A quasi-linear evolution equation and the method of Galerkin

Author:
R. W. Dickey

Journal:
Proc. Amer. Math. Soc. **37** (1973), 149-156

MSC:
Primary 35Q99

DOI:
https://doi.org/10.1090/S0002-9939-1973-0308620-3

MathSciNet review:
0308620

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Abstract: In this paper it is shown that under specified conditions on the initial data a certain infinite coupled system of ordinary differential equations has a solution satisfying an auxiliary convergence condition. The infinite system discussed is essentially the Galerkin expansion of the solution to a given quasi-linear wave equation. The results obtained suffice to prove the existence of a solution to this wave equation.

**[1]**R. Narasimha,*Non-linear vibrations of an elastic string*, J. Sound. Vib.**8**(1968), 134-146.**[2]**S. Woinowsky-Krieger,*The effect of axial force on the vibration of hinged bars*, J. Appl. Mech.**17**(1950), 35-36. MR**11**, 558. MR**0034202 (11:558f)****[3]**R. W. Dickey,*Free vibrations and dynamic buckling of the extensible beam*, J. Math. Anal. Appl.**29**(1970), 443-454. MR**40**#6831. MR**0253617 (40:6831)****[4]**-,*Infinite systems of nonlinear oscillation equations related to the string*, Proc. Amer. Math. Soc.**23**(1969), 459-468. MR**40**#458. MR**0247189 (40:458)****[5]**J. M. Ball,*Initial boundary value problems for an extensible beam*, J. Math. Anal. Appl. (to appear). MR**0319440 (47:7984)****[6]**E. A. Coddington and N. Levinson,*Theory of ordinary differential equations*, McGraw-Hill, New York, 1955. MR**16**, 1022. MR**0069338 (16:1022b)****[7]**N. J. Zabusky,*Exact solution for the vibrations of a nonlinear continuous model string*, J. Mathematical Phys.**3**(1962), 1028-1039. MR**26**#4067. MR**0146545 (26:4067)****[8]**P. D. Lax,*Development of singularities of solutions of nonlinear hyperbolic partial differential equations*, J. Mathematical Phys.**5**(1964), 611-613. MR**29**#2532. MR**0165243 (29:2532)****[9]**R. C. MacCamy and V. J. Mizel,*Existence and nonexistence in the large of solutions of quasilinear wave equations*, Arch. Rational Mech. Anal.**25**(1967), 299-320. MR**35**#7000. MR**0216165 (35:7000)**

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0308620-3

Article copyright:
© Copyright 1973
American Mathematical Society