Oscillation Theorems for Solutions of Hyperbolic Equations

Abstract

Abstract.It is shown that solutions of hyperbolic equations in cylindrical space time domains which vanish on the lateral boundary of the cylinder must have arbitrarily large zeros in the interior of the cylinder. In case the coefficients of the equation are time independent the solutions will have arbitrarily large zeros on any interior line of the cylinder. This paper concerns the oscillations of solutions of hyperbolic equations. More specifically, we shall consider solutions of the equationin some space-time cylinder Li x (0, co), where we assume that the operator on the right is uniformly elliptic:

On the zeros of solutions to nonlinear hyperbolic equations

“…Assume that Conditions Ai-Aj hold. Let t t be a number with t l > p and u be a solution to the problem (l)- (3). // there is a number t 2 > t x such that W(t ly t 2 ) = 0, then the solution u has a zero in Q(t…”

“…Let ? be any number with F > p and u be any solution to the problem (l)- (3). It follows from the hypothesis that W(t, t x ) < 0 and W(t, t 2 ) > 0 for some numbers f x and t 2 with Kt 1 <t 2 .…”

“…Oscillation theory for hyperbolic equations has been investigated by numerous authors. We refer the reader to [2,[4][5][6][7]9,10,12] for characteristic initial value problems for hyperbolic equations, and to [1,3,5,11,13] for initial boundary value problems for wave equations. Forced oscillations of solutions to hyperbolic equations were studied by Kreith et al [5] and Mishev [7].…”

Oscillatory properties for semilinear degenerate hyperbolic equations of second order

“…There are some results on (GOP), for example, see [2,8,9,13,19]. But one knows few results on (POP) and (NOP) in comparison with (GOP).…”

“…But one knows few results on (POP) and (NOP) in comparison with (GOP). To our knowledge, (POP) are treated in Theorem 2 of [9] for linear P − with time-independent coefficients in general space dimensions n 1, a few fine results on (POP) for semilinear wave equations are investigated in [1] plus for P (−) with time-dependent coefficients in [14] and [15] when n = 1. (NOP) is also discussed in [16] for n 3 only because of the lack of the positivity of fundamental solutions to wave equations in high space dimensions n 4.…”