Oscillatory properties of equations of mathematical physics with time-dependent coefficients (Q2714165)

The authors consider the equation of the type NEWLINE\[NEWLINEu_{tt}+2\alpha _0(t)u_t -2\alpha _1(t) \Delta u_t +2\alpha _2(t) \Delta ^2u_t +\beta _0(t)u -\beta _1(t) \Delta u + \beta _2(t) \Delta ^2u = 0NEWLINE\]NEWLINE with the boundary condition \(u=0\) (and \(\Delta u = 0\), respectively). Using the notation \(\gamma (t) = \alpha _0(t) + \lambda_1 \alpha _1(t) + \lambda_1^2\alpha _2(t)\) (where \(\lambda _1\) is the first eigenvalue of the Laplacian \(\Delta\)) they prove that under the condition NEWLINE\[NEWLINE\text{infess} (\beta _0(t) + \lambda_1 \beta _1(t) + \lambda_1^2\beta _2(t)-\gamma _t(t)-\gamma^2(t)) \geq \omega ^2 > 0NEWLINE\]NEWLINE any solution of the above problem is globally oscillatory and the oscillatory time is equal to \(\pi / \omega\).