Hyperbolic-hyperbolic singular limits. (English) Zbl 0629.35079

Roughly speaking, the limit \(\epsilon \to 0^+\) of the differential equation \(\epsilon M(u)+L(u)=0\) is called a singular limit if, M has a higher derivative in some variable than does L. Such a singular limit is said to be uniformly well-posed if the solution is bounded independently of \(\epsilon\) and converges as \(\epsilon\) tends to 0 to a solution of the reduced equation \(L(u)=0\). Singular limits of partial differential equations are categorized by the types of the full and reduced equations. For example, \(\epsilon (u_{tt}-c^ 2u_{xx})+Bu_ t+Du_ x-h=0\) is a hyperbolic-hyperbolic singular limit. In this paper the author shows that the conditions for the well-posedness of the hyperbolic-hyperbolic singular limit \(\epsilon A(u_{tt}-c^ 2\Delta u)+Bu_ t+D^ ju_{xj}-h=0,\) \(u(x,0)=f(x)\), \(\epsilon u_ t(0,x)=\epsilon g(x)\), where c is a scalar, u a vector, and A, B, and \(D^ j\) are matrices with A and B positive definite and B and \(D^ j\) symmetric, are \(B>0\) and \(| D| B^{-1}<| c|\). Further, the author explores the generalization of these conditions for systems of the form \[ \epsilon (Au_{tt}-Cu_{xx})+Bu_ t+Du_ x=0 \] where A, B, C, and D are constant symmetric matrices, and A, B, and C are positive definite.
Reviewer: E.C.Young


35L45 Initial value problems for first-order hyperbolic systems
35L15 Initial value problems for second-order hyperbolic equations
35B25 Singular perturbations in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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