12 1. INTRODUCTION Chapter 11 picks up from Chapter 3 the discussion of conservation laws, now systems of conservation laws. Unlike the general theoretical develop- ments in Chapters 5–9, for which Sobolev spaces provide the proper abstract framework, we are forced to employ here direct linear algebra and calculus computations. We pay particular attention to the solution of Riemann’s problem and to entropy criteria. Chapter 12, an introduction to nonlinear wave equations, is new with this edition. We provide long time and short time existence theorems for certain quasilinear wave equations and an in-depth examination of semilinear wave equations, especially for subcritical and critical power nonlinearities in three space dimensions. To complement these existence theorems, the final section identifies various criteria ensuring nonexistence of solutions. Appendices A–E provide for the reader’s convenience some background material, with selected proofs, on inequalities, linear functional analysis, measure theory, etc. The Bibliography is an updated and extensive listing of interesting PDE books to consult for further information. Since this is a textbook and not a reference monograph, I have mostly not attempted to track down and document the original sources for the myriads of ideas and methods we will encounter. The mathematical literature for partial differential equations is truly vast, but the books cited in the Bibliography should at least provide a starting point for locating the primary sources. (Citations to selected research papers appear throughout the text.) 1.5. PROBLEMS 1. Classify each of the partial differential equations in §1.2 as follows: (a) Is the PDE linear, semilinear, quasilinear or fully nonlinear? (b) What is the order of the PDE? 2. Let k be a positive integer. Show that a smooth function defined on Rn has in general n + k − 1 k = n + k − 1 n − 1 distinct partial derivatives of order k. (Hint: This is the number of ways of inserting n − 1 dividers | within a row of k symbols ◦: for example, ◦ ◦ || ◦ ◦ ◦ | ◦ | ◦ ◦ ◦ || ◦ ◦ ◦ ◦|. Explain why each such pattern corresponds to precisely one of the partial derivatives of order k.)

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