A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation

We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus, $$u_t+u{u}_x={\mathrm{\Lambda }}^{r}u-\epsilon {\mathrm{\Lambda }}^{s}u,x\in \mathbb{T},$$ where $\epsilon >0$, $s>1$, $r\in [-1,s)$. 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 ${\epsilon }_k=k^{r-s}$, $k\in \mathbb{N}$. Using the Crandall-Rabinowitz theorem we obtain smooth local curves of nontrivial equilibria emanating from each $({\epsilon }_k,0)$ 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 $\epsilon$-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.