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Spectral condition for instability. (English) Zbl 0960.47033
Bona, Jerry (ed.) et al., Nonlinear PDE’s, dynamics and continuum physics. 1998 AMS-IMS-SIAM joint summer research conference, Mount Holyoke College, MA, USA, July 10-23, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 255, 189-198 (2000).
Consider the evolution equation $du/dt= Lu+ F(u)$, $L$ linear, $F$ nonlinear, $F(0)= 0$, in some Banach space $X$. If $L$ generates a strongly continuous semigroup, the spectrum of $L$ meeting the right half plane, and if further $F: X\to X$ is strongly continuous such that there exist positive constants $\eta$, $\rho$, and $c$ such that $\|F(u)\|\le c\|u\|^{1+\eta}$ whenever $\|u\|< \rho$, then the zero solution is (nonlinearly) unstable. This result is proved in general and applied to a regularized Boussinesq equation. For the entire collection see [Zbl 0941.00019].

MSC:
47H20Semigroups of nonlinear operators
35B40Asymptotic behavior of solutions of PDE
35Q53KdV-like (Korteweg-de Vries) equations