Finite time blow up of the solutions to boussinesq equation with linear restoring force and arbitrary positive energy

Nonexistence of global solutions of abstract wave equations with high energies

“…By means of the orthogonal decomposition of the velocity, introduced in [ 10 ], where , we define the functional That is, This decomposition allows us to get sufficient conditions for nonexistence of global solutions for any positive value of the initial energy. Define where is the embedding constant of in .…”

“…Several recent works have studied some equations of the type ( 1 ) and showed blow-up of solutions with initial energies such that for two particular values and . See [ 5 , 7 , 10 – 12 ]. Here, we shall prove that for any initial energy such that , there exist initial data for which the corresponding solution is not global.…”

“…For high energies, previous results conclude qualitative properties based in part on the sign of , see [ 16 – 18 ]. By means of the invariance of some sets, along with the solution, it is proved in [ 5 , 10 , 12 ] that holds under sufficient conditions on that imply blow-up. Here, we do not have invariance properties, and we need to analyze when Theorem 3.1 implies .…”

“…This result improves the ones known in the literature. Indeed, in [ 10 ] blow-up is proved for initial energies . We proved that blow-up can occur for initial energies .…”

Blow-Up of Solutions to Semilinear Nonautonomous Wave Equations with Robin Boundary Conditions

“…A number of papers were addressed to the question of blow up of solutions with arbitrary positive initial energy of initial boundary value problems for various nonlinear wave equations (see, e.g. [3] , [4], [14] and references therein).…”