On the Existence and Uniqueness of Solutions of the Equation

John C. Cleménts

doi:10.4153/CMB-1975-036-1

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.

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'(u_x)u_xx+λu_xxt=p₀u_tt: 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

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Clements, John 1974. Existence Theorems for a Quasilinear Evolution Equation. SIAM Journal on Applied Mathematics, Vol. 26, Issue. 4, p. 745.

Ebihara, Yukiyoshi 1978. On some nonlinear evolution equations with the strong dissipation. Journal of Differential Equations, Vol. 30, Issue. 2, p. 149.

Ebihara, Yukiyoshi 1979. On some nonlinear evolution equations with the strong dissipation, II. Journal of Differential Equations, Vol. 34, Issue. 3, p. 339.

Pecher, Hartmut 1980. On global regular solutions of third order partial differential equations. Journal of Mathematical Analysis and Applications, Vol. 73, Issue. 1, p. 278.

Yamada, Yoshio 1981. Quasilinear wave equations and related nonlinear evolution equations. Nagoya Mathematical Journal, Vol. 84, Issue. , p. 31.

Ebihara, Yukiyoshi 1982. On some nonlinear evolution equations with strong dissipation, III. Journal of Differential Equations, Vol. 45, Issue. 3, p. 332.

Ang, Dang Dinh and Pham Ngoc Dinh, A. 1988. Strong Solutions of a Quasilinear Wave Equation with Nonlinear Damping. SIAM Journal on Mathematical Analysis, Vol. 19, Issue. 2, p. 337.

Hai, Dang Dinh 1988. On a quasilinear wave equation with nonlinear damping. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 110, Issue. 3-4, p. 227.

Dinh, A. Pham Ngoc and Ang, Dang Dinh 1989. Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications. p. 499.

Hai, Dang Dinh 1991. On a doubly damped quasilinear wave equation. Annali di Matematica Pura ed Applicata, Vol. 160, Issue. 1, p. 77.

Kawashima, S. and Shibata, Y. 1992. Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Communications in Mathematical Physics, Vol. 148, Issue. 1, p. 189.

Muñoz Rivera, Jaime E. 1995. Existence and decay of weak solutions for systems arising in thermodynamics. Nonlinear Analysis: Theory, Methods & Applications, Vol. 24, Issue. 6, p. 825.

Zhijian, Yang 2002. Blowup of solutions for a class of non‐linear evolution equations with non‐linear damping and source terms. Mathematical Methods in the Applied Sciences, Vol. 25, Issue. 10, p. 825.

Yacheng, Liu and Junsheng, Zhao 2004. Multidimensional viscoelasticity equations with nonlinear damping and source terms. Nonlinear Analysis: Theory, Methods & Applications, Vol. 56, Issue. 6, p. 851.

Keyantuo, Valentin and Lizama, Carlos 2006. Periodic solutions of second order differential equations in Banach spaces. Mathematische Zeitschrift, Vol. 253, Issue. 3, p. 489.

Benaissa, Abbès and Mokeddem, Soufiane 2007. Decay estimates for the wave equation of p‐Laplacian type with dissipation of m‐Laplacian type. Mathematical Methods in the Applied Sciences, Vol. 30, Issue. 2, p. 237.

Kalantarov, Varga and Zelik, Sergey 2009. Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. Journal of Differential Equations, Vol. 247, Issue. 4, p. 1120.

Sun, Fuqin Wang, Mingxin and Li, Huiling 2010. Global and blow-up solutions for a quasilinear hyperbolic equation with strong damping. Nonlinear Analysis: Theory, Methods & Applications, Vol. 73, Issue. 5, p. 1408.

Liu, Xiaopan and Apreutesei, Narcisa C. 2012. On Existence, Uniform Decay Rates, and Blow‐Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source. Abstract and Applied Analysis, Vol. 2012, Issue. 1,

Xu, Runzhang Liu, Jie Niu, Yi and Chen, Shaohua 2013. Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term. Boundary Value Problems, Vol. 2013, Issue. 1,

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On the Existence and Uniqueness of Solutions of the Equation

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