 28 , 58 , 73, 185 addition, 6 adjoint, 21 , 56, 63 adjoint problem, 40 adjoint space, 16 almost separablyvalued, 171 analytic, 27 , 117 , 169, 193 arcwiseconnected, 5 ArzelaAscoli Theorem, 5, 17, 31, 34, 90, 94, 150

Laplace operator, 70 Laplace transform, 116 Laplacian, 70 , 92, 118, 153, 155, 163, 185, 193 Legendre polynomials, 48 , 104 Liapunov function, 278 , 280 linear operator from X to Y, 11 linearly independent, 7 Lipschitz continuous, 4
