A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation

We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus, \[ u_t+u u_x=Λ^{r}u-\varepsilon Λ^{s}u,\qquad x\in\mathbb T, \] where $\varepsilon>0$, $s>1$, $r\in[-1,s)$. We first prove local and global well-posedness for initial data in $H^{3}(\mathbb T)$. We then investigate the steady-state problem and show that the trivial branch undergoes bifurcation at the critical values $\varepsilon_k=k^{\,r-s}$, $k\in\mathbb N$. Using the Crandall-Rabinowitz theorem we obtain smooth local curves of nontrivial equilibria emanating from each $(\varepsilon_k,0)$ and compute the bifurcation direction. To address the global continuation of these branches we derive global a priori bounds and apply a global alternative based on the Fitzpatrick-Pejsachowicz-Rabier degree for Fredholm maps of index zero. In particular, for the component bifurcating from the first critical point we prove that its $\varepsilon$-projection contains the interval $(2^{r-s},1)$, yielding the existence of nontrivial steady states for that parameter range. We complement the theory with numerical continuation results illustrating the bifurcation diagram and solution profiles.