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An application of semigroups of locally Lipschitz operators to Carrier equations with acoustic boundary conditions. (English) Zbl 1145.47044
Summary: A generation theorem for semigroups of locally Lipschitz operators on a subset of a real Banach space is given and applied to the problem of the well-posedness of the Carrier equation $$u_{tt} - \kappa (\| u\| ^{2})\Delta u+\gamma | u_t| ^{p - 1}u_t=0$$ in $$\varOmega \times (0,\infty )$$ with acoustic boundary condition, where $$p>2$$ and $$\varOmega$$ is a bounded domain in an arbitrary dimensional space.

MSC:
 47H20 Semigroups of nonlinear operators 35L70 Second-order nonlinear hyperbolic equations
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References:
 [1] Ph. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, thesis, Orsay, 1972 [2] Brezis, H., Équations et inéquations non linéaires dans LES espaces vectoriels en dualité, Ann. inst. Fourier, 18, 115-175, (1968) · Zbl 0169.18602 [3] Chambers, J.T.; Oharu, S., Semi-groups of local lipschitzians in a Banach space, Pacific J. math., 39, 89-112, (1971) · Zbl 0232.47080 [4] Crandall, M.G., Nonlinear semigroups and evolution governed by accretive operators, (), Part 1, pp. 305-337 · Zbl 0637.47039 [5] Crandall, M.G.; Evans, L.C., On the relation of the operator $$\partial / \partial s + \partial / \partial \tau$$ to evolution governed by accretive operators, Israel J. math., 21, 261-278, (1975) · Zbl 0351.34037 [6] Crandall, M.G.; Liggett, T.M., Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. math., 93, 265-298, (1971) · Zbl 0226.47038 [7] Frota, C.L.; Goldstein, J.A., Some nonlinear wave equations with acoustic boundary conditions, J. differential equations, 164, 92-109, (2000) · Zbl 0979.35105 [8] Kobayashi, Y., Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. math. soc. Japan, 27, 640-665, (1975) · Zbl 0313.34068 [9] Y. Kobayashi, On the generation of nonlinear contraction semigroups, in: Proceedings of the Symposium on Real Analysis and Functional Analysis, 1978, pp. 1-16 (in Japanese) [10] Kobayashi, Y.; Oharu, S., Semigroups of locally Lipschitzian operators in Banach spaces, Hiroshima math. J., 20, 573-611, (1990) · Zbl 0766.47038 [11] Kobayashi, Y.; Matsumoto, T.; Tanaka, N., Semigroups of locally Lipschitz operators associated with semilinear evolution equations, J. math. anal. appl., 330, 1042-1067, (2007) · Zbl 1123.34044 [12] Kobayashi, Y.; Tanaka, N., Semigroups of locally Lipschitz operators, Math. J. okayama univ., 44, 155-170, (2002) · Zbl 1033.47040 [13] Lakshmikantham, V.; Leela, S., Differential and integral inequalities, (1969), Academic Press New York · Zbl 0177.12403 [14] Pazy, A., The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. anal. math., 40, 239-262, (1981) · Zbl 0507.47042 [15] Walker, J.A., Dynamical systems and evolution equations, theory and applications, (1980), Plenum Press New York · Zbl 0421.34050
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