Nonlinear estimates for traveling wave solutions of reaction diffusion equations
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Nonlinear estimates for traveling wave solutions of reaction diffusion equations Li‑Chang Hung1 · Xian Liao2 Received: 13 October 2019 / Revised: 19 March 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract In this paper we will establish nonlinear a priori lower and upper bounds for the solutions to a large class of equations which arise from the study of traveling wave solutions of reaction–diffusion equations, and we will apply our nonlinear bounds to the Lotka–Volterra system of two and four competing species as examples. The idea used in a series of papers by the first author et al. for the establishment of the linear N-barrier maximum principle will also be used in the proof. Keywords Traveling wave solutions · Reaction diffusion equations · Lotka–Volterra system Mathematics Subject Classification 35B50 · 35C07 · 35K57
1 Introduction The present paper is devoted to nonlinear a priori upper and lower bounds for the solutions ui = ui (x) ∶ ℝ ↦ [0, ∞) , i = 1, … , n to the following boundary value problem of n equations { l di (ui )xx + 𝜃 (ui )x + uii fi (u1 , u2 , … , un ) = 0, x ∈ ℝ, i = 1, 2, … , n, (1) (u1 , u2 , … , un )(−∞) = 𝐞− , (u1 , u2 , … , un )(∞) = 𝐞+ . In the above, di , li > 0 , 𝜃 ∈ ℝ are parameters, fi ∈ C0 ([0, ∞)n ) are given functions and the boundary values 𝐞− , 𝐞+ take value in the following constant equilibria set
* Li‑Chang Hung [email protected] Xian Liao [email protected] 1
Department of Mathematics, National Taiwan University, Taipei, Taiwan
2
Institute of Analysis, Karlsruhe Institute of Technology, Karlsruhe, Germany
13
Vol.:(0123456789)
L-C. Hung, X. Liao
{
| l (u1 , … , un ) | uii fi (u1 , … , un ) = 0, |
ui ≥ 0,
} ∀i = 1, … , n .
(2)
Equations (1) arise from the study of traveling waves solutions of reaction–diffusion equations (see [16, 18]). A series of papers [2–7] by Hung et al. have been contributed to the linear (N-barrier) maximum principle for the n Eq. (1), and in particular the lower and upper bounds for any linear combination of the solutions n ∑
𝛼i ui (x),
∀(𝛼1 , … , 𝛼n ) ∈ (ℝ+ )n
i=1
have been established in terms of the parameters di , li , 𝜃 in (1). Here we aim to derive nonlinear estimates for the polynomials of the solutions: n ∏
(ui (x) + ki )𝛼i ,
∀(𝛼1 , … , 𝛼n ) ∈ (ℝ+ )n
i=1
for some ki ≥ 0 , which is related to the diversity indices of the species in ecology: ∑n Dq = ( i=1 (ui )q )1∕(1−q) , q ∈ (1, ∞). Observe that when either 𝐞+ = (0, … , 0) or 𝐞− = (0, … , 0) , the trivial lower ∏n 𝛼 ∏ bound of ni=1 (ui (x) + ki )𝛼i is i=1 ki i . For ki > 0 , the following lower bound for the upper solutions of (1) holds.
Theorem 1 (Lower bound) Suppose that (ui (x))ni=1 ∈ (C2 (ℝ))n with ui (x) ≥ 0 , ∀i = 1, … , n is an upper solution of (1): { l di (ui )xx + 𝜃(ui )x + uii fi (u1 , u2 , … , un ) ≤ 0, x ∈ ℝ, i = 1, 2, … , n, (3) (u1 , u2 , … , un )(−∞) = 𝐞− , (u1 , u2 , … , un )(∞) = 𝐞+ , and that there exist (ui )ni=1 ∈ (ℝ+ )n such that
fi (u1 , … , un ) ≥ 0, for all � � ∑n u (u1 , … , un ) ∈
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