ON GLOBAL EXISTENCE, ASYMPTOTIC STABILITY AND BLOWING UP OF SOLUTIONS FOR SOME DEGENERATE NON-LINEAR WAVE EQUATIONS OF KIRCHHOFF TYPE WITH A STRONG DISSIPATION

摘要

We study on the initial-boundary value problem for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation: u(t)t = del u(2y)Delta u - Delta u(t) = u(alpha)u, u(0) = u(0), u(t)(0) = u(1), u(theta Omega = 0.) When the initial energy E(u(0), u(1)) = u(1)(2) + 1/y+1del u(0)(2(y+1)) - 2/alpha+2u(0)(alpha+2)(alpha+2) 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(0), u(1)) is negative, the solution blows up at some finite time. In the proof we use the 'modified potential well' and 'Concavity' methods. [References: 41]