We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.
The Dirichlet initial-boundary value problem of a compressible reactive gas model equation with a nonlocal nonlinear source term is investigated. Under certain conditions, it can be proven that the blow-up rate is uniform in all compact subsets of the domain, and the blow-up rate is irrelative to the exponent of the diffusion term, however, relative to the exponent of the nonlocal nonlinear source.
We study the Cauchy problem of damped generalized Boussinesq equation u_(tt)-u_(xx)+(u_(xx)+f(u))_(xx)-αu_(xxt)= 0.First we give the local existence of weak solution and smooth solution.Then by using potential well method and convexity method we prove the global existence and finite time blow up of solution,then we obtain some sharp conditions for the well-posedness problem.
