Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

Abstract

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$\begin{aligned} \partial ^2_tu(t,x)- a(x,\lambda )^2\partial _x^2u(t,x)= b(x,\lambda ,u(t,x),u(t-\tau ,x),\partial _tu(t,x),\partial _xu(t,x)), \; x \in (0,1) \end{aligned}$$ ∂ t 2 u ( t , x ) - a ( x , λ ) 2 ∂ x 2 u ( t , x ) = b ( x , λ , u ( t , x ) , u ( t - τ , x ) , ∂ t u ( t , x ) , ∂ x u ( t , x ) ) , x ∈ ( 0 , 1 ) with smooth coefficient functions a and b such that $$a(x,\lambda )>0$$ a ( x , λ ) > 0 and $$b(x,\lambda ,0,0,0,0) = 0$$ b ( x , λ , 0 , 0 , 0 , 0 ) = 0 for all x and $$\lambda $$ λ . We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to t and x) and smooth dependence (on $$\tau $$ τ and $$\lambda $$ λ ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $$u=0$$ u = 0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter $$\tau $$ τ . To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics and then apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the “loss of derivatives” property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays $$\tau $$ τ .

Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs

“…Remark 1. 18 In many applications of forced frequency locking the T 0 -periodic solution to the unforced autonomous equation is born in a Hopf bifurcation from a stationary solution, and it is natural to ask how this Hopf bifurcation interacts with small T -periodic forcing for T ≈ T 0 . In ODE cases this is done, e.g.…”

Maximum principle for the weak solutions of the Cauchy problem for the fourth‐order hyperbolic equations

“…This theory got its present‐day development in the works by Grubb [12], Hormander [13], and Posilicano [20]. Later, the problem of well‐posedness of boundary‐value problems for various types of second‐order differential equations was studied by Burskii and Zhedanov [2, 3] which developed a method of traces associated with a differential operator and applied this method to establish the Poncelet, Abel, and Goursat problems, and by Kmit and Recke [14]. In the previous works of author (see [6]) there have been developed qualitative methods of studying Cauchy problems and nonstandard in the case of hyperbolic equations Dirichlet and Neumann problems for the linear fourth‐order equations (moreover, for an equation of any even order $$2\,m,\, m\ge 2$$,) with the help of operator methods (L‐traces, theory of extension, moment problem, method of duality equation‐domain and others) [4, 8].…”