Existence and non-existence of global solutions for the wave equations

JIN Shou-bo; ZHANG Zu-feng

doi:10.3969/j.issn.1000-5641.2018.02.001

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Mar. 2018

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JIN Shou-bo, ZHANG Zu-feng. Existence and non-existence of global solutions for the wave equations[J]. Journal of East China Normal University (Natural Sciences), 2018, (2): 1-10. doi: 10.3969/j.issn.1000-5641.2018.02.001

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JIN Shou-bo, ZHANG Zu-feng. Existence and non-existence of global solutions for the wave equations[J]. Journal of East China Normal University (Natural Sciences), 2018, (2): 1-10. doi: 10.3969/j.issn.1000-5641.2018.02.001

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Existence and non-existence of global solutions for the wave equations

doi: 10.3969/j.issn.1000-5641.2018.02.001

School of Mathematics and Statistics, Suzhou University, Suzhou Anhui 234000, China

Received Date: 2017-03-14
Publish Date: 2018-03-25

Abstract

Abstract

In this paper we investigated the initial boundary value problem for a class of higher order wave equations with two opposite source terms. Firstly, we introduced the latest research progress of the wave equations and defined some important generalized functionals and sets, then the properties of the functionals were discussed. Secondly, it was proved that these sets were invariant under the wave equation. Finally, we proved the existence of global weak solutions by the combination of Galerkin approximation method and potential well method, and obtained the conditions of the non-existence of global weak solutions by using the potential well method and the convexity. The optimal threshold results were given for the existence and non-existence of global weak solutions.
- wave equation,
- k-Laplace operator,
- existence,
- non-existence,
- potential well

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References(21)

References

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