zbMATH — the first resource for mathematics

On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. (English) Zbl 0878.35081
The author considers the Cauchy problem for the strongly damped Kirchhoff equation of degenerate type $u_{tt}-\left(\int_{\Omega}|\nabla u|^2 dx\right)^{2\gamma} \Delta u -\Delta u_t=|u|^{\alpha}u, \eqno (*)$ where $$\gamma \geq1$$, and $$\Omega$$ is a bounded domain in $${\mathbb{R}}^N$$, and proves the following result: If the initial data $$u_0\equiv u(0,x)\in H^2(\Omega)\cap H^1_0(\Omega)$$ and $$u_1\equiv u_t(0,x)\in H^1_0(\Omega)$$ are sufficiently small and $$\int |\nabla u_0|^{2(\gamma+1)}dx > \int |u_0|^{\alpha+2}dx$$, then the problem has a global solution provided $$2\gamma <\alpha \leq 4/(N-2)$$ if $$N> 3$$, or $$2\gamma <\alpha$$ if $$N=1,2$$.

MSC:
 35L80 Degenerate hyperbolic equations 35L75 Higher-order nonlinear hyperbolic equations 35L35 Initial-boundary value problems for higher-order hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
Keywords:
Kirchhoff equations
Full Text:
References:
 [1] Arosio, Math. Methods Appl. Sci. 14 pp 177– (1991) [2] and , ’Global solutions to the Cauchy problem for a nonlinear hyperbolic equation’, Nonlinear P.D.E. and their Applications, Collège de France Seminar, Eds (and , eds.), Research Notes Math., Vol VI, pp. 1-26. Pitman, Boston, (1984). [3] Bernstein, Izv. Akad. Nauk SSSR, Ser. Mat. 4 pp 17– (1940) [4] Crippa, Nonlinear Anal. 21 pp 565– (1993) [5] D’Ancona, Math. Methods Appl. Sci. 17 pp 477– (1994) [6] D’Ancona, Invent. Math. 108 pp 247– (1992) [7] and , ’On an abstract weakly hyperbolic equation modelling the non-linear vibrating string’, in: Developments in PDEs and Applications to Mathematical Physics ( and , eds.), Plenum Press, New York, 1992. [8] Dickey, SIAM J. Appl. Math. 19 pp 208– (1970) [9] Ebihara, Nonlinear Anal. 10 pp 27– (1986) [10] and , Elliptic Partial Differential Equations of Second Order, 2nd edn, Springer, Berlin 1983. · Zbl 0361.35003 [11] Hosoya, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 pp 239– (1991) [12] Ikehata, Differential Integral Equations 8 pp 607– (1995) [13] Vorlesungen über Mechanik, Teubner, Stuttgart, 1883. · JFM 66.1359.01 [14] Kobayashi, Math. Ann. 296 pp 215– (1993) [15] Levine, Tran. Amer. Math. Soc. 192 pp 1– (1974) [16] Levine, SIAM J. Math. Anal. 5 pp 138– (1974) [17] Levine, Math. Ann. 214 pp 205– (1975) [18] Matos, Funkcial. Ekvac. 34 pp 303– (1991) [19] Medeiro, Comput. Appl, Math. 6 pp 257– (1987) [20] Menzala, Nonlinear Anal. 3 pp 613– (1979) [21] ’The asymptotic behavior of solutions to the Krichhoff equation with viscous damping’, preprint. [22] Nakao, J. Math. Anal. Appl. 60 pp 542– (1977) [23] Nakao, J. Math. Soc. Japan 30 pp 747– (1978) [24] Nakao, Math. Z. 219 pp 289– (1995) [25] Nakao, Math. Z. 214 pp 325– (1993) [26] Nirenberg, Ann. Scuola Norm. Sup. Pisa 13 pp 115– (1959) [27] Nishihara, Tokyo J. Math. 7 pp 437– (1984) [28] Nishihara, Funkcial. Ekvac. 27 pp 125– (1984) [29] Nishihara, Nonlinear Anal. 21 pp 17– (1993) [30] Nishihara, Adv. Math. Sci. Appl. 4 pp 285– (1994) [31] Nishihara, Funkcial. Ekvac. 33 pp 151– (1990) [32] Ono, Adv. Math. Sci. Appl. 5 pp 457– (1995) [33] Ono, J. Math. Tokushima Univ. 29 pp 43– (1995) [34] Payne, Israel J. Math. 22 pp 273– (1975) [35] Pohožaev, Math. USSR Sbornik 25 pp 145– (1975) [36] Rivera, Appl. Anal. 10 pp 93– (1980) [37] Rivera, Ann. Fac. Sci. Toulouse Math. 1 pp 237– (1992) · Zbl 0783.47074 [38] Sattinger, Arch. Rat. Mech. Anal. 30 pp 148– (1968) [39] Tsutsumi, Math. Japan 17 pp 173– (1972) [40] Yamada, Nonlinear Anal. 11 pp 1155– (1987) [41] Yamazaki, Funkcial. Ekvac. 31 pp 439– (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.