1  9
 Vazquez, J. L. (Juan Luis)
 Oxford : Clarendon, 2007.
 Description
 Book — xxii, 624 p. : ill., ; 24 cm.
 Summary

 Preface
 1. Introduction
 PART 1
 2. Main applications
 3. Preliminaries and basic estimates
 4. Basic examples
 5. The Dirichlet problem I. Weak solutions
 6. The Dirichlet problem II. Limit solutions, very weak solutions and some other variants
 7. Continuity of local solutions
 8. The Dirichlet problem III. Strong solutions
 9. The Cauchy problem. L' theory
 10. The PME as an abstract evolution equation. Semigroup approach
 11. The Neumann problem and problems on manifolds
 PART 2
 12. The Cauchy problem with growing initial data
 13. Optimal existence theory for nonnegative solutions
 14. Propagation properties
 15. Onedimensional theory. Regularity and interfaces
 16. Full analysis of selfsimilarity
 17. Techniques of symmetrization and concentration
 18. Asymptotic behaviour I. The Cauchy problem
 19. Regularity and finer asymptotics in several dimensions
 20. Asymptotic behaviour II. Dirichlet and Neumann problems
 COMPLEMENTS
 21. Further applications
 22. Basic facts and appendices
 Bibliography
 Index.
 (source: Nielsen Book Data)
(source: Nielsen Book Data)
Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Stacks  
QA379 .V399 2007  Unknown 
 Vazquez, J. L. (Juan Luis)
 Oxford : Oxford University Press, 2006.
 Description
 Book — xiii, 234 p. : ill. ; 24 cm.
 Summary

 Preface
 PART I
 1. Preliminaries
 2. Smoothing effect and time decay. Data in L'(Rn) or M(Rn)
 3. Smoothing effect and time decay from Lp or Mp
 4. Lower bounds, contractivity, error estimates and continuity
 PART II
 5. Subcritical range of the FDE. Critical line. Extinction. Backward effect
 6. Improved analysis of the critical line. Delayed regularity
 7. Extinction rates and asymptotics for 0m c
 8. Logarithmic diffusion in 2d and intermediate 1d range
 9. Superfast FDE
 10. Summary of main results for the PME/FDE
 PART III
 11. Evolution equations of the pLaplacian type
 12. Appendices
 Bibliography
 Index.
 (source: Nielsen Book Data)
(source: Nielsen Book Data)
 Vazquez, J. L. (Juan Luis)
 Oxford : Oxford University Press, 2006.
 Description
 Book — xiii, 234 p. : ill.
5. A stability technique for evolution partial differential equations : a dynamical systems approach [2004]
 Galaktionov, Victor A.
 Boston : Birkhäuser, ©2004.
 Description
 Book — 1 online resource (xix, 377 pages) : illustrations Digital: text file.PDF.
 Summary

 1. Stability Theorem: A Dynamical Systems Approach. 1.1 Perturbed dynamical systems. 1.2 Some concepts from dynamical systems. 1.3 The three hypotheses. 1.4 The STheorem: Stability of omegalimit sets. 1.5 Practical stability assumptions. 1.6 A result on attractors. Remarks and comments on the literature.
 2. Nonlinear Heat Equations: Basic Models and Mathematical Techniques. 2.1 Nonlinear heat equations. 2.2 Basic mathematical properties. 2.3 Asymptotics. 2.4 The Lyapunov method. 2.5 Comparison techniques. 2.5.1 Intersection comparison and Sturm's theorems. 2.5.2 Shifting comparison principle (SCP). 2.5.3 Other comparisons. Remarks and comments on the literature.
 3. Equation of Superslow Diffusion. 3.1 Asymptotics in a bounded domain. 3.2 The Cauchy problem in one dimension. Remarks and comments on the literature.
 4. Quasilinear Heat Equations with Absorption. The Critical Exponent. 4.1 Introduction: Diffusionabsorption with critical exponent. 4.2 First mass analysis. 4.3 Sharp lower and upper estimates. 4.4 ?limits for the perturbed equation. 4.5 Extended mass analysis: Uniqueness of stable asymptotics. 4.6 Equation with gradientdependent diffusion and absorption. 4.7 Nonexistence of fundamental solutions. 4.8 Solutions with L1 data. 4.9 General nonlinearity. 4.10 Dipolelike behaviour with critical absorption exponents in a half line and related problems. Remarks and comments on the literature.
 5. Porous Medium Equation with Critical Strong Absorption. 5.1 Introduction and results: Strong absorption and finitetime extinction. 5.2 Universal a priori bounds. 5.3 Explicit solutions on twodimensional invariant subspace. 5.4 L?estimates on solutions and interfaces. 5.5 Eventual monotonicity and on the contrary. 5.6 Compact support. 5.7 Singular perturbation of firstorder equation. 5.8 Uniform stability for semilinear HamiltonJacobi equations. 5.9 Local extinction property. 5.10 Onedimensional problem: first estimates. 5.11 Bernstein estimates for singularly perturbed firstorder equations. 5.12 Onedimensional problem: Application of the STheorem. 5.13 Empty extinction set: A KPP singular perturbation problem. 5.14 Extinction on a sphere. Remarks and comments on the literature.
 6. The Fast Diffusion Equation with Critical Exponent. 6.1 The fast diffusion equation. Critical exponent. 6.2 Transition between different selfsimilarities. 6.3 Asymptotic outer region. 6.4 Asymptotic inner region. 6.5 Explicit solutions and eventual monotonicity. Remarks and comments on the literature.
 7. The Porous Medium Equation in an Exterior Domain. 7.1 Introduction. 7.2 Preliminaries. 7.3 Nearfield limit: The inner region. 7.4 Selfsimilar solutions. 7.5 Farfield limit: The outer region. 7.6 Selfsimilar solutions in dimension two. 7.7 Farfield limit in dimension two. Remarks and comments on the literature.
 8. Blowup FreeBoundary Patterns for the NavierStokes Equations. 8.1 Freeboundary problem. 8.2 Preliminaries, local existence. 8.3 Blowup: The first, stable monotone pattern. 8.4 Semiconvexity and first estimates. 8.5 Rescaled singular perturbation problem. 8.6 Freeboundary layer. 8.7 Countable set of nonmonotone blowup patterns on stable manifolds. 8.8 Blowup periodic and globally decaying patterns. Remarks and comments on the literature.
 9. Equation ut = uxx + u ln2u: Regional Blowup. 9.1 Regional blowup via HamiltonJacobi equation. 9.2 Exact solutions: Periodic global blowup. 9.3 Lower and upper bounds: Method of stationary states. 9.4 Semiconvexity estimate. 9.5 Lower bound for blowup set and asymptotic profile. 9.6 Localization of blowup. 9.7 Minimal asymptotic behaviour. 9.8 Minimal blowup set. 9.9 Periodic blowup solutions. Remarks and comments on the literature.
 10. Blowup in Quasilinear Heat Equations Described by HamiltonJacobi Equations. 10.1 General models with blowup degeneracy. 10.2 Eventual monotonicity of large solutions. 10.3 L?bounds: Method of stationary states. 10.4 Gradient bound and singlepoint blowup. 10.5 Semiconvexity estimate and global blowup. 10.6 Singular perturbation problem. 10.7 Uniform stability for HamiltonJacobi equation. Asymptotic profile. 10.8 Blowup finaltime profile. Remarks and comments on the literature. Remarks and comments on the literature.
 11. A Fully Nonlinear Equation from Detonation Theory. 11.1 Mathematical formulation of the problem. 11.2 Outline of results. 11.3 On local existence, regularity and quenching. 11.4 Singlepoint quenching and first sharp estimate. 11.5 Fundamental estimates: Dynamical system of inequalities. 11.6 Asymptotic profile near the quenching time. Remarks and comments on the literature.
 12. Further Applications to Second and HigherOrder Equations. 12.1 A homogenization problem for heat equations. 12.2 Stability of perturbed nonlinear parabolic equations with Sturmian property. 12.3 Global solutions of a 2mthorder semilinear parabolic equation in the supercritical range. 12.4 The critical exponent for 2mthorder semilinear parabolic equations with absorption. 12.5 Regional blowup for 2mthorder semilinear parabolic equations... Remarks and comments on the literature. References.
 (source: Nielsen Book Data)
(source: Nielsen Book Data)
 Galaktionov, Victor A.
 Boston, MA : Birkhäuser Boston, 2004.
 Description
 Book — 1 online resource (xix, 377 pages 10 illustrations).
 Summary

 1. Stability Theorem: A Dynamical Systems Approach. 1.1 Perturbed dynamical systems. 1.2 Some concepts from dynamical systems. 1.3 The three hypotheses. 1.4 The STheorem: Stability of omegalimit sets. 1.5 Practical stability assumptions. 1.6 A result on attractors. Remarks and comments on the literature.
 2. Nonlinear Heat Equations: Basic Models and Mathematical Techniques. 2.1 Nonlinear heat equations. 2.2 Basic mathematical properties. 2.3 Asymptotics. 2.4 The Lyapunov method. 2.5 Comparison techniques. 2.5.1 Intersection comparison and Sturm's theorems. 2.5.2 Shifting comparison principle (SCP). 2.5.3 Other comparisons. Remarks and comments on the literature.
 3. Equation of Superslow Diffusion. 3.1 Asymptotics in a bounded domain. 3.2 The Cauchy problem in one dimension. Remarks and comments on the literature.
 4. Quasilinear Heat Equations with Absorption. The Critical Exponent. 4.1 Introduction: Diffusionabsorption with critical exponent. 4.2 First mass analysis. 4.3 Sharp lower and upper estimates. 4.4 ?limits for the perturbed equation. 4.5 Extended mass analysis: Uniqueness of stable asymptotics. 4.6 Equation with gradientdependent diffusion and absorption. 4.7 Nonexistence of fundamental solutions. 4.8 Solutions with L1 data. 4.9 General nonlinearity. 4.10 Dipolelike behaviour with critical absorption exponents in a half line and related problems. Remarks and comments on the literature.
 5. Porous Medium Equation with Critical Strong Absorption. 5.1 Introduction and results: Strong absorption and finitetime extinction. 5.2 Universal a priori bounds. 5.3 Explicit solutions on twodimensional invariant subspace. 5.4 L?estimates on solutions and interfaces. 5.5 Eventual monotonicity and on the contrary. 5.6 Compact support. 5.7 Singular perturbation of firstorder equation. 5.8 Uniform stability for semilinear HamiltonJacobi equations. 5.9 Local extinction property. 5.10 Onedimensional problem: first estimates. 5.11 Bernstein estimates for singularly perturbed firstorder equations. 5.12 Onedimensional problem: Application of the STheorem. 5.13 Empty extinction set: A KPP singular perturbation problem. 5.14 Extinction on a sphere. Remarks and comments on the literature.
 6. The Fast Diffusion Equation with Critical Exponent. 6.1 The fast diffusion equation. Critical exponent. 6.2 Transition between different selfsimilarities. 6.3 Asymptotic outer region. 6.4 Asymptotic inner region. 6.5 Explicit solutions and eventual monotonicity. Remarks and comments on the literature.
 7. The Porous Medium Equation in an Exterior Domain. 7.1 Introduction. 7.2 Preliminaries. 7.3 Nearfield limit: The inner region. 7.4 Selfsimilar solutions. 7.5 Farfield limit: The outer region. 7.6 Selfsimilar solutions in dimension two. 7.7 Farfield limit in dimension two. Remarks and comments on the literature.
 8. Blowup FreeBoundary Patterns for the NavierStokes Equations. 8.1 Freeboundary problem. 8.2 Preliminaries, local existence. 8.3 Blowup: The first, stable monotone pattern. 8.4 Semiconvexity and first estimates. 8.5 Rescaled singular perturbation problem. 8.6 Freeboundary layer. 8.7 Countable set of nonmonotone blowup patterns on stable manifolds. 8.8 Blowup periodic and globally decaying patterns. Remarks and comments on the literature.
 9. Equation ut = uxx + u ln2u: Regional Blowup. 9.1 Regional blowup via HamiltonJacobi equation. 9.2 Exact solutions: Periodic global blowup. 9.3 Lower and upper bounds: Method of stationary states. 9.4 Semiconvexity estimate. 9.5 Lower bound for blowup set and asymptotic profile. 9.6 Localization of blowup. 9.7 Minimal asymptotic behaviour. 9.8 Minimal blowup set. 9.9 Periodic blowup solutions. Remarks and comments on the literature.
 10. Blowup in Quasilinear Heat Equations Described by HamiltonJacobi Equations. 10.1 General models with blowup degeneracy. 10.2 Eventual monotonicity of large solutions. 10.3 L?bounds: Method of stationary states. 10.4 Gradient bound and singlepoint blowup. 10.5 Semiconvexity estimate and global blowup. 10.6 Singular perturbation problem. 10.7 Uniform stability for HamiltonJacobi equation. Asymptotic profile. 10.8 Blowup finaltime profile. Remarks and comments on the literature. Remarks and comments on the literature.
 11. A Fully Nonlinear Equation from Detonation Theory. 11.1 Mathematical formulation of the problem. 11.2 Outline of results. 11.3 On local existence, regularity and quenching. 11.4 Singlepoint quenching and first sharp estimate. 11.5 Fundamental estimates: Dynamical system of inequalities. 11.6 Asymptotic profile near the quenching time. Remarks and comments on the literature.
 12. Further Applications to Second and HigherOrder Equations. 12.1 A homogenization problem for heat equations. 12.2 Stability of perturbed nonlinear parabolic equations with Sturmian property. 12.3 Global solutions of a 2mthorder semilinear parabolic equation in the supercritical range. 12.4 The critical exponent for 2mthorder semilinear parabolic equations with absorption. 12.5 Regional blowup for 2mthorder semilinear parabolic equations... Remarks and comments on the literature. References.
 (source: Nielsen Book Data)
(source: Nielsen Book Data)
7. Degenerate diffusions [1993]
 New York : SpringerVerlag, c1993.
 Description
 Book — 228 p. : ill. ; 24 cm.
 Summary

This volume is the proceedings of the IMA workshop "Degenerate Diffusions" held at the University of Minnesota from May 13May 18, 1991. The workshop consisted of two parts. The emphasis of the first four days was on current progress or new problems in nonlinear diffusions involving free boundaries or sharp interfaces. Analysts and geometers will find some of the mathematical models described in this volume interesting; and the papers of more pure mathematical nature included here should provide applied mathematicians with powerful methods and useful techniques in handling singular perturbation problems as well as free boundary problems. The last two days of the workshop were a celebration of James Serrin's 65th birthday. A wide range of topics was covered in this part of the workshop. As a consequence, the scope of this book is much broader than what the title Degenerate Diffusions might suggest.
(source: Nielsen Book Data)
 Online
SAL3 (offcampus storage)
SAL3 (offcampus storage)  Status 

Stacks  Request (opens in new tab) 
QA377 .D555 1993  Available 
 Providence, R.I. : American Mathematical Society, c2006.
 Description
 Book — viii, 123 p. : ill. ; 26 cm.
 Summary

 Steepest descent flows and applications to spaces of probability measures by L. Ambrosio Hypocoercivity: the example of linear transport by L. Desvillettes A hybrid system of PDE's arising in multistructure interaction: Coupling of wave equations in $n$ and $n1$ space dimensions by H. Koch and E. Zuazua Some rigorous results for vortex patterns in BoseEinstein condensates by A. Aftalion Qualitative properties of some Boltzmann like equations which do not fulfill a detailed balance condition by M. Escobedo and S. Mischler.
 (source: Nielsen Book Data)
(source: Nielsen Book Data)
SAL3 (offcampus storage)
SAL3 (offcampus storage)  Status 

Stacks  Request (opens in new tab) 
QA374 .R4235 2006  Available 
 CIME Course on Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (2016 : Cetraro, Italy)
 Cham : Springer : Firenze : Fondazione CIME, Roberto Conti, Centro Internazionale Matematico Estivo International Mathematical Summer Center, [2017]
 Description
 Book — ix, 278 pages : illustrations (partly color) ; 24 cm.
 Summary

 1. The geometry of diffusing and selfattracting particles in a onedimensional faircompetition regime / Vincent Calvez, José Antonio Carrillo, and Franca Hoffman
 2. Bubbling blowup in critical parabolic problems / Manuel del Pino
 3. Regularity theory for local and nonlocal minimal surfaces: an overview / Matteo Cozzi and Alessio Figalli
 4. Short tales from nonlinear CalderónZygmund theory / Giuseppe Mingione
 5. The mathematical theories of diffusion: nonlinear and fractional diffusion / Juan Luis Vázquez.
(source: Nielsen Book Data)
Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Serials  
Shelved by Series title V.2186  Unknown 
Articles+
Journal articles, ebooks, & other eresources
Guides
Course and topicbased guides to collections, tools, and services.