The complex Ginzburg-Landau equation with data in Sobolev spaces of negative indices (Q2705860)

The paper deals with the initial value problem for the complex Ginzburg-Landau equations of the form NEWLINE\[NEWLINE \left\{\begin{align*}{ &\partial_t u=Au+(a+i\nu)\Delta u-(b+i\mu)|u|^{2\sigma}u,\qquad (x,t)\in {\Bbb R}^n\times (0,\infty),\cr &u(x,0)=u_0(x)\in H^r_p\cr}\end{align*}\right. NEWLINE\]NEWLINE with real constants \(\sigma>0,\) \(A\geq 0,\) \(a>0,\) \(\nu,\) \(\mu.\) Local well-posedness is established for \(r\) and \(p\) satisfying NEWLINE\[NEWLINE 1<p<\infty,\qquad {{1}\over {\sigma(2\sigma+1)}}\leq {n\over p}<{1\over\sigma} NEWLINE\]NEWLINE NEWLINE\[NEWLINE r_0<r<-2\sigma r_0,\qquad r_0={n\over p}-{1\over \sigma} .NEWLINE\]