Global existence and blow-up results for an hyperbolic problem on IR^N

Abstract

We discuss the asymptotic behaviour of solutions for the non-local hyperbolic problemwith initial conditions and , in the case where and is a positive function lying in . When the initial energy which corresponds to the problem, is non-negative and small, there exists a unique global solution in time. When the initial energy is negative, the solution blows-up in finite time. A combination of the modified potential well method and the concavity method is widely used.

Keywords:

• quasilinear hyperbolic equations
• global solution
• blow-up
• potential well
• concavity method
• unbounded domains
• generalised Sobolev spaces

AMS Subject Classifications:

• 35A07
• 35B30
• 35B40
• 35L15
• 35L70
• 35L80
• 47F05
• 47H20