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Well-posedness for a nonlinear shallow spherical shell. (English) Zbl 0962.35117
The well-posedness problem for the the nonlinear shallow spherical shell is considered as well-posedness problem for a system of two nonlinear second-order partial differential equations with initial and boundary conditions in a domain \((0, \rho_0)\times(0, \infty)\). These equations are rewriten in the form of the second-order abstract Cauchy problem \[ M u_{tt}(t)+Au(t)+B\partial\Phi B^*u_t(t)+Bf(u(t))\ni{\mathcal F}(u),\quad t>0, \]
\[ u(0)=u_0,\;u'(0)=u_1 \] studied in [I. Lasiecka, Nonlinear Anal., Theory Methods Appl. 23, No. 6, 797-823 (1994; Zbl 0822.35096)]. The operators used in this formulation depend on the linearized version of the system studied in [I. Lasiecka, R. Triggiani and V. Valente, Adv. Differ. Equ. 1, No. 4, 635-674 (1996; Zbl 0857.35017)] and are defined in some weighted spaces. On the basis of this reduction it is proved that for a class of initial values there exists a global, unique, mild solution of the problem.
MSC:
35L55 Higher-order hyperbolic systems
35L90 Abstract hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
74K25 Shells
35L20 Initial-boundary value problems for second-order hyperbolic equations
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