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Solvability of a mixed problem with the third boundary condition for nonstationary second order equations with operator coefficients and the asymptotic behaviour of solutions with respect to the parameters. (Russian) Zbl 0825.35079
Solvability and the asymptotic behaviour with respect to the parameters of solutions of a mixed problem with homogeneous third boundary condition is studied for the following equation: \[ A_ 1(x,\partial /\partial x)u_{tt}+\eta _ 1A_ 2(x,\partial /\partial x)u_ t+\eta _ 2A_ 3(x,\partial /\partial x)u=f(t,x), (t,x)\in Q_ T, \] where \(\eta _ i=\text{const}>0\), \(i=1,2\) are parameters, and the operator coefficients \(A_ j(x,\partial /\partial x)\), \(j=1,2,3\) are \[ A_ j(x,\partial /\partial x)u\equiv -\sum _{k,i=1}^ n(a_{ki}(x)u_{x_ k})_{x_ i}+a(x)u \] or \(A_ j(x,\partial /\partial x)u\equiv p(x)u\).
MSC:
35M10 PDEs of mixed type
35B40 Asymptotic behavior of solutions to PDEs
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