A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients. (February 2020) - Sun, Ningkui
Abstract: We consider a reaction–diffusion–advection equation of the form: u t = u x x − β ( t ) u x + f ( t, u ) for x ∈ [ 0, h ( t ) ), where β ( t ) is a T -periodic function, f ( t, u ) is a T -periodic Fisher–KPP type of nonlinearity with a ( t ) ≔ f u ( t, 0 ) changing sign, h ( t ) is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers c ̄ and B ( β ̃ ) with B ( β ̃ ) > c ̄ > 0, β ̄ ≔ 1 T ∫ 0 T β ( t ) d t and β ̃ ( t ) ≔ β ( t ) − β ̄, such that a vanishing–spreading dichotomy result holds when | β ̄ | < c ̄ ; a vanishing–transition–virtual spreading trichotomy result holds when β ̄ ∈ [ c ̄, B ( β ̃ ) ) ; all solutions vanish when β ̄ ⩾ B ( β ̃ ) or β ̄ ⩽ − c ̄ .