LEADER 04299nam 22005295 450 001 9910299765903321 005 20200706083144.0 010 $a3-319-21873-5 024 7 $a10.1007/978-3-319-21873-1 035 $a(CKB)3710000000467527 035 $a(EBL)4178441 035 $a(SSID)ssj0001546815 035 $a(PQKBManifestationID)16140993 035 $a(PQKBTitleCode)TC0001546815 035 $a(PQKBWorkID)14795902 035 $a(PQKB)11383427 035 $a(DE-He213)978-3-319-21873-1 035 $a(MiAaPQ)EBC4178441 035 $a(PPN)188457925 035 $a(EXLCZ)993710000000467527 100 $a20150831d2015 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aLectures on Nonlinear Evolution Equations $eInitial Value Problems /$fby Reinhard Racke 205 $a2nd ed. 2015. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2015. 215 $a1 online resource (315 p.) 300 $aDescription based upon print version of record. 311 $a3-319-21872-7 320 $aIncludes bibliographical references and index. 327 $aIntroduction -- 1. Global solutions to wave equations - existence theorems -- 2. L^p - L^q-decay estimates for the linear we equation -- 3. Linear symmetric hyperbolic systems -- 3.1 Energy estimates -- 3.2 A global existence theorem -- 3.3 Remarks on other methods -- 4. Some inequalities -- 5. Local existence for quasilinear symmetric hyperbolic -- 6. High energy estimates -- 7. Weighted  a priori estimates -- 8. Global solutions to wave equations - proofs -- 8.1 Proof of Theorem 1.1 -- 8.2 Proof ot Theorem 1.2 -- 9. Other methods -- 10. Development of singularities -- 11. More evolutions equations -- 11.1 Equations of elasiticity -- 11.1.1 Initially isotropic media in R^3 -- 11.1.2 Initially cubic media in R^3 -- 11.2 Heat equations -- 11.3 Equations of thermoelasticity -- 11.4 Schrödinger equations -- 11.5 Klein-Gordon equations -- 11.6 Maxwell equations -- 11.7 Plate equations -- 12. Further aspects and questions -- 13. Evolution equations in waveguides -- 13.1 Nonlinear wave equations -- 13.1.1 Linear part -- 13.1.2 Nonlinear part -- 13.2. Schrödinger equations -- 13.3. Equations of elasticity and Maxwell equations -- 13.4 General waveguides -- Appendix -- A. Interpolation -- B. The Theorem of Cauchy-Kowalevsky -- C. A local existence theorem for hyperbolic-parabolic systems References Notation Index. 330 $aThis book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered. 606 $aDifferential equations, Partial 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 608 $aEvolution equations, Nonlinear.$2fast 608 $aInitial value problems.$2fast 615 0$aDifferential equations, Partial. 615 14$aPartial Differential Equations. 676 $a510 700 $aRacke$b Reinhard$4aut$4http://id.loc.gov/vocabulary/relators/aut$065680 906 $aBOOK 912 $a9910299765903321 996 $aLectures on nonlinear evolution equations$91522739 997 $aUNINA