| Abstract: |
| In this talk we consider the following complex Ginzburg-Landau equation
\begin{equation}
\tag*{(CGL)\(_p\)}
\frac{\partial u}{\partial t}(t,x) - (\lambda + i\alpha)\Delta_p u + (\kappa + i\beta)|u|^{q-2}u - \gamma u = f,\quad (t,x) \in [0,T]\times\Omega,
\end{equation}
subject to the homogeneous Dirichlet boundary condition and suitable initial values.
Here \(-\Delta_pu = -\mathop{{\rm div}}(|\nabla u|^{p-2}\nabla u)\) denotes the p-Laplace operator.
The parameters in our equation are \(\lambda, \kappa > 0\) and \(\alpha,\beta,\gamma \in \mathbb{R}\); \(i = \sqrt{-1}\) is the imaginary unit; \(\Omega \subset \mathbb{R}^N\) is a possibly unbounded domain and \(f: [0,T]\times\Omega \to \mathbb{C}\) is an external force with \(T>0\).
This equation is a generalization of the well-known case \(p=2\).
We establish the global existence of solutions for (CGL)\(_p\) under \(\max\{1,2N/(N+2)\} < p\), \(2 \leq q\) and the suitable conditions on parameters \(\lambda, \kappa, \alpha, \beta\) subject to \(q\). |
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