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(b▽yu)+(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 solutionũ on the center manifold such that ‖u(x)-ũ(x)‖ Σ ⩽Ce -α(1-|x|) for allx, whereC andα 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aziz, A. K., Bell, J. B., and Schneider, M. (1982). Existence and uniqueness for an equation of mixed type in a rectangle.Math. Meth. Appl. Sci. 4, 307–316.
Bates, P. W., and Jones, C. K. R. T. (1989). Invariant manifolds for semilinear partial differential equations.Dynam. Report.2, 1–37.
Fischer, G. (1984). Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen.Math. Nachrichten 115, 137–157.
Grisvard, P. (1985).Elliptic Poblems in Nonsmooth Domains, Pitman Advanced, Boston-London-Melbourne.
Keyfitz, B. (1991). Admissability conditions for shocks in conservation laws that change type.SIAM Math. Anal.22, 1284–1292.
Kirchgässner, K. (1982). Wave solutions of reversible systems and applications.J. Diff. Eq.45, 113–127.
Manwell, A. R. (1977).The Tricomi Equation with Applications to the Theory of Plane Transonic Flow, Research Notes in Mathematics 35, Pitman, San Francisco-London-Melbourne.
Mielke, A. (1986). A reduction principle for nonautonomuous systems in finite-dimensional spaces.J. Diff. Eq. 65, 68–88.
Mielke, A. (1987). Über maximale LpRegularität für Differentialgleichungen in Banach- und Hilbert-Räumen.Math. Annal.277, 121–133.
Mielke, A. (1990). Normal hyperbolicity of center manifolds and Saint-Venant's principle.Arch. Rat. Mech. Anal. 119, 353–372.
Mielke, A. (1991).Hamiltonian and Lagrangian Flows on Center Manifolds and Applications to Elliptic Variational Problems, Lecture Notes in Mathematics Vol. 1489, Springer Verlag, Berlin.
Scarpellini, B. (1990). Center manifolds of infinite dimensions I & II.J. Appl. Math. Phys. (ZAMP) 42, 1–32, 280–314.
Schmilz, H. (1989). On Mellin transforms of generalized functions, Manuscript, Universität Stuttgart, Stuttgart.
Schneider, M. (1984). Eindeutigkeitsaussagen für das Tricomi-Problem im ℝ2 für eine Klasse nichlinearer Gleichungen gemischten Typs.Monatshefte Math.97, 63–73.
Titchmarsh, E. C. (1946).Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford University Press, Oxford.
Triebel, H. (1972).Höhere Analysis, Deutscher Verlag der Wissenschaften, Berlin.
Triebel, H. (1978).Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam.
Vanderbauwhede, A., and van Gils, S. A. (1987). Center manifolds and contractions on a scale of Banach space.J. Funct. Anal.72, 209–224.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mielke, A. On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds. J Dyn Diff Equat 4, 419–443 (1992). https://doi.org/10.1007/BF01053805
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01053805
Key words
- Mixed type
- nonlinear elliptic-hyperbolic problem
- bounded solutions
- center manifolds
- weak normal hyperbolicity