Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems

Please always quote using this URN: urn:nbn:de:0296-matheon-11981
  • We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j  + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that $b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$. We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on $\la$ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction. The proof is done by means of a Liapunov-Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used. There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs: First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters is much more delicate. And second, a sufficient amount of dissipativity is needed in the system, and a priori it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions.

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Metadaten
Author:Irina Kmit, Lutz Recke
URN:urn:nbn:de:0296-matheon-11981
Referee:Alexander Mielke
Language:English
Date of first Publication:2013/02/08
Institute:Humboldt-Universität zu Berlin
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B10 Periodic solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B32 Bifurcation [See also 37Gxx, 37K50]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Lxx Hyperbolic equations and systems [See also 58J45] / 35L50 Initial-boundary value problems for first-order hyperbolic systems
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Lxx Hyperbolic equations and systems [See also 58J45] / 35L60 Nonlinear first-order hyperbolic equations
Preprint Number:1000