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Article

Keywords:
nonlinear evolution equation; mixed problem; asymptotic stability of solutions
Summary:
We establish the asymptotic stability of solutions of the mixed problem for the nonlinear evolution equation $(|u_t|^{r-2}u_t)_t-\Delta u_{tt}-\Delta u-\delta\Delta u_t=f(u)$.
References:
[1] Love A.H.: A Treatise on Mathematical Theory of Elasticity. Dover, New York, 1944. MR 0010851
[2] Ferreira J., Rojas-Medar M.: On global weak solutions of a nonlinear evolution equation in noncylindrical domain. in Proceedings of the 9th International Colloquium on Differential Equations, D. Bainov (Ed.), VSP, 1999, pp.155-162. Zbl 0943.35054
[3] Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J.: Existence and uniform decay for a nonlinear viscoelastic equation with strong damping. Math. Meth. Appl. Sci. 24 (2001), 1043-1053. MR 1855298 | Zbl 0988.35031
[4] Nakao M., Ono K.: Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. Math. Z. 214 (1993), 325-342. MR 1240892 | Zbl 0790.35072
[5] Ono K.: On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation. J. Math. Anal. Appl. 216 (1997), 321-342. MR 1487267 | Zbl 0893.35078
[6] Park J.Y., Bae J.J.: On solutions of quasilinear wave equations with nonlinear damping terms. Czechoslovak Math. J. 50 (2000), 565-585. MR 1777478 | Zbl 1079.35533
[7] Levine H.A., Pucci P., Serrin J.: Some remarks on global nonexistence for nonautonomous abstract evolution equations. Contemporary Mathematics 208 (1997), 253-263. MR 1467010 | Zbl 0882.35081
[8] Pucci P., Serrin J.: Stability for abstract evolution equations. in Partial Differential Equation and Applications, P. Marcellimi, et al. (Eds.), Marcel Dekker, 1996, pp.279-288. MR 1371599 | Zbl 0879.47027
[9] Pucci P., Serrin J.: Asymptotic stability for nonautonomous wave equation. Comm. Pure Appl. Math. XLXX (1996), 177-216. MR 1371927
[10] Payne L.E., Sattinger D.H.: Saddle points and unstability of nonlinear hyperbolic equations. Israel J. Math. 22 (1975), 273-303. MR 0402291
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