TY - JOUR

T1 - Coincidence sets associated with second-order ordinary differential equations of logistic type

AU - Takeuchi, Shingo

PY - 2009/5/1

Y1 - 2009/5/1

N2 - This paper concerns the formation of a coincidence set for the positive solution of the boundary-value problem -ε(φp(u x))x = φq(u)f(c(x) - u) in I = (-1,1) with u(±1) = 0, where ε is a positive parameter, φp(s) = |s|p-2 s, 1 < q < p < ∞, f(s) ∼ φθ+1(s) (s → 0), 0 < θ < ∞ and c(x) is a positive smooth function satisfying (φp(cx))x = 0 in I. The positive solution uε(x) converges to c(x) uniformly on any compact subset of I as ε → 0. It is known that if c(x) is constant and θ < p - 1, then the solution coincides with c(x) somewhere in I for sufficiently small ε and the coincidence set Iε = {x ∈ I: uε(x) = c(x)} converges to I as |I \ Iε| ∼ ε1/p (ε → 0). It is proved in this paper that even if c(x) is variable and θ < 1, then Iε has a positive measure and converges to I with order O(εκ) as ε → 0, where κ = min{1/p, 1/2}. Moreover, it is also shown that, if θ ≥ 1, then Iε is empty for every ε. The proofs rely on comparison principles and an energy method for obtaining local comparison functions.

AB - This paper concerns the formation of a coincidence set for the positive solution of the boundary-value problem -ε(φp(u x))x = φq(u)f(c(x) - u) in I = (-1,1) with u(±1) = 0, where ε is a positive parameter, φp(s) = |s|p-2 s, 1 < q < p < ∞, f(s) ∼ φθ+1(s) (s → 0), 0 < θ < ∞ and c(x) is a positive smooth function satisfying (φp(cx))x = 0 in I. The positive solution uε(x) converges to c(x) uniformly on any compact subset of I as ε → 0. It is known that if c(x) is constant and θ < p - 1, then the solution coincides with c(x) somewhere in I for sufficiently small ε and the coincidence set Iε = {x ∈ I: uε(x) = c(x)} converges to I as |I \ Iε| ∼ ε1/p (ε → 0). It is proved in this paper that even if c(x) is variable and θ < 1, then Iε has a positive measure and converges to I with order O(εκ) as ε → 0, where κ = min{1/p, 1/2}. Moreover, it is also shown that, if θ ≥ 1, then Iε is empty for every ε. The proofs rely on comparison principles and an energy method for obtaining local comparison functions.

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M3 - Review article

AN - SCOPUS:79960926581

SN - 0893-4983

VL - 22

SP - 587

EP - 600

JO - Differential and Integral Equations

JF - Differential and Integral Equations

IS - 5-6

ER -