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Beam evolution equation with variable coefficients. (English) Zbl 1067.35129
Summary: We investigate the following initial-boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping: $(a(x,t)u'(x,t))'+ \Delta(b(x,t) \Delta u(x,t))- M\biggl( x,t, \int_\Omega |\nabla u(x,t)|^2\,dx\biggr) \Delta u(x,t)+ \delta u'(x,t)= 0 \text{ in }Q,$ $u(x,t)= \frac {\partial u}{\partial\nu} (x,t)= 0\quad\text{on }\Sigma,$ $u(x,0)= u_0(x), \qquad u'(x,0)= u_1(x) \quad\text{in }\Omega,$ where $$\Omega$$ is a non-empty bounded open set of $$\mathbb R^n$$, for $$n\geq 1$$, with $$C^2$$ boundary $$\Gamma$$, $$Q$$ is the cylinder $$\Omega\times ]0,T[$$ of $$\mathbb R^{n+1}$$, for $$T>0$$, $$|\nabla u(x,t)|$$ is the norm in $$\mathbb R^n$$ of the vector $$\nabla u(x,t)$$. We show the existence of a unique global weak solution and that the energy associated with this solution has a decay rate estimate. Besides, we prove the existence and uniqueness of non-local strong solutions.

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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