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ویرایش: 2nd ed. 2019 نویسندگان: Prof. Dr. Pavol Quittner, Prof. Dr. Philippe Souplet سری: Birkhäuser Advanced Texts Basler Lehrbücher ISBN (شابک) : 3030182207, 9783030182205 ناشر: Birkhäuser سال نشر: 2019 تعداد صفحات: 738 زبان: English فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود) حجم فایل: 5 مگابایت
کلمات کلیدی مربوط به کتاب مسائل سهموی فوق خطی: انفجار، وجود جهانی و حالت های ثابت (متن های پیشرفته بیرخاوزر Basler Lehrbücher): است
در صورت تبدیل فایل کتاب Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States (Birkhäuser Advanced Texts Basler Lehrbücher) به فرمت های PDF، EPUB، AZW3، MOBI و یا DJVU می توانید به پشتیبان اطلاع دهید تا فایل مورد نظر را تبدیل نمایند.
توجه داشته باشید کتاب مسائل سهموی فوق خطی: انفجار، وجود جهانی و حالت های ثابت (متن های پیشرفته بیرخاوزر Basler Lehrbücher) نسخه زبان اصلی می باشد و کتاب ترجمه شده به فارسی نمی باشد. وبسایت اینترنشنال لایبرری ارائه دهنده کتاب های زبان اصلی می باشد و هیچ گونه کتاب ترجمه شده یا نوشته شده به فارسی را ارائه نمی دهد.
این کتاب به مطالعه کیفی حل معادلات و سیستم های دیفرانسیل جزئی
بیضوی فوق خطی و سهموی اختصاص دارد. این دسته از مسائل به طور
خاص شامل تعدادی از سیستم های واکنش- انتشار است که در مدل های
مختلف ریاضی به ویژه در شیمی، فیزیک و زیست شناسی به وجود می
آیند.
دو فصل اول به معرفی این زمینه و خواننده را قادر می سازد تا با بررسی مسائل مدل ساده، به ترتیب از نوع بیضوی و سهموی، با ایده های اصلی آشنا شود. سه فصل بعدی به مسائلی با ساختار پیچیدهتر اختصاص یافته است. یعنی سیستم های بیضوی و سهموی، معادلات با غیرخطی های وابسته به گرادیان، و معادلات غیرمحلی. آنها شامل بسیاری از پیشرفت ها هستند که جنبه های مختلفی از تحقیقات فعلی را منعکس می کنند. اگرچه تکنیکهای معرفیشده در دو فصل اول ابزارهای کارآمدی برای حمله به برخی از جنبههای این مشکلات ارائه میکنند، اما اغلب پدیدههای جدید و بهویژه رفتارهای متفاوتی را به نمایش میگذارند که مطالعه آنها نیاز به ایدههای جدیدی دارد. بسیاری از مسائل باز ذکر شده و نظر داده شده است.
کتاب مستقل و به روز است، کیفیت آموزشی بالایی دارد. این کتاب به مشکلاتی اختصاص دارد که به شدت مورد مطالعه قرار گرفته اند، اما تاکنون به طور عمیق در ادبیات کتاب به آن پرداخته نشده است. مخاطبان مورد نظر شامل دانشجویان تحصیلات تکمیلی و کارشناسی ارشد و محققینی است که در زمینه معادلات دیفرانسیل جزئی و ریاضیات کاربردی کار می کنند.
ویرایش اول این کتاب به یکی از مراجع استاندارد در این زمینه
تبدیل شده است. این ویرایش دوم متن اصلاح شده ای را ارائه می
دهد و حاوی تعدادی به روز رسانی است که منعکس کننده پیشرفت های
مهم اخیر است که از اولین نسخه در این زمینه رو به رشد ظاهر شده
است.
This book is devoted to the qualitative study of solutions of
superlinear elliptic and parabolic partial differential
equations and systems. This class of problems contains, in
particular, a number of reaction-diffusion systems which
arise in various mathematical models, especially in
chemistry, physics and biology.
The first two chapters introduce to the field and enable the reader to get acquainted with the main ideas by studying simple model problems, respectively of elliptic and parabolic type. The subsequent three chapters are devoted to problems with more complex structure; namely, elliptic and parabolic systems, equations with gradient depending nonlinearities, and nonlocal equations. They include many developments which reflect several aspects of current research. Although the techniques introduced in the first two chapters provide efficient tools to attack some aspects of these problems, they often display new phenomena and specifically different behaviors, whose study requires new ideas. Many open problems are mentioned and commented.
The book is self-contained and up-to-date, it has a high didactic quality. It is devoted to problems that are intensively studied but have not been treated so far in depth in the book literature. The intended audience includes graduate and postgraduate students and researchers working in the field of partial differential equations and applied mathematics.
The first edition of this book has become one of the standard
references in the field. This second edition provides a
revised text and contains a number of updates reflecting
significant recent advances that have appeared in this
growing field since the first edition.
Contents Introduction to the first edition Introduction to the second edition List of the main new results proved in the second edition 1. Preliminaries General Domains Functions of space and time Banach spaces and linear operators Function spaces Eigenvalues and eigenfunctions Further frequent notation Chapter I Model Elliptic Problems 2. Introduction 3. Classical and weak solutions 4. Isolated singularities 5. Pohozaev’s identity and nonexistence results 6. Homogeneous nonlinearities 7. Minimaxmethods 8. Liouville-type results 8.1. Statements of the Liouville-type results 8.2. Proofs of Liouville-type theorems for elliptic inequalities 8.3. Proof of Theorem 8.1(i) based on integral bounds, and related singularity estimates 8.4. Proofs of Liouville-type theorems based on moving planes 9. Positive radial solutions of Δu + up =0 in Rn 10. A priori bounds via themethod of Hardy-Sobolev inequalities 11. A priori bounds via bootstrap in Lpδ-spaces 12. A priori bounds via the rescalingmethod 13. A priori bounds viamoving planes and Pohozaev’s identity Chapter II Model Parabolic Problems 14. Introduction 15. Well-posedness in Lebesgue spaces 16. Maximal existence time. Uniformbounds from Lp-estimates 17. Blow-up 18. Fujita-type results 19. Global existence for the Dirichlet problem 19.1. Small data global solutions Asymptotic stability of the zero solution Potential well theory 19.2. Structure of global solutions in bounded domains 19.3. Diffusion eliminating blow-up 20. Global existence for the Cauchy problem 20.1. Small data global solutions 20.2. Global solutions with exponential spatial decay 20.3. Asymptotic profiles for small data solutions 20.4. Small data in scale-invariant Morrey spaces 20.5. Blow-up for large Morrey norm and the separation problem 21. Parabolic Liouville-type results 22. A priori bounds 22.1. A priori bounds in the subcritical case 22.2. Boundedness of global solutions in the supercritical case 22.3. Global unbounded solutions in the critical case 22.4. Estimates for nonglobal solutions 22.5. Partial results in the supercritical case for nonconvex domains 23. Blow-up rate 23.1. The lower estimate 23.2. The upper estimate: summary 23.3. The upper estimate for time-increasing solution 23.4. The upper estimate in the subcritical case: the method of backward similarity variables 23.5. The upper estimate for ps ≤ p < pJL: intersection-comparison 23.6. Some other applications of backward similarity variables 24. Blow-up set and space profile 24.1. Single-point blow-up for radial decreasing solutions and first estimates of the space profile 24.2. Properties of the blow-up set 24.3. Refined single-point blow-up space profiles 25. Self-similar blow-up behavior 25.1. Space-time profile in similarity variables in the subcritical case 25.2. Refined space-time blow-up behavior for radially decreasing solutions. 25.3. Other blow-up profiles in the suband supercritical cases 26. Universal bounds and initial blow-up rates 27. Complete blow-up 28. Applications of a priori and universal bounds 28.1. A nonuniqueness result 28.2. Existence of periodic solutions 28.3. Existence of optimal controls 28.4. Transition from global existence to blow-up and stationary solutions 28.5. Decay of the threshold solution of the Cauchy problem 28.6. Parabolic Liouville-type theorems for radial solutions 29. Decay and grow-up of threshold solutions in the super-supercritical case Chapter III Systems 30. Introduction 31. Elliptic systems 31.1. A priori bounds by the method of moving planes and Pohozaev-type identities 31.2. Liouville-type results for the Lane-Emden system 31.2a. Liouville-type results for other systems 31.3. A priori bounds by the rescaling method 31.4. A priori bounds by the Lpδ alternate bootstrap method 32. Parabolic systems coupled by power source terms 32.1. Well-posedness and continuation in Lebesgue spaces 32.2. Blow-up and global existence 32.3. Fujita-type results 32.4. Blow-up asymptotics 33. The role of diffusion in blow-up 33.1. Diffusion preserving global existence Systems with dissipation of mass Systems of Gierer-Meinhardt type Systems with dissipation of mass, equal diffusions and mixed boundary conditions 33.2. Diffusion inducing blow-up Systems with dissipation of mass and unequal diffusions Systems with equal diffusions and homogeneous Neumann boundary conditions Diffusion-induced blow-up for other systems 33.3. Diffusion eliminating blow-up Chapter IV Equations with Gradient Terms 34. Introduction 35. Well-posedness and gradient bounds 36. Perturbations of themodel problem: blow-up and global existence 37. Fujita-type results 38. A priori bounds and blow-up rates 39. Blow-up sets and profiles 40. Diffusive Hamilton-Jacobi equations and gradient blow-up on the boundary 40.1. Gradient blow-up and global existence 40.2. Asymptotic behavior of global solutions 40.3. Space profile of gradient blow-up 40.4. Time rate of gradient blow-up 41. An example of interior gradient blow-up Chapter V Nonlocal Problems 42. Introduction 43. Problems involving space integrals (I) 43.1. Blow-up and global existence 43.2. Blow-up rates, sets and profiles 43.3. Uniform bounds from Lq-estimates 43.4. Universal bounds for global solutions 44. Problems involving space integrals (II) 44.1. Transition from single-point to global blow-up 44.2. A problem with control of mass 44.3. A problem with variational structure 44.4. A problem arising in the modeling of Ohmic heating 45. Fujita-type results for problems involving space integrals 46. A problem with memory term 46.1. Blow-up and global existence 46.2. Blow-up rate Appendices 47. Appendix A: Linear elliptic equations 47.1. Elliptic regularity 47.2.Lp-Lq-estimates 47.3. Some elliptic operators in weighted Lebesgue spaces (I) 47.4. Some elliptic operators in weighted Lebesgue spaces (II) 48. Appendix B: Linear parabolic equations 48.1. Parabolic regularity 48.2. Heat semigroup,Lp-Lq-estimates, decay, gradient estimates 48.3. Weak and integral solutions 49. Appendix C: Linear theory in Lp δ -spaces and in uniformly local spaces 49.1. The Laplace equation in L p δ -spaces 49.2. The heat semigroup in L p δ L-spaces 49.3. Some pointwise boundary estimates for the heat equation 49.4. Proof of Theorems 49.2, 49.3 and 49.7 49.5. The heat equation in uniformly local Lebesgue spaces 49.6. The heat equation in Morrey spaces 50. Appendix D: Poincar´e, Hardy-Sobolev, and other useful inequalities 50.1. Basic inequalities 50.2. The Poincar´e inequality 50.3. Hardy and Hardy-Sobolev inequalities 51. Appendix E: Local existence, regularity and stability for semilinear parabolic problems 51.1. Analytic semigroups and interpolation spaces 51.2. Local existence and regularity for regular data 51.3. Stability of equilibria 51.4. Self-adjoint generators with compact resolvent 51.5. Singular initial data 51.6. Uniform bounds from Lq-estimates 51.7. An elementary proof of local well-posedness for problem (14.1) in 52. Appendix F:Maximumand comparison principles. Zero number 52.1. Maximum principles for the Laplace equation 52.2. Comparison principles for classical and strong solutions 52.3. Comparison principles via the Stampacchia method 52.4. Comparison principles via duality arguments 52.5. Monotonicity of radial solutions 52.6. Monotonicity of solutions in time 52.7. Systems and nonlocal problems 52.8. Zero number 53. Appendix G: Dynamical systems 53a. Appendix Ga: Summary of positive radial steady states and self-similar profiles of (18.1) 54. Appendix H:Methodological notes A. Background tools B. Main classes of techniques C. Some other techniques I. METHODS FOR ELLIPTIC PROBLEMS M1. Methods to prove existence of solutions M2. Methods to prove nonexistence of solutions M3. Methods to prove nonexistence in elliptic Liouville-type results M4. Methods to study regularity and singularities of solutions M5. Methods to prove a priori estimates II. METHODS FOR PARABOLIC PROBLEMS M6. Methods for local well-posedness M7. Methods to prove global existence (and also asymptotic behavior, boundedness, decay, stability) M8. Methods to prove blow-up M9. Methods to prove nonexistence in parabolic Fujitaand Liouvilletype results M10. Methods to prove boundedness of global solutions and parabolic a priori estimates M11. Methods to prove universal bounds of positive solutions and initial blow-up rates M12. Methods to establish blow-up rates M13. Methods to study blow-up sets and profiles 55. Appendix I: Selection of open problems 1. Model elliptic problem 2. Model parabolic problem 3. Systems 4. Problems involving gradient terms 5. Nonlocal problems Bibliography List of Symbols Index