04458nam 22005895 450 991081909790332120240516010750.01-4612-0873-410.1007/978-1-4612-0873-0(CKB)3400000000089300(SSID)ssj0001298441(PQKBManifestationID)11686519(PQKBTitleCode)TC0001298441(PQKBWorkID)11241627(PQKB)10351096(DE-He213)978-1-4612-0873-0(MiAaPQ)EBC3073820(PPN)238032868(EXLCZ)99340000000008930020121227d1994 u| 0engurnn|008mamaatxtccrShock Waves and Reaction—Diffusion Equations /by Joel Smoller2nd ed. 1994.New York, NY :Springer New York :Imprint: Springer,1994.1 online resource (XXIII, 634 p.) Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,0072-7830 ;258Bibliographic Level Mode of Issuance: Monograph0-387-94259-9 1-4612-6929-6 Includes bibliographical references and index.1 Ill-Posed Problems -- 2 Characteristics and Initial-Value Problems -- 3 The One-Dimensional Wave Equation -- 4 Uniqueness and Energy Integrals -- 5 Holmgren’s Uniqueness Theorem -- 6 An Initial-Value Problem for a Hyperbolic Equation -- 7 Distribution Theory -- 8 Second-Order Linear Elliptic Equations -- 9 Second-Order Linear Parabolic Equations -- 10 Comparison Theorems and Monotonicity Methods -- 11 Linearization -- 12 Topological Methods -- 13 Bifurcation Theory -- 14 Systems of Reaction-Diffusion Equations -- 15 Discontinuous Solutions of Conservation Laws -- 16 The Single Conservation Law -- 17 The Riemann Problem for Systems of Conservation Laws -- 18 Applications to Gas Dynamics -- 19 The Glimm Difference Scheme -- 20 Riemann Invariants, Entropy, and Uniqueness -- 21 Quasi-Linear Parabolic Systems -- 22 The Conley Index -- 23 Index Pairs and the Continuation Theorem -- 24 Travelling Waves -- 25 Recent Results -- Author Index.For this edition, a number of typographical errors and minor slip-ups have been corrected. In addition, following the persistent encouragement of Olga Oleinik, I have added a new chapter, Chapter 25, which I titled "Recent Results." This chapter is divided into four sections, and in these I have discussed what I consider to be some of the important developments which have come about since the writing of the first edition. Section I deals with reaction-diffusion equations, and in it are described both the work of C. Jones, on the stability of the travelling wave for the Fitz-Hugh-Nagumo equations, and symmetry-breaking bifurcations. Section II deals with some recent results in shock-wave theory. The main topics considered are L. Tartar's notion of compensated compactness, together with its application to pairs of conservation laws, and T.-P. Liu's work on the stability of viscous profiles for shock waves. In the next section, Conley's connection index and connection matrix are described; these general notions are useful in con­ structing travelling waves for systems of nonlinear equations. The final sec­ tion, Section IV, is devoted to the very recent results of C. Jones and R. Gardner, whereby they construct a general theory enabling them to locate the point spectrum of a wide class of linear operators which arise in stability problems for travelling waves. Their theory is general enough to be applica­ ble to many interesting reaction-diffusion systems.Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,0072-7830 ;258Mathematical analysisAnalysis (Mathematics)Analysishttps://scigraph.springernature.com/ontologies/product-market-codes/M12007Mathematical analysis.Analysis (Mathematics).Analysis.515Smoller Joelauthttp://id.loc.gov/vocabulary/relators/aut57297MiAaPQMiAaPQMiAaPQBOOK9910819097903321Shock Waves and Reaction—Diffusion Equations3971674UNINA