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On the well-posedness of the Kirchhoff string. (English) Zbl 0858.35083
Let $\Omega$ be a bounded open subset of $\bbfR^n$ with smooth boundary $\partial\Omega$. (The cases $\Omega=\bbfR^n$ or $\bbfR^n_+$ are also permitted.) Let $m:[0,\infty)\to (0,\infty)$ be locally Lipschitz and coercive at $\infty$, in the sense that $m(s)>0$, $\forall s>0$, and $\int^\infty_0 m(s)ds=\infty$. The authors study the well-posedness in the Hadamard sense of the Cauchy-Dirichlet problem $$u_{tt}-m\Biggl( \int_\Omega|\nabla_xu|^2 dx \Biggr)\Delta_xu=f(x,t) \qquad (x\in\Omega,\ t>0),\tag P$$ $$u(x,0)= u_0(x), \quad u_t(x,0)= u_1(x)\quad (x\in\Omega), \qquad u(.,t)|_{\partial\Omega}=0,$$ where $(u_0,u_1)\in V_\alpha(\Omega)\times V_{\alpha-1}(\Omega)$ and $f\in L^1_{\text{loc}} ([0,\infty); V_{\alpha-1}(\Omega))$, $\alpha\geq3/2$ ($V$ stands for $D(A^{\alpha/2})$ with $A=-\Delta$, $D(A)= H^1_0(\Omega)\cap H^2(\Omega))$. They key idea is to view (P) as a special case of an abstract second order Cauchy problem of the form $$u''+m(\langle Au,u\rangle)Au=f \quad (t>0), \qquad u(0)=u_0, \quad u'(0)=u_1$$ in a Hilbert space $H$ with inner product $\langle.,.\rangle$, where $A$ is a selfadjoint nonnegative linear operator in $H$.

35L70Nonlinear second-order hyperbolic equations
45K05Integro-partial differential equations
74K05Strings (solid mechanics)
35L20Second order hyperbolic equations, boundary value problems
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