Bifurcating positive stable steady-states for a system of damped wave equations. (English) Zbl 1041.35046

This paper concerns the homogeneous Dirichlet problem for a class of systems of damped wave equations. There is a one-dimensional bifurcation parameter, and it is supported that for all bifurcation parameters there exists a trivial stationary solution. The local Crandall-Rabinowitz theorem is used to show that from the trivial solution branch bifurcates a branch of positive steady-states, and the spectrum of the linearization in these steady-states is discussed. Then it is shown that the properties of this spectrum imply nonlinear stability of the positive steady-states. This is a nontrivial result, because the linearization does not generate an analytic semigroup, but only a strongly continuous one. Finally, results about the global behavior of the corresponding connected component of nontrivial steady states are proved.


35L70 Second-order nonlinear hyperbolic equations
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
35L55 Higher-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
47D06 One-parameter semigroups and linear evolution equations