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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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