Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-10-31T23:07:59.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

Published online by Cambridge University Press:  14 November 2011

Dao-Qing Dai  and
Wei Lin
Dao-Qing Dai
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, P.R. China
Wei Lin
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, P.R. China
Get access Rights & Permissions [Opens in a new window]

Synopsis

An initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variables

The complex function H is measurable on ℂ ×I × ℂ5, with I being an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 3-4 , 1992 , pp. 203 - 217
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Begehr, H.. Entire solutions of quasilinear pseudoparabolic equations. Demonstratio Math. 18 (1985) 673–685.CrossRefGoogle Scholar
2Begehr, H.. Gan ze Lösungen fastlinearer pseudoparabolischer Gleichungen. Mathematical papers given on the occasion of Ernst Mohr's 75-th birthday, pp. 15–22 (Berlin: Tech. Univ. Berlin, 1985).Google Scholar
3Begehr, H. and Hile, G. N.. Riemann boundary value problems for nonlinear elliptic systems. Complex Variables Theory Appl. 1 (1983), 239–261.Google Scholar
4Begehr, H. and Gilbert, R. P.. Piecewise continuous solutions of pseudoparabolic equations in two space dimension. Proc. Roy. Soc. Edinburgh Sect. A 81 (1978), 153–173.CrossRefGoogle Scholar
5Begehr, H. and Dai, D. Q.. The initial boundary value problem for nonlinear pseudoparabolic equations (to be submitted).Google Scholar
6Brown, P. M., Gilbert, R. P. and Hsiao, G. C.. Constructive function theoretic methods for fourth order pseudoparabolic equations in two space variables. Rend. Mat. 8 (1975), 935–951.Google Scholar
7Colton, D.. Integral operators and the first initial boundary value problem for pseudoparabolic equations with analytic coefficients. J. Differential Equations 13 (1973), 506–522.CrossRefGoogle Scholar
8Chazarain, J. and Piriou, A.. Introduction to the Theory of Linear Partial Differential Equations (Gröningen: Noordhoff, 1982).Google Scholar
9Dai, D. Q.. On an initial boundary value problem for nonlinear pseudoparabolic equations with two space variables. Complex Variables Theory Appl. 14 (1990), 139–151.Google Scholar
10Gilbert, R. P. and Hsiao, G. C.. Constructive function theoretic methods for higher order pseudoparabolic equations, Lecture Notes in Mathematics 561 (Berlin: Springer, 1977).Google Scholar
11Gilbert, R. P. and Jensen, J.. A computational approach for constructing solutions of one- dimensional pseudoparabolic and metaparabolic equations. SIAM J. Sci. Stat. Comput. 3 (1982), 111–125.CrossRefGoogle Scholar
12Gilbert, R. P. and Schneider, M.. Generalized meta and pseudoparabolic equations in the plane. In the Moscow Academy of Science volume honouring I. N. Vekua, 160–172 (Moscow: Nauka, 1978).Google Scholar
13Muskhelishvili, N. I.. Singular Integral Equations (Gröningen: Noordhoff, 1953).Google Scholar
14Pluschke, V.. Solution of nonlinear pseudoparabolic equations by semidiscretization in time. Complex Variables Theory Appl. 7 (1987), 321–336.Google Scholar
15Vekua, I. N..Generalized Analytic Functions (Oxford: Pergamon, 1962).Google Scholar
16Wen, G. C., Li, Z. Z. and Li, H. Z.. Some properties of the solutions for first order nonlinear equations in the all-plane and their applications. J. Hebei Univ. (Natural Science ed.) 1 (1981), No. 2, 28–35 (in Chinese).Google Scholar