Some results on the stability and bifurcation of stationary solutions of delay-diffusion equations
Nonlinear delay-diffusion equations of the form \[ u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau)+ g(x,u(x,t)) \] are studied and stability and bifurcation results are obtained by studying the perturbation of the spectrum of the linearization \[ u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau) \] caused by \(g\). It is shown that a centre manifold of any dimension exists for some \(a\), \(b\), and \(\tau\). Moreover, it is shown that if \(\pm i\omega_0,\pm i\omega_1,\dots,\pm i\omega_{m-1}\) are the \(2m\) pure imaginary eigenvalues of the linearized equation, then the numbers \(\omega_0,\omega_1,\dots,\omega_{m-1}\) are rationally independent and as \(\tau\) increases through the critical value, the nonlinear equation undergoes a bifurcation from \(2\ell\) eigenvalues, where \(\ell\) is the sum of the multiplicities of the \(m\) eigenvalues.
scientific article; zbMATH DE number 1005572
scientific article