Solvability of complex Ginzburg-Landau equations with non-dissipative terms (Theory of evolution equations and applications to nonlinear problems)

Abstract

In this paper, we consider the following complex Ginzburg-Landau equation. (CGL) left{begin{aray}{l} u{t}-($lambda$+i $alpha$) $Delta$ u-( $kapa$+i $beta$)|u|^{mathrm{q}-2}u- $gamma$ u=f & (t, x)in[0, T] times $Omega$, u(t, x)=0 & (t, x)in[0, T]timespartial $Omega$, mathrm{u}(0, x)=u{0}(x) & xin $Omega$, end{aray}right. where $Omega$ subset mathbb{R}^{N} is a smooth bounded domain. Parameters $lambda$, $kappa$ are positive, while $alpha$, $beta$, $gamma$ in mathbb{R} are real parameters and i=sqrt{-1} is the imaginary unit. We assume that q is Sobolev sub-critical, i.e., 2 < q < +infty when N= 1, 2 and 2 < q < displaystyle frac{2N}{N-2} when N geq 3. We study the local well-posedness of (CGL) and the global continuation of local solutions for small data.