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Multi-clustered solutions for a singularly perturbed forced pendulum equation

  • Salomé Martínez (a1) and Dora Salazar (a2)

Abstract

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely

$$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$
Using a limiting energy function, we describe the behaviour of the solutions as the parameter ε approaches zero. We also prove the existence of a family of solutions having a prescribed asymptotic profile and exhibiting a highly rotatory behaviour alternated with a highly oscillatory behaviour in some open subsets of the domain. The proof relies on a combination of the Nehari finite dimensional reduction with the topological degree theory.

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References

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Multi-clustered solutions for a singularly perturbed forced pendulum equation

  • Salomé Martínez (a1) and Dora Salazar (a2)

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