| Abstract: |
| In this talk, we consider the complex Ginzburg-Landau equation (CGL) with non-dissipative nonlinearities of the form:
\[
\frac{\partial u}{\partial t}(t,x)-(\lambda+i\alpha)\Delta u-(\kappa+i\beta)|u|^{q-2}u-\gamma u=f(t,x)
\]
on bounded domains \(\Omega\) with smooth boundaries.
Here \(i=\sqrt{-1}\) denotes the imaginary unit; parameters \(\lambda\) and \(\kappa\) are positive; \(\alpha\), \(\beta\) and \(\gamma\) are real.
Under the Sobolev subcritical condition on \(q\) and a suitable assumption on \(\gamma\) depending on \(\lambda\) and the first eigenvalue of the Laplace operator with the homogeneous Dirichlet boundary condition on \(\Omega\), we show that there exists a periodic solution with period \(T>0\) of (CGL) for given sufficiently small external forces \(f\) defined on \([0,T]\).
Our method is based on the theory developed in \^Otani (1984), which shows a large applicability to ensure the existence of periodic solutions for nonlinear evolution equations of parabolic type.
The theory employs Schauder`s fixed point theorem based on the strong relative boundedness of perturbations compared with the principal term, while the term \(-i\alpha\Delta u\) is not dominated by the parabolic principal term \(-\lambda\Delta u\). So that we use the monotonicity of \(-i\alpha\Delta u\) in order to deal with this difficulty. |
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