On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds |
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Authors: | Alexander Mielke |
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Affiliation: | (1) Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, W-7000 Stuttgart 80, Germany |
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Abstract: | We consider the equation a(y)uxx+divy(b(y)yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L2() of the operatorAu= (1/a) divy(byu)+(c/a) in the case ofb changing sign. This leads to the abstract problem uxx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, ux) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x)Ce-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear. |
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Keywords: | Mixed type nonlinear elliptic-hyperbolic problem bounded solutions center manifolds weak normal hyperbolicity |
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