LEADER 04935nam 22006613 450 001 9910960017603321 005 20231110223118.0 010 $a9781470471699 010 $a1470471698 035 $a(MiAaPQ)EBC29379018 035 $a(Au-PeEL)EBL29379018 035 $a(CKB)24267685500041 035 $a(OCoLC)1336954728 035 $a(EXLCZ)9924267685500041 100 $a20220721d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aType II Blow up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (88 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.278 311 08$aPrint version: Burzio, Stefano Type II Blow up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on Providence : American Mathematical Society,c2022 9781470453466 327 $aCover -- Title page -- Chapter 1. Introduction -- 1.1. The type II blow up solutions of [33], [32] -- 1.2. The effect of symmetries on the solutions of Theorem 1.1 -- 1.3. Conditional stability of type II solutions -- 1.4. Spectral theory associated with the linearisation ? -- 1.5. Description of the data perturbation in terms of the distorted Fourier transform -- 1.6. Outline of the main result from [26] -- 1.7. Figures -- Chapter 2. The main theorem and outline of the proof -- 2.1. The main theorem -- 2.2. Outline of the proof -- Chapter 3. Construction of a two parameter family of approximate blow up solutions -- 3.1. Step 0: the bulk term -- 3.2. Step 1: choice of the first correction ? -- 3.3. Step 2: the ? error -- 3.4. Step 3: choice of second correction ? -- 3.5. Step 4: the ? error -- 3.6. Step 5: inductive step -- 3.7. Step 6: choice of _{ ?, }, =1,2 -- Chapter 4. Modulation theory -- determination of the parameters _{1,2}. -- 4.1. Re-scalings and the distorted Fourier transform -- 4.2. The effect of scaling the bulk part -- Chapter 5. Iterative construction of blow up solution almost matching the perturbed initial data -- 5.1. Formulation of the perturbation problem on Fourier side -- 5.2. The proof of Theorem 5.1 -- 5.3. Translation to original coordinate system -- Chapter 6. Proof of Theorem 2.1 -- Chapter 7. Outlook -- Bibliography -- Index -- Back Cover. 330 $a"We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation on constructed in Krieger, Schlag, and Tartaru ("Slow blow-up solutions for the critical focusing semilinear wave equation", 2009) and Krieger and Schlag ("Full range of blow up exponents for the quintic wave equation in three dimensions", 2014) are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter is sufficiently close to the self-similar rate, i. e., is sufficiently small. This result is qualitatively optimal in light of the result of Krieger, Nakamishi, and Schlag ("Center-stable manifold of the ground state in the energy space for the critical wave equation", 2015). The paper builds on the analysis of Krieger and Wong ("On type I blow-up formation for the critical NLW", 2014)"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aNonlinear wave equations 606 $aBlowing up (Algebraic geometry) 606 $aPerturbation (Mathematics) 606 $aAsymptotic expansions 606 $aIterative methods (Mathematics) 606 $aFourier transformations 606 $aPartial differential equations -- Hyperbolic equations and systems -- Wave equation$2msc 606 $aPartial differential equations -- Qualitative properties of solutions -- Asymptotic behavior of solutions$2msc 615 0$aNonlinear wave equations. 615 0$aBlowing up (Algebraic geometry) 615 0$aPerturbation (Mathematics) 615 0$aAsymptotic expansions. 615 0$aIterative methods (Mathematics) 615 0$aFourier transformations. 615 7$aPartial differential equations -- Hyperbolic equations and systems -- Wave equation. 615 7$aPartial differential equations -- Qualitative properties of solutions -- Asymptotic behavior of solutions. 676 $a515/.353 676 $a515.353 686 $a35L05$a35B40$2msc 700 $aBurzio$b Stefano$01800729 701 $aKrieger$b Joachim$01071019 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910960017603321 996 $aType II Blow up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on$94345650 997 $aUNINA