Global classical solutions for quasilinear hyperbolic systems.

*(English)*Zbl 0841.35064
Research in Applied Mathematics. 32. Chichester: Wiley. Paris: Masson. viii, 315 p. (1994).

In this book for first-order quasilinear hyperbolic systems (FOQHS) the following two problems are studied.

(P1) Under what conditions does the problem in consideration (Cauchy problem, boundary value problem, etc.) for FOQHS admit a unique global solution? Based on this problem the regularity and the global behaviour of the solution as \(t\to \infty\) is further studied.

(P2) Under what conditions does the classical solution to the considered problem blow up in a finite time? This problem often leads to further investigation on estimates of the life span of classical solution as well as on the behaviour of blow-up phenomena.

All considerations are for one-dimensional FOQHS. The approach of problems of type (P1) or (P2) is primarily based on the theory of the local solution. The author constantly uses the book: the author and Yu Wen-ci, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series 5 (1985; Zbl 0627.35001).

The organization of the book is based on its division into eight chapters. Cauchy problems for single first-order equations, reducible quasilinear hyperbolic systems, general FQHS and quasilinear hyperbolic systems with dissipation are considered in chapters 1, 2, 3, 4, respectively. In the last chapters (5-8) boundary value and Riemann problems are discussed.

The greater part of the book represents the results of the academic research done by the author and his colloborators.

In conclusion, this is a good book adressed to graduate students as well as researches and specialists in the field of partial differential equations.

(P1) Under what conditions does the problem in consideration (Cauchy problem, boundary value problem, etc.) for FOQHS admit a unique global solution? Based on this problem the regularity and the global behaviour of the solution as \(t\to \infty\) is further studied.

(P2) Under what conditions does the classical solution to the considered problem blow up in a finite time? This problem often leads to further investigation on estimates of the life span of classical solution as well as on the behaviour of blow-up phenomena.

All considerations are for one-dimensional FOQHS. The approach of problems of type (P1) or (P2) is primarily based on the theory of the local solution. The author constantly uses the book: the author and Yu Wen-ci, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series 5 (1985; Zbl 0627.35001).

The organization of the book is based on its division into eight chapters. Cauchy problems for single first-order equations, reducible quasilinear hyperbolic systems, general FQHS and quasilinear hyperbolic systems with dissipation are considered in chapters 1, 2, 3, 4, respectively. In the last chapters (5-8) boundary value and Riemann problems are discussed.

The greater part of the book represents the results of the academic research done by the author and his colloborators.

In conclusion, this is a good book adressed to graduate students as well as researches and specialists in the field of partial differential equations.

Reviewer: L.G.Vulkov (Russe)

##### MSC:

35L60 | First-order nonlinear hyperbolic equations |

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

35B40 | Asymptotic behavior of solutions to PDEs |