## Asymptotics for Dissipative Nonlinear EquationsMany 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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### Contents

I | 1 |

II | 2 |

IV | 3 |

VII | 4 |

IX | 5 |

X | 13 |

XI | 17 |

XII | 26 |

LVII | 287 |

LVIII | 290 |

LIX | 298 |

LX | 302 |

LXI | 316 |

LXII | 319 |

LXIII | 323 |

LXIV | 330 |

XIV | 37 |

XV | 41 |

XVI | 46 |

XVII | 51 |

XVIII | 66 |

XIX | 67 |

XX | 74 |

XXI | 80 |

XXII | 82 |

XXIII | 92 |

XXIV | 109 |

XXV | 111 |

XXVI | 112 |

XXVII | 116 |

XXVIII | 124 |

XXIX | 125 |

XXX | 133 |

XXXI | 140 |

XXXII | 143 |

XXXIII | 146 |

XXXIV | 151 |

XXXV | 157 |

XXXVI | 160 |

XXXVII | 163 |

XXXVIII | 169 |

XXXIX | 175 |

XL | 179 |

XLI | 194 |

XLIII | 198 |

XLIV | 205 |

XLV | 206 |

XLVI | 209 |

XLVII | 210 |

XLVIII | 218 |

XLIX | 221 |

L | 222 |

LI | 223 |

LII | 231 |

LIII | 242 |

LIV | 254 |

LV | 268 |

LVI | 286 |

LXV | 333 |

LXVI | 336 |

LXVII | 339 |

LXVIII | 342 |

LXIX | 349 |

LXX | 351 |

LXXI | 367 |

LXXII | 372 |

LXXIV | 381 |

LXXV | 382 |

LXXVI | 395 |

LXXVII | 398 |

LXXVIII | 402 |

LXXIX | 407 |

LXXX | 409 |

LXXXI | 411 |

LXXXII | 413 |

LXXXIII | 427 |

LXXXIV | 431 |

LXXXV | 456 |

LXXXVI | 457 |

LXXXVII | 462 |

LXXXVIII | 465 |

LXXXIX | 468 |

XC | 474 |

XCI | 476 |

XCII | 487 |

XCIII | 491 |

XCIV | 492 |

XCV | 494 |

XCVI | 498 |

XCVII | 499 |

XCVIII | 506 |

XCIX | 510 |

C | 513 |

CI | 514 |

CII | 522 |

CIII | 529 |

CIV | 539 |

541 | |

554 | |

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### Common terms and phrases

apply Theorem asymptotic behavior asymptotic formulas asymptotic representation behavior of solutions Cauchy problem complete metric space contraction mapping damped wave data uq G decay estimate define Denote estimate is true estimate of Lemma exists a unique Fourier transformation function global existence Green operator heat kernel Hence Holder inequality initial data uq integral equation integrating with respect Korteweg-de Vries-Burgers equation L2 norm large time asymptotic linear operator Naumkin and Shishmarev nonlinear heat equation nonlinear operator norm odd function oo uniformly operator C satisfy operator Q optimal time decay proof of Theorem right-hand side self-similar self-similar solution small initial data Sobolev spaces solution u G C subcritical sufficiently small Suppose symbol Theorem 3.2 ty2v uniformly with respect unique global solution unique solution wave equation Young inequality