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Decay of solutions of some nonlinear wave equations. (English) Zbl 0689.35081

The authors present numerous results on solutions to the equations \[ (1)\quad u_ t+u_ x+uu_ x-\nu u_ x-u_{xxt}=0, \] and \[ (2)\quad u_ t+u_ x+uu_ x-\nu u_{xx}+u_{xxx}\quad (plus\quad initial\quad conditions) \] which are called regularized long-wave equation with a Burgers-type dissipative term appended and Korteweg-de Vries-Burgers equation respectively. Many results on (sharp) decay rates as \(t\to \infty\) for solutions to these equations with respect to various norms \((L^ 2,L^{\infty},...)\) are given. The proofs are detailed and written in a broad readable style with lots of comments starting right from the extensive introduction.
Section 2 mainly recalls the well-posedness of (1), (2), section 3 discusses (1) giving decay rates of solutions in various norms. Sect. 4 studies the linearized equation associated to (1) and presents results on the asymptotic behaviour of the slutions using the Fourier transform. Sect. 5 improves the results from sect. 3 using those on the linarized part from sect. 4. Equation (2) is mentioned in sect. 6 the previous considerations for (1) should (all but a few) carry over. Finally the relation to the heat equation is shortly discussed, an interpretation is given and hints to related other interesting equations are given.
Reviewer: R.Racke

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
35K05 Heat equation
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