Global existence and blowup of solutions for a class of nonlinear higher-order wave equations

Abstract

In this paper, we consider a class of nonlinear higher-order wave equation with nonlinear damping. We show that the solution is global in time under some conditions without the relation between p and q and we also show that the local solution blows up in finite time if q > p with some assumptions on initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blow-up solution are given. This extend the recent results of Ye (J Ineq Appl, 2010).

Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term

“…For general m ≥ 2 and g(s) = a|s| r−2 s (r ≥ 2), problem (1) was studied in [16,18]. Ye [16] showed the solution exists global if the initial energy is sufficiently small.…”

“…Ye [16] showed the solution exists global if the initial energy is sufficiently small. Zhou et al [18] proved the global existence result without the relation between p and r and showed that the energy functional decays algebraically by the method introduced by Nakao [10]. Moreover, the blow-up properties of the local solution with non-positive initial energy as well as small positive initial energy were also established.…”

Global existence and exponential decay of solutions for higher-order parabolic equation with logarithmic nonlinearity

“…Also, He et al [5] proved the decay and the finite time blow-up for weak solutions of the equation, by using the potential well technique and concave technique. Recently many other authors investigated higher-order hyperbolic and parabolic type equation [2,3,6,[11][12][13][14][15]. Ishige et al [6] studied the Cauchy problem for nonlinear higher-order heat equation as follows…”

Existence, Decay, and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Delay Term

“…on suitable Banach space, and they proved some global 2 Journal of Function Spaces nonexistence of solutions. Some other authors studied related problems (see [41][42][43][44][45]). Motivated by the above works, we deal with the existence, decay, and blow-up results for the higher-order Kirchhoff type equation ( 1) with delay term and source term.…”