##
**On the Cauchy problem.**
*(English)*
Zbl 0616.35002

Notes and Reports in Mathematics in Science and Engineering, Vol. 3. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers); Beijing, The People’s Republic of China: Science Press. VIII, 177 p. $ 36.00 (1985).

Each chapter in this book is an expanded form of a lecture from the series given in 1983 at Wuhan University; additional material and a number of Appendices have been included to make the account more self contained.

In Chapter 1 general linear partial differential equations of evolution type are introduced and illustrated in terms of three standard models; the Wave, Heat and Schrödinger equations. In this chapter the notions of \(C^{\infty}\)- and \(H^{\infty}\)-wellposedness are defined and various essential differences between the three model equations are discussed. The chapter ends with a comprehensive historical account indicating how the study of these equations has developed since the pioneering work of Hadamard.

Chapter 2 is devoted to a detailed discussion of \(H^{\infty}\)- wellposedness for equations in which the coefficients depend only on t the time variable. Conditions are obtained, mainly necessary and sufficient, for \(H^{\infty}\)-wellposedness and they are illustrated by considering in detail a specific equation.

Chapter 3 begins by introducing the Lax-Mizohata Theorem together with a brief historical background. The main part of the chapter is concerned with obtaining, by a localisation technique, conditions which are mainly only necessary for problems involving equations with coefficients depending on both x and t to be \(H^{\infty}\)-wellposed. Basic facts concerning pseudodifferential operators are given and briefly proved in an Appendix.

In Chapter 4 the concept of a Gevrey class, \(\gamma^{(s)}\), is introduced and it is indicated how the Cauchy problem can be defined in such classes. Distinctions are drawn between strictly hyperbolic and nonstrictly hyperbolic (weakly hyperbolic) equations and it is pointed out that strictly hyperbolic problems are \(C^{\infty}\)-wellposed but nonstrictly hyperbolic problems are, in general, not \(H^{\infty}\)- wellposed unless additional (Levi) conditions are imposed on the lower order terms. However if the coefficients and all the given data belong to suitable Gevrey classes then it is pointed out that unique solvability can be obtained without any conditions being imposed on lower order terms. It is shown that problems involving nonstrictly hyperbolic equations with constant multiplicity can be reduced to the study of a certain elementary operator. Conditions are obtained indicating when \(H^{\infty}\)-wellposedness can be obtained. For clarity proofs of many results are relegated to Appendices.

The next two chapters are concerned with microlocal analysis in Gevrey classes. The starting point for the notion of microlocal analysis used here is the concept of a wave front set (or singular spectrum). This is first defined in the \(C^{\infty}\) sense and a Fundamental Theorem characterizing the wave front set is given. The main aim of chapters 5 and 6 is to show that this Fundamental Theorem also holds in Gevrey classes. It is shown that a straightforward extension of the \(C^{\infty}\) case is not possible. However the so called (\(\alpha\),\(\beta)\) method developed in Chapter 3 can be adapted for analysis in Gevrey classes. The arguments are often quite lengthy and again, for clarity, appear in Appendices.

The final chapter begins with some general remarks on evolution equations and the notion of Koveleskian evolution equations is introduced to distinguish between hyperbolic and nonhyperbolic equations. Non- Koveleskian equations which, it is pointed out, present quite delicate problems of analysis, include parabolic equations and, in particular, Schrödinger type equations. These classes of equation are characterized in terms of their various types of wellposedness (e.g. \(C^{\infty}\)- or \(H^{\infty}\)-) in directions of t. The remainder of the chapter is concerned with an investigation of Schrödinger type equations and in particular with conditions governing their \(L^ 2\)-wellposedness and a necessary and a sufficient condition are given.

The account is well motivated and contains references to open problems.

In Chapter 1 general linear partial differential equations of evolution type are introduced and illustrated in terms of three standard models; the Wave, Heat and Schrödinger equations. In this chapter the notions of \(C^{\infty}\)- and \(H^{\infty}\)-wellposedness are defined and various essential differences between the three model equations are discussed. The chapter ends with a comprehensive historical account indicating how the study of these equations has developed since the pioneering work of Hadamard.

Chapter 2 is devoted to a detailed discussion of \(H^{\infty}\)- wellposedness for equations in which the coefficients depend only on t the time variable. Conditions are obtained, mainly necessary and sufficient, for \(H^{\infty}\)-wellposedness and they are illustrated by considering in detail a specific equation.

Chapter 3 begins by introducing the Lax-Mizohata Theorem together with a brief historical background. The main part of the chapter is concerned with obtaining, by a localisation technique, conditions which are mainly only necessary for problems involving equations with coefficients depending on both x and t to be \(H^{\infty}\)-wellposed. Basic facts concerning pseudodifferential operators are given and briefly proved in an Appendix.

In Chapter 4 the concept of a Gevrey class, \(\gamma^{(s)}\), is introduced and it is indicated how the Cauchy problem can be defined in such classes. Distinctions are drawn between strictly hyperbolic and nonstrictly hyperbolic (weakly hyperbolic) equations and it is pointed out that strictly hyperbolic problems are \(C^{\infty}\)-wellposed but nonstrictly hyperbolic problems are, in general, not \(H^{\infty}\)- wellposed unless additional (Levi) conditions are imposed on the lower order terms. However if the coefficients and all the given data belong to suitable Gevrey classes then it is pointed out that unique solvability can be obtained without any conditions being imposed on lower order terms. It is shown that problems involving nonstrictly hyperbolic equations with constant multiplicity can be reduced to the study of a certain elementary operator. Conditions are obtained indicating when \(H^{\infty}\)-wellposedness can be obtained. For clarity proofs of many results are relegated to Appendices.

The next two chapters are concerned with microlocal analysis in Gevrey classes. The starting point for the notion of microlocal analysis used here is the concept of a wave front set (or singular spectrum). This is first defined in the \(C^{\infty}\) sense and a Fundamental Theorem characterizing the wave front set is given. The main aim of chapters 5 and 6 is to show that this Fundamental Theorem also holds in Gevrey classes. It is shown that a straightforward extension of the \(C^{\infty}\) case is not possible. However the so called (\(\alpha\),\(\beta)\) method developed in Chapter 3 can be adapted for analysis in Gevrey classes. The arguments are often quite lengthy and again, for clarity, appear in Appendices.

The final chapter begins with some general remarks on evolution equations and the notion of Koveleskian evolution equations is introduced to distinguish between hyperbolic and nonhyperbolic equations. Non- Koveleskian equations which, it is pointed out, present quite delicate problems of analysis, include parabolic equations and, in particular, Schrödinger type equations. These classes of equation are characterized in terms of their various types of wellposedness (e.g. \(C^{\infty}\)- or \(H^{\infty}\)-) in directions of t. The remainder of the chapter is concerned with an investigation of Schrödinger type equations and in particular with conditions governing their \(L^ 2\)-wellposedness and a necessary and a sufficient condition are given.

The account is well motivated and contains references to open problems.

Reviewer: G.Roach

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35K05 | Heat equation |

35L05 | Wave equation |

35J10 | Schrödinger operator, Schrödinger equation |

35B65 | Smoothness and regularity of solutions to PDEs |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |