ASYMPTOTICS FOR DISSIPATIVE NONLINEAR EQUATIONS

By J. MANAN

CONTENTS PRELIMINARY RESULTS NOTATIONS DE?NITIONS ASYMPTOTIC NOTATIONS H¨OLDER INEQUALITY YOUNG INEQUALITY SOBOLEV IMBEDDING INEQUALITY AN INTERPOLATION INEQUALITY CONTRACTION MAPPING PRINCIPLE GRONWALL’S LEMMA LOCAL EXISTENCE SMALL INITIAL DATA LARGE INITIAL DATA ESTIMATES FOR LINEAR SEMIGROUPS ESTIMATES IN WEIGHTED LEBESGUE SPACES ESTIMATES IN THE L2 - THEORY ESTIMATES IN FOURIER SPACES ESTIMATES FOR LARGE X AND T WEAK NONLINEARITY GENERAL APPROACH ASYMPTOTICS FOR LARGE X AND T NONLINEAR HEAT EQUATION DAMPED WAVE EQUATION SOBOLEV TYPE EQUATIONS SMALL DATA LARGE DATA WHITHAM TYPE EQUATION A MODEL EQUATION LOCAL EXISTENCE AND SMOOTHING E?ECT WEAK DISSIPATION, STRONG DISPERSION PRELIMINARY LEMMAS PROOF OF THEOREM 2 59 A SYSTEM OF NONLINEAR EQUATIONS GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR COMMENTS CRITICAL NONCONVECTIVE EQUATIONS FRACTIONAL EQUATIONS COMPLEX LANDAU-GINZBURG EQUATION PRELIMINARIES PROOF OF THEOREM 3 22 FOR RE ?(?, ?) > 0 PROOF OF THEOREM 3 22 FOR ? = 0, µ > 0 PROOF OF THEOREM 3 22 FOR ? = 0, µ = 0, ? > 0 ASYMPTOTIC EXPANSION WHITHAM TYPE EQUATIONS PROOF OF THEOREM 3 56 CRITICAL CONVECTIVE EQUATIONS WHITHAM EQUATION PROOF OF THEOREM 4 9 SELF-SIMILAR SOLUTIONS PROOF OF THEOREM 4 13 KDV-B EQUATION LEMMAS PROOF OF THEOREM 4 18 SECOND TERM OF ASYMPTOTICS PROOF OF THEOREM 4 25 BBM-B EQUATION PROOF OF THEOREM 4 28 PROOF OF THEOREM 4 29 PROOF OF THEOREM 4 36 PROOF OF THEOREM 4 38 SUBCRITICAL NONCONVECTIVE EQUATIONS FRACTIONAL HEAT EQUATIONS PROOF OF THEOREM 5 19 OSCILLATING SOLUTIONS TO NONLINEAR HEAT EQUATION PROOF OF THEOREM 5 37 SUBCRITICAL CONVECTIVE EQUATIONS BURGERS TYPE EQUATIONS RAREFACTION WAVE SHOCK WAVE ZERO BOUNDARY CONDITIONS