Global solutions and finite time blow up for damped semilinear wave equations

Abstract

A class of damped wave equations with superlinear source term is considered. It is shown that every global solution is uniformly bounded in the natural phase space. Global existence of solutions with initial data in the potential well is obtained. Finally, not only finite time blow up for solutions starting in the unstable set is proved, but also high energy initial data for which the solution blows up are constructed.

Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level

“…We also obtained the finite time blow-up result for certain solutions whose initial data have arbitrary high initial energy, see [20][21][22][23]. For details of the study of other kinds of evolution equations, we refer the reader to [24][25][26][27][28][29][30][31][32][33][34][35].…”

Global existence and blow-up of solutions for a nonlinear wave equation with memory

“…The proof follows from a directly application of the Galerkin method as in [22,30], thus we omit it here.…”

“…Next we will prove the main blow-up result by the concavity method of Levine [35,36] and the estimates similar as [30]. …”

The initial-boundary value problems for a class of sixth order nonlinear wave equation

“…The equation (4) was first introduced by Boussinesq [2]. Later, extensive research has been carried out to study the Boussinesq equation by different views.…”

“…Later, extensive research has been carried out to study the Boussinesq equation by different views. Cho and Ozawa [3] established the global existence and the scattering of a small amplitude solution to the Cauchy problem of the equation (4). Considering the effect of damping, Varlamov [15,16] considered the following damped Boussinesq equation…”