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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)

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35L15 Initial value problems for second-order hyperbolic equations 74K05 Strings
##### Keywords:
global existence; decay properties; blow up
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