On nonlinear evolution operators associated with some nonlinear dispersive equations

Authors:
Shinnosuke Oharu and Tadayasu Takahashi

Journal:
Proc. Amer. Math. Soc. **97** (1986), 139-145

MSC:
Primary 35Q20; Secondary 47H20, 58D25

MathSciNet review:
831403

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The initial-boundary value problem for a nonlinear dispersive system with time-dependent boundary condition is discussed in the Sobolev space from the point of view of the theory of nonlinear evolution operators. A notion of weak solution to the problem is introduced and the associated family of solution operators is constructed in such a way that it gives rise to a nonlinear evolution operator with time-dependent domain. Various qualitative properties as well as regularity of the weak solutions are investigated through those of the constructed evolution operator.

**[1]**T. B. Benjamin, J. L. Bona, and J. J. Mahony,*Model equations for long waves in nonlinear dispersive systems*, Philos. Trans. Roy. Soc. London Ser. A**272**(1972), no. 1220, 47–78. MR**0427868****[2]**J. L. Bona and P. J. Bryant,*A mathematical model for long waves generated by wavemakers in non-linear dispersive systems*, Proc. Cambridge Philos. Soc.**73**(1973), 391–405. MR**0339651****[3]**Jerry L. Bona and Vassilios A. Dougalis,*An initial and boundary value problem for a model equation for propagation of long waves*, J. Math. Anal. Appl.**75**(1980), no. 2, 503–522. MR**581837**, 10.1016/0022-247X(80)90098-0**[4]**M. G. Crandall and A. Pazy,*Nonlinear evolution equations in Banach spaces*, Israel J. Math.**11**(1972), 57–94. MR**0300166****[5]**L. C. Evans,*Nonlinear evolution equations in an arbitrary Banach space*, Israel J. Math.**26**(1977), no. 1, 1–42. MR**0440431****[6]**T. Iwamiya, S. Oharu, and T. Takahashi,*On the semigroup approach to some nonlinear dispersive equations*, Numerical analysis of evolution equations (Kyoto, 1978) Lecture Notes Numer. Appl. Anal., vol. 1, Kinokuniya Book Store, Tokyo, 1979, pp. 95–134. MR**690439****[7]**Kazuo Kobayasi, Yoshikazu Kobayashi, and Shinnosuke Oharu,*Nonlinear evolution operators in Banach spaces*, Osaka J. Math.**21**(1984), no. 2, 281–310. MR**752464****[8]**-,*Nonlinear evolution operators in Banach spaces*. II, Hiroshima Math. J. (to appear).**[9]**L. A. Medeiros and M. Milla Miranda,*Weak solutions for a nonlinear dispersive equation*, J. Math. Anal. Appl.**59**(1977), no. 3, 432–441. MR**0466924****[10]**R. E. Showalter,*Sobolev equations for nonlinear dispersive systems*, Applicable Anal.**7**(1977/78), no. 4, 297–308. MR**504616**, 10.1080/00036817808839200**[11]**Nicolae H. Pavel,*Nonlinear evolution equations governed by 𝑓-quasidissipative operators*, Nonlinear Anal.**5**(1981), no. 5, 449–468. MR**613054**, 10.1016/0362-546X(81)90094-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
35Q20,
47H20,
58D25

Retrieve articles in all journals with MSC: 35Q20, 47H20, 58D25

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1986-0831403-1

Keywords:
Nonlinear dispersive equations,
long waves of small amplitude,
nonlinear evolution operators

Article copyright:
© Copyright 1986
American Mathematical Society