TY - JOUR

T1 - A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations

AU - Liang, Xing

AU - Lin, Xiaotao

AU - Matano, Hiroshi

PY - 2010/11/1

Y1 - 2010/11/1

N2 - Abstract. We consider the equation ut = uxx + b(x)u(1 - u), x ε R, where b(x) is a nonnegative measure on R that is periodic in x. In the case where b(x) is a smooth periodic function, it is known that there exists a travelling wave (more precisely a "pulsating travelling wave") with average speed c if and only if c ≥ c*(b), where c*(b) is a certain positive number depending on b. This constant c*(b) is called the "minimal speed". In this paper, we first extend this theory by showing the existence of the minimal speed c*(b) for any nonnegative measure b with period L. Next we study the question of maximizing c*(b) under the constraint ∫[0,L) b(x)dx = αL, where a is an arbitrarily given positive constant. This question is closely related to the problem studied by mathematical ecologists in late 1980s but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac's delta functions αL ΣkεZ δ(x + kL).

AB - Abstract. We consider the equation ut = uxx + b(x)u(1 - u), x ε R, where b(x) is a nonnegative measure on R that is periodic in x. In the case where b(x) is a smooth periodic function, it is known that there exists a travelling wave (more precisely a "pulsating travelling wave") with average speed c if and only if c ≥ c*(b), where c*(b) is a certain positive number depending on b. This constant c*(b) is called the "minimal speed". In this paper, we first extend this theory by showing the existence of the minimal speed c*(b) for any nonnegative measure b with period L. Next we study the question of maximizing c*(b) under the constraint ∫[0,L) b(x)dx = αL, where a is an arbitrarily given positive constant. This question is closely related to the problem studied by mathematical ecologists in late 1980s but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac's delta functions αL ΣkεZ δ(x + kL).

UR - http://www.scopus.com/inward/record.url?scp=77954620658&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2010-04931-1

DO - 10.1090/S0002-9947-2010-04931-1

M3 - Article

AN - SCOPUS:77954620658

VL - 362

SP - 5605

EP - 5633

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -