Stability results for multi-soliton solutions of the Korteweg-de Vries equation are stated and proved. The theory developed here contributes to earlier discussions of this issue by Maddocks and Sachs, Martel, Merle and Tsai and Schuur.

An energy inequality is derived for a strictly hyperbolic operator under some assumptions related to the low regularity of the coefficients and its oscillating character.

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P.~Fife and J.B.~McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.

We consider the system \[ -\Delta u = \lambda f(u,v); \, x \in \Omega \] \[ -\Delta v = \lambda g(u,v); \, x \in \Omega \] \[ u = 0 = v; \, x \in \partial\Omega, \] where $\Omega$ is a ball in $ R^{N}, N \geq 1$ and $\partial\Omega$ is its boundary, $\lambda $ is a positive parameter, and $f$ and $g$ are smooth functions that are negative at the origin (semipositone system) and satisfy certain linear growth conditions at infinity. We establish nonexistence of positive solutions when $\lambda$ is large. Our proofs depend on energy analysis and comparison methods.

We consider the removability of a level set for solutions to fully nonlinear elliptic and parabolic equations. We prove that if a $C^1$ function $u$ is a viscosity solution to the fully nonlinear equation $F(x,u,Du,D^2u)=0$ or $u_t + F(t,x,u,Du,D^2u)=0$ in a domain outside the zero-level set of $u$, then $u$ is indeed a viscosity solution to the same equation in the whole domain, under some hypotheses on $F$. We also establish the removability result for singular fully nonlinear equations.