Existence of additional unstable equilibria for some monostable systems

Abstract

We prove the existence of additional unstable equilibria E ± for a class of “monostable” reaction–diffusion systems u t = A u x x + f ( u ) , u ( x , t ) ∈ R N , having known equilibria S < T with S “stable” and T “unstable” such that the derivative d f [ T ] has an even number of unstable eigenvalues. The proofs use positive-operator theory and degree theory. Systems in the class considered are known to admit monotone increasing travelling-wave solutions w ( x − c t ) connecting ordered stable and unstable equilibria for all c greater than or equal to a positive critical value and the existence of E ± > S yields two one-parameter families of travelling waves connecting S to E ± in addition to the family connecting S to T .