×

Oscillation properties for systems of hyperbolic differential equations of neutral type. (English) Zbl 0964.35166

The author obtains several groups of sufficient conditions for the oscillation of the system of partial functional differential equations \[ \begin{aligned} \frac{\partial^2}{\partial t} \Biggl[p(t)u_i(x,t)+\sum_{r=1}^d \lambda_r (t)u_i(x,t-\tau_r)\Biggr]&= a_i(t)\Delta u_i(x,t)\\ +\sum_{j=1}^m \sum_{k=1}^sa_{ijk}\Delta u_j(x,\rho_k(t))-q_i(x,t)u_i(x,t) &- \sum_{j=1}^m\sum_{h=1}^l q_{ijh}(x,t)u_j(x,\sigma_h(x,t)),\\ (x,t)\in\Omega\times[0,+\infty)&\equiv G,\;i=1,2,\cdots,m \end{aligned}\tag{1} \] with the boundary condition \[ \frac{\partial u_i(x,t)}{\partial n}+g_i(x,t)u_i(x,t)=0,\quad (x,t)\in \partial \Omega\times [0,+\infty), \tag{2} \] or \[ u_i(x,t)=0,\quad (x,t)\in \partial \Omega\times [0,+\infty), \tag{3} \] where \(\Delta\) is the Laplacian operator; \(n\) is the unit exterior normal vector to \(\partial \Omega\); \(\partial \Omega\) is piecewise smooth. The technique of the proof and the sufficient conditions for oscillation are similar to that in [the author and B. T. Cui, Bull. Inst. Math., Acad. Sin. 28, No. 3, 189-200 (2000; Zbl 0964.35164), cf. the preceding review].

MSC:

35R10 Partial functional-differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 0964.35164
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Mishev, D. P.; Bainov, D. D., Oscillation of the solutions of parabolic differential equations of neutral type, Appl. Math. Comput., 28, 97-111 (1988) · Zbl 0673.35037
[2] Fu, X. L.; Zhuang, W., Oscillation of neutral delay parabolic equations, J. Math. Anal. Appl., 191, 473-489 (1995) · Zbl 0829.34057
[3] Lalli, B. S.; Yu, Y. H.; Cui, B. T., Oscillation of hyperbolic equations with functional arguments, Appl. Math. Comput., 53, 97-110 (1993) · Zbl 0798.35151
[4] Li, W. N.; Cui, B. T., A necessary and sufficient condition for oscillation of parabolic equations of neutral type, Math. Appl., 12, 50-53 (1999)
[5] Cui, B. T., Oscillation properties of the solutions of hyperbolic equations with deviating arguments, Demonstratio Math., 29, 61-68 (1996) · Zbl 0860.35137
[6] Bainov, D.; Cui, B. T.; Minchev, E., Forced oscillation of solutions of certain hyperbolic equations of neutral type, J. Comput. Appl. Math., 72, 309-318 (1996) · Zbl 0866.35130
[7] Li, Y. K., Oscillation of systems of hyperbolic differential equations with deviating arguments, Acta Math. Sinica, 40, 100-105 (1997) · Zbl 0881.34079
[8] Li, W. N.; Cui, B. T., Oscillations of systems of neutral delay parabolic equations, Demonstratio Math., 31, 813-824 (1998) · Zbl 0929.35166
[9] Li, W. N.; Cui, B. T., Oscillation for systems of parabolic equations of neutral type, Southeast Asian Bull. Math., 23, 447-456 (1999) · Zbl 0940.35022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.