On Global Existence, Asymptotic Stability and Blowing Up of Solutions for Some Degenerate Non-linear Wave Equations of Kirchhoff Type with a Strong Dissipation

✍ Scribed by Kosuke Ono

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 335 KB
• Volume: 20
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/(sici)1099-1476(19970125)20:2<151::aid-mma851>3.0.co;2-0

✦ Synopsis

We study on the initial-boundary value problem for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation:

When the initial energy

associated with the equations is non-negative and small, a unique (weak) solution exists globally in time and has some decay properties. When the initial energy E(u , u

) is negative, the solution blows up at some finite time. In the proof we use the 'modified potential well' and 'Concavity' methods. '0. Concerning the solvability of (1.1), the analytic case is rather well known in general dimension, see for example the works of Bernstein [3], Pohoz\ aev [35], Nishihara [27], Arosio and Spagnolo [2], D'Ancona and Spagnolo [6,7], D'Ancona and Shibata [5], etc. On the other hand, in the case of Sobolev space we know only the local solutions in time solvability (see [1,4,8,9,19,20,36,40,41] and the references cited therein). So for,