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Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. (English) Zbl 0879.35110
The equation of the Kirchhoff string is the following: \[ \rho h {\partial^2u\over\partial t^2}= \Biggl\{p_0+{Eh\over 2L}\int^L_0 \Biggl({\partial u\over\partial x}\Biggr) dx\Biggr\} {\partial^2u\over\partial x^2} u(0,x)= u_0(x), \] for \(0<x<L\), \(t\geq 0\), where \(u=u(x,t)\) is the lateral displacement at the space coordinate \(x\) and the time \(t\), \(E\) the Young modulus, \(\rho\) the mass density, \(h\) the cross-section area, \(L\) the length, \(p_0\) the initial axis tension. In this paper, the author considers the initial value problem for the second-order hyperbolic equations generalizing the Kirchhoff strings: \[ u''+ M(|A^{1/2}u|^2)Au+\delta u'= f(u)\quad\text{in }\Omega\times [0,\infty), \]
\[ u(x,0)= u_0(x),\;u'(x,0)= u_1(x)\quad\text{and}\quad u(x,t)|_{\partial\Omega}=0, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \('=\partial_t\equiv=\partial/\partial t\); \(A=-\Delta\) is the Laplace operator with domain \({\mathcal D}={\mathcal H}^\varepsilon(\otimes)\cap{\mathcal H}^\infty_{\prime}(\otimes)\), \(|\cdot|\) is the norm of \(L^2(\Omega)\), \(\delta\geq 0\), \(f(u)=|u|^\alpha u\) with \(\alpha>0\), \(M(r)\) is a nonnegative locally Lipschitz function for \(r\geq 0\) like \(M(r)a+ br^\gamma\) with \(a\geq 0\), \(b\geq 0\), \(a+b>0\), and \(\gamma>0\).
He gives the local existence theorem and the global existence and decay properties of solutions for degenerate (i.e. \(a=0\)) and non-degenerate (i.e. \(a>0\)) equations with a dissipative term, respectively. And he also studies the blow up problem in cases of initial energy being nonpositive and positive, respectively.
Reviewer: A.Tsutsumi (Osaka)

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
74K05 Strings
Full Text: DOI
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