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On the Existence and Uniqueness of Solutions of the Equation

Published online by Cambridge University Press:  20 November 2018

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Abstract

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The existence and uniqueness of strong global solutions of initial-boundary value problems for the quasilinear equation utt—∂σi(uxi)/∂xi—ΔNut= f is established for functions σi(ξ), i = 1, …, N, satisfying: σi,(ξ) ∊ C1(-∞, ∞), σi(0) = 0 and for some constant K0.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N., Linear andquasilinear equations of parabolic type, A.M.S. Trans, of Math. Monographs, Vol. 23.Google Scholar
2. Ladyzenskaja, O. A. and Vralcera, N. N., Linear andquasilinear elliptic equations, Math, in Sc. and Eng. Vo., 46 (1968).Google Scholar
3. Lions, J. L., Quelques methods de résolution des problémes aux limits non linéaires, Dunod, Gauthier-Villars, Paris 1969.Google Scholar
4. Lions, J. L. and Strauss, W. A., Some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965), 43-96.Google Scholar
5. MacCamy, R. C. and Mizel, V. J., Existence and non-existence in the large of solutions to quasilinear wave equations, Arch. Rational Mech. Anal., 25 (1967) 299-320.Google Scholar
6. MacCamy, R. C., Mizel, V. J. and Greenberg, J. M., On the existence, uniqueness and stability of solutions of the equation σ'(ux)uxx+λuxxt=p0utt: Jour. Math, and Mech., 17 (1968), 707-728.Google Scholar
7. Sather, Jerome, The existence of a global classical solution of the initial-boundary value problem for ☐u+u3=f, Arch. Rational Mech. Anal. 22 (1966), 292-307.Google Scholar
8. Stoker, J. J., Topics in nonlinear elasticity, Courant Inst, of Math. Se, N.Y. Univ., 1964.Google Scholar
9. Tsutsumi, M., Some nonlinear evolution equations of second order, Proc. Japan Acad., 47 (1971), 950-955.Google Scholar