A nonlinear beam equation arising in the theory of elastic bodies.

*(English)*Zbl 0902.35066The authors study a nonlinear Cauchy problem which arises in the theory of oscillations in elastic bodies under external forces. More precisely, they consider the problem
\[
\partial_{t}^{2}w + \partial_{\xi}^{4}w + \gamma_{1}\partial_{t} \partial_{\xi}^{4}w - \gamma_{2}\partial_{\xi}^{2}w - \gamma_{3} \left ( \int_{0}^{1} (\partial_{\xi}w)^{2} \;d \xi \right) \partial_{\xi}^{2}w - \gamma_{4}\partial_{t}w - \gamma_{5}(\partial_{t}w)^{3} = 0,
\]

\[ w(\xi,0)= w_{0}(\xi), \quad \partial_{t}w(\xi,0) = w_{1}(\xi),\quad w(0,t) = w(1,t) = \partial_{\xi}^{2}w(0,t) = \partial_{\xi}^{2} w(1,t) = 0,\tag{1} \] where \((\xi,t)\in(0,1)\times[0,+\infty)\), \(\gamma_{1},...,\gamma_{5}\) are real parameters, \(\gamma_{1} \geq 0.\)

The type of equation which appears in (1) has been suggested by Blevins (R. D. Blevins, Flow-induced vibrations (Van Nostrand Reinhold Comp., New York 1977; Zbl 0385.73001)]. The initial conditions in (1) are usual, whereas the boundary one (which describe the so-called simply supported ends), have been proposed by some studies in engineering [see W. Herfort and H. Troger, Robust modelling of flow induced oscillations of bluff bodies. Math. Modelling 8, 251-255 (1987)]. These boundary conditions cause some additional difficulties, due to the fact that classical Sobolev spaces can not be used.

The authors show the global solvability of (1). To prove this, they reformulate the problem as a nonlinear Cauchy problem in an appropriate Sobolev space which contains the boundary conditions. Then, they study the associated linear problem (by using strongly continuous contraction semigroups) and obtain some estimates for the nonlinear part. Next, this Cauchy problem is transformed into an integral equation which gives rise to a fixed point problem. By applying some previous results [see C. Foiaş, G. Gussi, and V. Poenaru, Rev. Roum. Math. Pures Appl. 3, 283-304 (1958; Zbl 0087.11604)], it is proved the existence and uniqueness of a local solution. Finally, after some uniform boundedness estimations, this solution becomes a global one.

\[ w(\xi,0)= w_{0}(\xi), \quad \partial_{t}w(\xi,0) = w_{1}(\xi),\quad w(0,t) = w(1,t) = \partial_{\xi}^{2}w(0,t) = \partial_{\xi}^{2} w(1,t) = 0,\tag{1} \] where \((\xi,t)\in(0,1)\times[0,+\infty)\), \(\gamma_{1},...,\gamma_{5}\) are real parameters, \(\gamma_{1} \geq 0.\)

The type of equation which appears in (1) has been suggested by Blevins (R. D. Blevins, Flow-induced vibrations (Van Nostrand Reinhold Comp., New York 1977; Zbl 0385.73001)]. The initial conditions in (1) are usual, whereas the boundary one (which describe the so-called simply supported ends), have been proposed by some studies in engineering [see W. Herfort and H. Troger, Robust modelling of flow induced oscillations of bluff bodies. Math. Modelling 8, 251-255 (1987)]. These boundary conditions cause some additional difficulties, due to the fact that classical Sobolev spaces can not be used.

The authors show the global solvability of (1). To prove this, they reformulate the problem as a nonlinear Cauchy problem in an appropriate Sobolev space which contains the boundary conditions. Then, they study the associated linear problem (by using strongly continuous contraction semigroups) and obtain some estimates for the nonlinear part. Next, this Cauchy problem is transformed into an integral equation which gives rise to a fixed point problem. By applying some previous results [see C. Foiaş, G. Gussi, and V. Poenaru, Rev. Roum. Math. Pures Appl. 3, 283-304 (1958; Zbl 0087.11604)], it is proved the existence and uniqueness of a local solution. Finally, after some uniform boundedness estimations, this solution becomes a global one.

Reviewer: A.Cañada (Granada)

##### MSC:

35L35 | Initial-boundary value problems for higher-order hyperbolic equations |

35L75 | Higher-order nonlinear hyperbolic equations |

47H20 | Semigroups of nonlinear operators |

74B20 | Nonlinear elasticity |