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Comparison theorems for solutions of hyperbolic equations. (English. Russian original) Zbl 0678.35063
Math. USSR, Sb. 62, No. 2, 349-371 (1989); translation from Mat. Sb., Nov. Ser. 134(176), No. 3(11), 353-374 (1987).
At first, the authors complain why it is useful to introduce a quasi- asymptotics for studying the behaviour of solutions of hyperbolic initial-boundary value problem as \(t\to \infty\). A function h(t,x) has uniform quasi-asymptotics \(\omega\) (x) if, roughly speaking its integral mean in t of m-th order of a certain type has the limit \(\omega\) (x) as \(t\to \infty\) uniformly in \(x\in \Omega\). Then the authors consider the following boundary value problem \[ (*)\quad p(x)u_{tt}- \sum^{n}_{i,j=1}(a_{ij}(x)u_{x_ i})_{x_ j}=f(t,x),\quad \frac{\partial u}{\partial N}|_{\partial \Omega}=0,\quad u|_{t=0}=u_ t|_{t=0}=0, \] with bounded coefficients p, \(a_{ij}\) and \(p(x)\geq p,=const>0\). They establish so-called comparison theorems connecting the existence of the quasi-asymptotics to (*) with its existence for the problem (*) with \(p=1\); the last was studied earlier [see the authors, Mat. Sb. Nov. Ser. 131(173), No.4(12), 419-437 (1986; Zbl 0635.35056)]. So they prove theorems on necessary and sufficient conditions under which there exists such a quasi-asymptotics. The case of \(n=1\) is separately considered. To establish the theorems the authors study an initial problem in a Banach space, that covers the above problem, and establish corresponding theorems for it.
Reviewer: L.Lebedev

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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