##
**Nonlinear wave equations. Translated from the Chinese by Yachun Li.**
*(English)*
Zbl 1411.35004

Series in Contemporary Mathematics 2. Berlin: Springer; Shanghai: Shanghai Scientific and Technical Publishers (ISBN 978-3-662-55723-5/hbk; 978-3-662-55725-9/ebook). xiv, 391 p. (2017).

This work is devoted to the derivation of estimates on the maximal time of existence of solutions, in a given Lorentz frame, of nonlinear perturbations of the wave equation in Minkowski space, with smooth, compactly supported Cauchy data of the form \((\epsilon\varphi,\epsilon\psi)\). The equations have the form \(Lu=F(Z)\), where \(Z=(u,Du,D_xDu)\), where \(L\) is the usual wave operator. The nonlinearity \(F\) depends on the solution \(u\) and its first and second derivatives with respect to time and space, excluding \(u_{tt}\), and is assumed to vanish like the power \(1+\alpha\) of some norm of \(Z\). Thus, \(\alpha=1\) for quadratic nonlinearities. The main goal is the estimation of the rate at which the maximum time \(T(\epsilon,\alpha)\) on which a classical solution exists tends to infinity, as \(\epsilon\to 0+\), for fixed \((\varphi,\psi)\). This time may be called the blow-up time, or time when, in this particular Lorentz frame, an observer would see the appearance of a first singularity.

The book is a sequel to two books by T.-T. Li and Y. Chen, in 1989 (in Chinese) and in 1992 (in English: [Global classical solutions for nonlinear evolution equations. Harlow etc.: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. (1992; Zbl 0787.35002)]), that had dealt in addition with heat and Schrödinger equations; the present work streamlines and refines their results relative to perturbations of the wave equation. The method consists in all cases in devising a suitable norm with weights involving the variable \(t\), such that the desired solution appears as the fixed point of a contraction for this norm. The authors consider an inequality to be sharp if the blow-up rate cannot be improved; for instance, the inequality \(T\geq b\epsilon^{-2}\), with some constant \(b\), is sharp if one may find data and a constant \(a\) for which \(T\leq a\epsilon^{-2}\).

The introduction (Chapter 1) is followed by two preliminary chapters, on the method of spherical means and on Sobolev-type inequalities. Chapter 4 gives estimates on solutions of the wave equations, and Chapter 5, on estimates on products and compositions of functions. Although Nirenberg is mentioned, the contributions of Gagliardo and Moser are not. Chapter 6 solves the Cauchy problem in \(H^s\) spaces with high \(s\), by a Galerkin-type method. In the following one, the equation is reduced to a quasilinear system for \(u\) and its first-order derivatives. Chapters 8 to 11 deal with lower estimates of \(T\), and Chapter 12 proves global existence when the nonlinearity satisfies the null condition. Sharpness is discussed in Chapter 13 and 14. The fifteenth and last chapter very briefly addresses related issues for other types of equations. References (4 pages) and an index (5 pages) close the volume.

The work was written in Chinese, and translated by Yachun Li. It is apparent that it was primarily meant for a Chinese-speaking audience without access to other sources: thus, for background information on the hypergeometric function, we are only referred to a work in Chinese from 1979. Also, the work focuses exclusively on the authors’ approach: for instance, the method of symmetric-hyperbolic systems is not found in the index. The Preface states that this field was “initiated by F. John in the late 1970s and early 1980s when he gave some examples to reveal the blow-up phenomenon of solutions to nonlinear wave equations.” Actually, this statement would better fit J. B. Keller’s work [Commun. Pure Appl. Math. 10, 523–530 (1957; Zbl 0090.31802)].

Walter Strauss pointed out, in his review of the 1992 monograph by Li and Chen [W. Strauss, Bull. AMS, 29, No. 2, 265–269 (1993; Zbl 1416.00022)], that he had approached these problems via the contraction mapping theorem as early as 1974. Now, Strauss’ review is quoted, but his papers, and his standard monograph on nonlinear wave equations [W. A. Strauss, Nonlinear wave equations. Expository lectures from the CBMS regional conference held at George Mason University, January 16–22, 1989. Providence, RI: American Mathematical Society (1989; Zbl 0714.35003)] are not. His review had also noted that Lorentz invariance had not been given its due place. And indeed, the main focus of the monograph, the blow-up time, is not a Lorentz invariant (see the reviewer’s [Fuchsian reduction. Applications to geometry, cosmology and mathematical physics. Basel: Birkhäuser (2007; Zbl 1169.35002)], Section 10.1.1).

The book is a sequel to two books by T.-T. Li and Y. Chen, in 1989 (in Chinese) and in 1992 (in English: [Global classical solutions for nonlinear evolution equations. Harlow etc.: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. (1992; Zbl 0787.35002)]), that had dealt in addition with heat and Schrödinger equations; the present work streamlines and refines their results relative to perturbations of the wave equation. The method consists in all cases in devising a suitable norm with weights involving the variable \(t\), such that the desired solution appears as the fixed point of a contraction for this norm. The authors consider an inequality to be sharp if the blow-up rate cannot be improved; for instance, the inequality \(T\geq b\epsilon^{-2}\), with some constant \(b\), is sharp if one may find data and a constant \(a\) for which \(T\leq a\epsilon^{-2}\).

The introduction (Chapter 1) is followed by two preliminary chapters, on the method of spherical means and on Sobolev-type inequalities. Chapter 4 gives estimates on solutions of the wave equations, and Chapter 5, on estimates on products and compositions of functions. Although Nirenberg is mentioned, the contributions of Gagliardo and Moser are not. Chapter 6 solves the Cauchy problem in \(H^s\) spaces with high \(s\), by a Galerkin-type method. In the following one, the equation is reduced to a quasilinear system for \(u\) and its first-order derivatives. Chapters 8 to 11 deal with lower estimates of \(T\), and Chapter 12 proves global existence when the nonlinearity satisfies the null condition. Sharpness is discussed in Chapter 13 and 14. The fifteenth and last chapter very briefly addresses related issues for other types of equations. References (4 pages) and an index (5 pages) close the volume.

The work was written in Chinese, and translated by Yachun Li. It is apparent that it was primarily meant for a Chinese-speaking audience without access to other sources: thus, for background information on the hypergeometric function, we are only referred to a work in Chinese from 1979. Also, the work focuses exclusively on the authors’ approach: for instance, the method of symmetric-hyperbolic systems is not found in the index. The Preface states that this field was “initiated by F. John in the late 1970s and early 1980s when he gave some examples to reveal the blow-up phenomenon of solutions to nonlinear wave equations.” Actually, this statement would better fit J. B. Keller’s work [Commun. Pure Appl. Math. 10, 523–530 (1957; Zbl 0090.31802)].

Walter Strauss pointed out, in his review of the 1992 monograph by Li and Chen [W. Strauss, Bull. AMS, 29, No. 2, 265–269 (1993; Zbl 1416.00022)], that he had approached these problems via the contraction mapping theorem as early as 1974. Now, Strauss’ review is quoted, but his papers, and his standard monograph on nonlinear wave equations [W. A. Strauss, Nonlinear wave equations. Expository lectures from the CBMS regional conference held at George Mason University, January 16–22, 1989. Providence, RI: American Mathematical Society (1989; Zbl 0714.35003)] are not. His review had also noted that Lorentz invariance had not been given its due place. And indeed, the main focus of the monograph, the blow-up time, is not a Lorentz invariant (see the reviewer’s [Fuchsian reduction. Applications to geometry, cosmology and mathematical physics. Basel: Birkhäuser (2007; Zbl 1169.35002)], Section 10.1.1).

Reviewer: Satyanad Kichenassamy (Reims)

### MSC:

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

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35L15 | Initial value problems for second-order hyperbolic equations |

35L72 | Second-order quasilinear hyperbolic equations |

35B44 | Blow-up in context of PDEs |