ON A QUASILINEAR WAVE EQUATION WITH MEMORY AND NONLINEAR SOURCE TERMS

P. A. OGBIYELE

Abstract


In this paper, we consider a quasilinear wave equation with memory and nonlinear source terms

\[u_{tt} - \Delta u_t -\sum^n_{i=1} \frac{\partial}{\partial x_i} \sigma_i(u_{x_i}) + \int^t_0 m(t-s) \Delta u ds = g(u) . \]

In the absence of the nonlinear damping term, and under certain polynomial growth conditions on the nonlinear functions $\sigma_i, (i= 1, 2, ..., n)$ and $g$, we obtain existence and uniqueness of solution, using Galerkin approach and monotonicity method. The finite time blow up result was obtained using the concavity method.


Full Text:

PDF

References


bibitem{andrews} G. Andrews, On the existence of solutions to the equation $u_{tt}= u_{xxt}+ sigma(u_x)_x$, J. Differential Equations 35 (1980)

-231.

bibitem{AndrewsB} G. Andrews, J.M. Ball, Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations 44 (1982) 306 -341.

bibitem{AngD1} D.D. Ang, A.P.N. Dinh, On the strongly damped wave equation $u_{tt} -Delta u - Delta u_t + f(u) =0$ SIAM J. Math. Anal. 19 (1988) 1409 -1418.

bibitem{AngD2}D.D. Ang, A.P.N. Dinh, Strong solutions of a quasi-linear wave equation with nonlinear damping term,

SIAM J. Math. Anal. 19 (1988) 337 -347.

bibitem{GHS} G. Chen, H Yue, S. Wang, The initial boundary value problem for quasilinear wave equation with viscous damping. J. Math. Anal. Appl. 331 (2007) 823-839

bibitem{clements} J. Clements, Existence theorems for a quasi-linear evolution equation, SIAM J. Appl. Math. 26 (1974) 745 -752.

bibitem{Gothel} R. G"{o}thel and D. S. Jones Faedo-Galerkin approximation in equations of evolution. Math. Meth. Appl. Sci., 6 (1984) 41-54.

bibitem{GT} V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term, J. Diļ¬€erential Equations 109 (1994), 295 -308.

bibitem{GreenMM} J.M. Greenberg, R.C. MacCamy, V.J. Mizel, On the existence, uniqueness and stability of solutions of the equation $sigma^{prime}(u_x)u_{xx} + lambda u_{xxt}=rho_0 u_{tt}$, J. Math. Mech. 17 (1968) 707 -728.

bibitem{lev1} H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form, Trans. Amer. Math. Soc. 192 (1974), 1 -21.

bibitem{lev2} H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138 -146.

bibitem{Lions} J. -L. Lions, Quelques M'{e}thodes de r'{e}solution des problemex aux limites nonlin'{e}aires, Dunod-Gauthier villars, paris, (1969).

bibitem{yamada} Y. Yamada, some remarks on the equation $Y_{tt} -sigma(Y_x)Y_{xx} -Y_{xtx} = f $ Osaka J. Math. 17 (1980) 303 -323.

bibitem{yang1} Z. Yang, Cauchy problem for quasi-linear wave equations with viscous damping, J. Math. Anal. Appl. 320. (2006) 859-881

bibitem{yang3} Z. Yang, Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with nonlinear damping and source terms, Math. Methods Appl. Sci. 25 (2002) 795 -814.

bibitem{yang2} Z. Yang, Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms, J. Math. Anal. App. 300 (2004) 218-243

bibitem{yang4} Z. Yang, Blowup of solutions for a class of nonlinear evolution equations with nonlinear damping and source terms, Math. Methods Appl. Sci. 25 (2002) 825 -833.

bibitem{YangG} Z. Yang, G. Chen, Global existence of solutions for quasi-linear wave equations with viscous damping, J. Math. Anal. Appl. 285 (2003) 606 -620.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2018 Journal of the Nigerian Mathematical Society

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.