Format:
Online-Ressource (VIII, 447 p. 5 illus., 1 illus. in color, digital)
ISBN:
9783319001258
Series Statement:
Springer Proceedings in Mathematics & Statistics 44
Content:
Preface -- Global Existence and Energy Decay of Solutions for a Nondissipative Wave Equation with a Time-Varying Delay Term -- Non-uniqueness and uniqueness in the Cauchy problem of elliptic and backward-parabolic equations -- On internal regularity of solutions to the initial value problem for the Zakharov–Kuznetsov equation -- Singular semilinear elliptic equations with subquadratic gradient terms -- On the parabolic regime of a hyperbolic equation with weak dissipation: the coercive case -- H¥ well-posedness for degenerate p-evolution models of higher order with time-dependent coefficients -- On the global solvability for semilinear wave equations with smooth time dependent propagation speeds -- Filippov Solutions to Systems of Ordinary Differential Equations with Delta Function Terms as Summands -- Resolvent estimates and scattering problems for Schr¨odinger, Klein-Gordon and wave equations -- On an Optimal Control Problem for the Wave Equation in One Space Dimension Controlled by Third Type Boundary Data -- Critical exponent for the semilinear wave equation with time or space dependent damping -- A note on a class of conservative, well-posed linear control systems -- Recent progress in smoothing estimates for evolution equations -- Differentiability of Inverse Operators -- Quasi-symmetrizer and hyperbolic equations -- Solution of the Cauchy problem for generalized Euler-Poisson-Darboux equation by the method of fractional integrals -- Global Solutions of Semilinear System of Klein-Gordon Equations in de Sitter Spacetime.
Content:
Progress in Partial Differential Equations is devoted to modern topics in the theory of partial differential equations. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. The contributors were participants of the 8th ISAAC congress in Moscow in 2011 or are members of the PDE interest group of the ISAAC society. This volume is addressed to graduate students at various levels as well as researchers in partial differential equations and related fields. The reader will find this an excellent resource of both introductory and advanced material. The key topics are: • Linear hyperbolic equations and systems (scattering, symmetrisers) • Non-linear wave models (global existence, decay estimates, blow-up) • Evolution equations (control theory, well-posedness, smoothing) • Elliptic equations (uniqueness, non-uniqueness, positive solutions) • Special models from applications (Kirchhoff equation, Zakharov-Kuznetsov equation, thermoelasticity).
Note:
Description based upon print version of record
,
Progress in Partial Differential Equations; Preface; Contents; Chapter 1: Global Existence and Energy Decay of Solutions for a Nondissipative Wave Equation with a Time-Varying Delay Term; 1.1 Introduction; 1.2 Preliminaries and Main Results; 1.3 Global Existence; The First Estimate; The Second Estimate; 1.3.1 Analysis of the Nonlinear Term; 1.4 Asymptotic Behavior; Appendix; References; Chapter 2: Non-uniqueness and Uniqueness in the Cauchy Problem of Elliptic and Backward-Parabolic Equations; 2.1 Introduction; 2.1.1 Modulus of Continuity and Related Oscillation Conditions; 2.2 Non-uniqueness
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2.2.1 Non-uniqueness Under Global Conditions2.2.2 Non-uniqueness Under Local Conditions; 2.2.3 Non-uniqueness Under a Mixed Condition; 2.2.4 Scheme of the Construction of the Counterexamples; Step 1: Auxiliary Functions and Sequences; Step 2: Construction of a Solution u on [an,an+1]; Step 3: Construction of a Suitable a(t); Step 4: The Regularity of a(t); Step 5: Definition of Lower-Order Coefficients; 2.3 Uniqueness; 2.3.1 Uniqueness Under Global Conditions; 2.3.2 Uniqueness Under Local Conditions; 2.3.3 Degenerate Operators and Local Conditions; 2.3.4 Open Problems and Further Developments
,
2.4 Continuous Dependence for Backward-Parabolic OperatorsReferences; Chapter 3: On Internal Regularity of Solutions to the Initial Value Problem for the Zakharov-Kuznetsov Equation; 3.1 Introduction. Description of Main Results; 3.2 Sobolev Derivatives; 3.3 Fundamental Solution to the Linearized Equation; 3.4 Continuous Derivatives; References; Chapter 4: Singular Semilinear Elliptic Equations with Subquadratic Gradient Terms; 4.1 Introduction; 4.2 Main Results; 4.3 Proof of Theorem 1; 4.4 Proof of Theorem 2; 4.5 Proof of Theorem 3; 4.6 Proof of Theorem 4 and Theorem 5
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4.7 Proof of Theorem 6References; Chapter 5: On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case; 5.1 Introduction; 5.2 Statements; 5.2.1 Previous Works; 5.2.2 Main Results; Open Problem 2.7; 5.2.3 Heuristics; 5.2.4 Linearization; 5.3 Proofs; 5.3.1 Proof of Theorem 2.8; 5.3.2 Comparison Results for ODEs; 5.3.3 Proof of Theorem 2.9; Equivalence Between Energies; Differential Inequality for Eepsilon; Differential Inequality for Fepsilon; Differential Inequality for Gepsilon; 5.3.4 Singular Perturbation: Preliminary Estimates; 5.3.5 Proof of Theorem 2.10
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Differential Inequality for EepsilonDifferential Inequality for Fepsilon; Differential Inequality for Gepsilon; 5.3.6 Proof of Theorem 2.11; 5.3.7 Proof of Theorems 2.1, 2.2, 2.3, 2.4; References; Chapter 6: Hinfty Well-Posedness for Degenerate p-Evolution Models of Higher Order with Time-Dependent Coefficients; 6.1 Introduction; 6.2 General Notation and Main Theorem; 6.3 Proof; 6.3.1 First Step of the Proof; 6.3.2 Symbol Classes and Zones; 6.3.3 Treatment in the Pseudo-differential Zone; 6.3.4 Treatment in the Evolution Zone; 6.3.5 Verification; 6.4 Outlook; 6.4.1 About Optimality-C1-Theory
,
6.4.2 About Optimality-C2-theory
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Preface -- Global Existence and Energy Decay of Solutions for a Nondissipative Wave Equation with a Time-Varying Delay Term -- Non-uniqueness and uniqueness in the Cauchy problem of elliptic and backward-parabolic equations -- On internal regularity of solutions to the initial value problem for the Zakharov-Kuznetsov equation -- Singular semilinear elliptic equations with subquadratic gradient terms -- On the parabolic regime of a hyperbolic equation with weak dissipation: the coercive case -- H¥ well-posedness for degenerate p-evolution models of higher order with time-dependent coefficients -- On the global solvability for semilinear wave equations with smooth time dependent propagation speeds -- Filippov Solutions to Systems of Ordinary Differential Equations with Delta Function Terms as Summands -- Resolvent estimates and scattering problems for Schr¨odinger, Klein-Gordon and wave equations -- On an Optimal Control Problem for the Wave Equation in One Space Dimension Controlled by Third Type Boundary Data -- Critical exponent for the semilinear wave equation with time or space dependent damping -- A note on a class of conservative, well-posed linear control systems -- Recent progress in smoothing estimates for evolution equations -- Differentiability of Inverse Operators -- Quasi-symmetrizer and hyperbolic equations -- Solution of the Cauchy problem for generalized Euler-Poisson-Darboux equation by the method of fractional integrals -- Global Solutions of Semilinear System of Klein-Gordon Equations in de Sitter Spacetime.
Additional Edition:
ISBN 9783319001241
Additional Edition:
Buchausg. u.d.T. Progress in partial differential equations Cham : Springer, 2013 ISBN 9783319001241
Language:
English
Keywords:
Hyperbolische Differentialgleichung
;
Evolutionsgleichung
;
Randwertproblem
;
Optimale Kontrolle
;
Partielle Differentialgleichung
;
Aufsatzsammlung
DOI:
10.1007/978-3-319-00125-8
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Reissig, Michael 1958-
Author information:
Ruzhansky, Michael 1972-
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