Dynamics of nonlinear hyperbolic equations of Kirchhoff type

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation u(tt) - (a integral(Omega) vertical bar del u vertical bar(2)dx + b) Delta u = lambda u + vertical bar u vertical bar(p-1)u, where a, b > 0, p > 1, lambda is an element of R and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of a, b, lambda and the initial data for the following cases: (i) 1 < p < 3, (ii) p = 3 and a > 1/Lambda, (iii) p = 3, a <= 1/Lambda and lambda < b lambda(1), (iv) p = 3, a < 1/Lambda and lambda > b lambda(1), (v) p > 3 and lambda <= b lambda(1), (vi) p > 3 and lambda > b lambda(1), where lambda(1) = inf {parallel to del u parallel to(2)(2): u is an element of H-0(1)(Omega) and parallel to u parallel to(2) = 1}, and Lambda = inf {parallel to del u parallel to(4)(2): u is an element of H-0(1)(Omega) and parallel to u parallel to(4) = 1}. Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable a, b and lambda, and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for a > 0 and a = 0.