Some Studies in the Nonlinear Wave Equations

Abstract

在這篇論文中，我們將考慮2個具有初值及邊界值的非線性波方程。首先，考慮一個具有某種阻尼項 (強阻尼、 線性阻尼及非線性阻尼) 的積分--微分方程。我們利用 Fadeo-Galerkin及 Contraction Mapping Principle的方法來建立局部存在性和唯一性，並且使用 Nako 的不等式 ([40]) 來討論解的長時間存在 (global existence) 及漸進行為( asymptotic behavior) 。至於在解的有限時間爆增 (finite time blow-up) 方面，我們使用直接方法 ([33]) 來探討具有強阻尼及線性阻尼的問題。另一方面，我們利用能量法 (energy method) 來討論非線性阻尼問題的有限時間爆增現象。其次，我們考慮一個具有特殊邊界值的 Kirchhoff方程， 我們利用擾動的能量法 (perturbed energy method) ([56]) 來研究解的漸進行為，並且使用直接方法 ([33]) 來探討解的有限時間暴增問題。最後，我們提出一些與本文相關的有趣問題以作為未來的研究。

In this thesis, we shall consider two initial-boundary value problems for nonlinear wave equations. First, we consider a nonlinear integro-\r\ndifferential equation with some kind of damping terms - the strong damping term or the linear damping term or the nonlinear damping term. We establish the existence and uniqueness of local solutions by using Faedo-Galerkin method and Contraction Mapping Principle. We shall discuss the asymptotic behavior of global solutions by using Nako’s inequality ([40]). Moreover, the blow-up properties of local solutions with non-positive initial energy and small positive initial energy for strong or linear damping case are obtained by using direct method ([33]). On the other hand, for the nonlinear damping case, we apply the energy method to deduce the blow-up of local solutions with negative initial energy, vanishing initial energy and small positive initial energy. The estimates of lifespan of solutions are also given in each case. Secondly, we shall consider an initial-boundary value problem for a wave equation of Kirchhoff type with a linear boundary damping term. The asymptotic behavior of global solutions is investigated by using perturbed energy method ([56]). Moreover, the blow-up phenomena with the initial energy being non-positive and positive and the estimates for the blow-up time are obtained by direct approach ([33]). Finally, a list of some interesting problems related to our model is posed for further research.