Asymptotics for Dissipative Nonlinear Equations : Lecture Notes in Mathematics - Nakao Hayashi

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Asymptotics for Dissipative Nonlinear Equations

By: Nakao Hayashi, Elena I. Kaikina, Pavel Naumkin

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Paperback | 21 April 2006

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Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations Science Physics

Preliminary results	p. 1
Notations	p. 1
Definitions	p. 1
Asymptotic notations	p. 2
Holder inequality	p. 2
Young inequality	p. 3
Sobolev imbedding inequality	p. 3
An interpolation inequality	p. 3
Contraction mapping principle	p. 4
Gronwall's Lemma	p. 4
Local existence	p. 5
Small initial data	p. 13
Large initial data	p. 17
Estimates for linear semigroups	p. 26
Estimates in weighted Lebesgue spaces	p. 26
Estimates in the L² - theory	p. 37
Estimates in Fourier spaces	p. 41
Estimates for large x and t	p. 46
Weak Nonlinearity	p. 51
General approach	p. 51
Asymptotics for large x and t	p. 66
Small initial data	p. 67
Large initial data	p. 74
Nonlinear heat equation	p. 80
Damped wave equation	p. 82
Large initial data	p. 92
Sobolev type equations	p. 109
Local existence	p. 111
Small data	p. 112
Large data	p. 116
Whitham type equation	p. 124
A model equation	p. 124
Local existence and smoothing effect	p. 125
Small initial data	p. 133
Large initial data	p. 140
Weakdissipation, strong dispersion	p. 143
Preliminary Lemmas	p. 146
Proof of Theorem 2.59	p. 151
A system of nonlinear equations	p. 157
Local existence and smoothing effect	p. 160
Global existence and asymptotic behavior	p. 163
Large initial data	p. 169
Comments	p. 175
Critical Nonconvective Equations	p. 179
General approach	p. 179
Fractional equations	p. 194
Small data	p. 194
Large data	p. 198
Asymptotics for large x and t	p. 205
Small initial data	p. 206
Large initial data	p. 209
Nonlinear heat equation	p. 210
Complex Landau-Ginzburg equation	p. 218
Preliminaries	p. 221
Proof of Theorem 3.22 for Red(a,ß) > 0	p. 222
Proof of Theorem 3.22 for ? = 0,µ > 0	p. 223
Proof of Theorem 3.22 for ? = 0, µ = 0, ? > 0	p. 231
Asymptotic expansion	p. 242
Damped wave equation	p. 254
Small initial data	p. 254
Large initial data	p. 268
Sobolev type equations	p. 286
Small initial data	p. 287
Large initial data	p. 290
Whitham type equations	p. 298
Preliminary Lemmas	p. 302
Proof of Theorem 3.56	p. 316
Comments	p. 319
Critical Convective Equations	p. 323
General approach	p. 323
Whitham equation	p. 330
Preliminary Lemmas	p. 333
Proof of Theorem 4.9	p. 336
Self-similar solutions	p. 339
Proof of Theorem 4.13	p. 342
KdV-B equation	p. 349
Lemmas	p. 351
Proof of Theorem 4.18	p. 367
Second term of asymptotics	p. 372
Proof of Theorem 4.25	p. 372
BBM-B equation	p. 381
Preliminaries	p. 382
Proof of Theorem 4.28	p. 395
Proof of Theorem 4.29	p. 398
A system of nonlinear equations	p. 402
Preliminary Lemmas	p. 407
Proof of Theorem 4.36	p. 409
Self-similar solutions	p. 411
Proof of Theorem 4.38	p. 413
Comments	p. 427
Subcritical Nonconvective Equations	p. 431
General approach	p. 431
Fractional heat equations	p. 456
Small initial data	p. 457
Large initial data	p. 462
Whitham type equations	p. 465
Preliminary Lemmas	p. 468
Proof of Theorem 5.19	p. 474
Damped wave equation	p. 476
Small initial data	p. 476
Large data	p. 487
Sobolev type equations	p. 491
Small data	p. 492
Large data	p. 494
Oscillating solutions to nonlinear heat equation	p. 498
Lemmas	p. 499
Proof of Theorem 5.37	p. 506
Comments	p. 510
Subcritical Convective Equations	p. 513
Burgers type equations	p. 513
Rarefaction wave	p. 514
Shockwave	p. 522
Zero boundary conditions	p. 529
Comments	p. 539
References	p. 541
Index	p. 555
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Asymptotics for Dissipative Nonlinear Equations

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