Growth of Fourier–Lebesgue norms for mKdV

We demonstrate inflation of Fourier–Lebesgue norms for solutions to the focusing modified Korteweg–de Vries equation posed on the real line. For p≠ 2 p ≠ 2 and all s∈ R s ∈ R , we construct a sequence of solutions uₙ u n whose initial data uₙ(0) u n ( 0 ) converges to zero in the Fourier–Lebesgue spaces FLᵖₛ(R) F L s p ( R ) , but whose evolutions at later times tₙ t n diverge to infinity.