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Keywords:
Petrovsky equation; Hartree type nonlinearity; damping; potential well; global existence; finite time blow up
Petrovsky equation; Hartree type nonlinearity; damping; potential well; global existence; finite time blow up
Summary:
We study a damped Petrovsky equation with a Hartree type nonlocal source in a bounded domain. We establish a suitable energy framework, prove the local existence of weak solutions, and introduce the associated Nehari manifold and potential well structure. For weak solutions whose Nehari functional is continuous in time, we prove global continuation and uniform bounds for stable initial data below the potential well depth. The time derivative is square integrable on the positive half-line. For solutions satisfying the energy inequality on every time interval, we also obtain monotonicity and a finite limiting energy. We also prove a finite time blow up result under negative initial energy and suitable assumptions on the initial data and the parameters.
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