Global Solutions for an 𝑚-Component System of Activator-Inhibitor Type

Abstract

This paper deals with a reaction-diffusion system with fractional reactions modeling 𝑚-substances into interaction following activator-inhibitor's scheme. The existence of global solutions is obtained via a judicious Lyapunov functional that generalizes the one introduced by Masuda and Takahashi.

1. Introduction

In this paper, we are concerned with the existence of global solutions to a reaction-diffusion system with 𝑚 components generalizing the activator-inhibitor system: 𝜕_𝑡𝑢₁−𝑎₁Δ𝑢₁=𝑓₁(𝑢)=𝜎−𝑏₁𝑢₁+𝑢^𝑝11₁∏^𝑚_𝑗=2𝑢^𝑝1𝑗_𝑗,𝜕_𝑡𝑢_𝑖−𝑎_𝑖Δ𝑢_𝑖=𝑓_𝑖(𝑢)=−𝑏_𝑖𝑢_𝑖+𝑢^𝑝𝑖1₁∏^𝑚_𝑗=2𝑢^𝑝𝑖𝑗_𝑗,𝑖=2,…,𝑚,𝑥∈Ω,𝑡>0, (1) supplemented with Neumann boundary conditions 𝜕𝑢_𝑖{}𝜕𝜂=0,on𝜕Ω×𝑡>0,𝑖=1,…,𝑚, (2) and the positive initial data 𝑢_𝑖(𝑥,0)=𝜑_𝑖(𝑥)onΩ,𝑖=1,…,𝑚. (3)

Here 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑚), Ω is an open bounded domain of class 𝐶¹ in ℝ^𝑁, with boundary 𝜕Ω, and 𝜕/𝜕𝜂 denotes the outward normal derivative on 𝜕Ω.

Throughout the paper, we make the following hypotheses:

The indexes 𝑎_𝑖, 𝑝_𝑖𝑗 are nonnegative for all 𝑖,𝑗=1,…,𝑚, with 𝜎>0: 0<𝑝₁₁−1<max_{𝑘=2,3,…,𝑚}𝑝_𝑘1𝑝min1,^1𝑘𝑝_𝑘𝑘,𝑝+1𝑝^1𝑗_𝑘𝑗,𝑗=2,…,𝑚,𝑗≠𝑘; (4) we set 𝐴_𝑖𝑗=(𝑎_𝑖+𝑎_𝑗)/(2𝑎_𝑖𝑎_𝑗) for all 𝑖,𝑗=1,…,𝑚. Let 𝛼_𝑖,𝑖=1,…,𝑚, be positive constants such that 𝛼₁>2max1,^𝑚_𝑖=1𝑏_𝑖𝑏₁,𝑆^𝑙_𝑙>0,𝑙=2,…,𝑚, (5)(6) where 𝑆^𝑟_𝑙=𝑆^𝑟−1_𝑟−1⋅𝑆_𝑙^𝑟−1−𝐻_𝑙^𝑟−12𝐻,𝑟=3,…,𝑙,_𝑟𝑙=det_{1≤𝑖,𝑗≤𝑙}𝑎_{𝑖,𝑗𝑖≠𝑙,…𝑟+1𝑗≠𝑙−1,..𝑟}^{𝑘=𝑟−2}_𝑘=1(det[𝑘])^{2(𝑟−𝑘−2)},𝑆𝑟=3,…,𝑙−1,_2𝑙=𝛼²₁𝛼²_𝑙𝑎₁𝑎_𝑙12𝛼_𝑙−𝐴²_1𝑙,𝐻²_𝑙=𝛼²₁𝛼₂𝛼_𝑙𝑎₁√𝑎₂𝑎_𝑙𝛼₁−1𝛼₁𝐴_2𝑙−𝐴₁₂𝐴_1𝑙, (7) where det_{1≤𝑖,𝑗≤𝑙}((𝑎_𝑖,𝑗)_{𝑖≠𝑙,…,𝑟+1,𝑗≠𝑙−1,…,𝑟}) stands for the determinant of the 𝑟-square symmetric matrix obtained from the matrix (𝑎_𝑖,𝑗)_{1≤𝑖,𝑗≤𝑚} by removing the (𝑟+1)th,(𝑟+2)th,…,𝑙th rows and the 𝑟th,(𝑟+1)th,…,(𝑙−1)th columns, where det[1],…,det[𝑚] are the minors of the matrix (𝑎_𝑖𝑗)_{1≤𝑖,𝑗≤𝑚}. The elements of the matrix are as follows: 𝑎₁₁=𝑎₁𝛼₁𝛼₁,𝑎−1_1𝑖=−𝛼₁𝛼_𝑖𝑎₁+𝑎_𝑖2𝑎,𝑖=2,…,𝑚,_𝑖𝑖=𝑎_𝑖𝛼_𝑖𝛼_𝑖𝑎+1,𝑖=2,…,𝑚,_𝑖𝑗=𝛼_𝑖𝛼_𝑗𝑎_𝑖+𝑎_𝑗2,𝑖,𝑗=2,…,𝑚,𝑖≠𝑗. (8)

The main result of the paper is the following.

Theorem 1. Assume that condition (4) is satisfied. Let 𝑢 be a solution of (1)–(3) with positive and bounded initial data, and let 𝐿(𝑡)=_Ω𝑢^𝛼1₁(𝑡,𝑥)∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗(𝑡,𝑥)𝑑𝑥. (9) Then the functional 𝐿 is uniformly bounded on the interval [0,𝑇^∗], 𝑇^∗<𝑇_max, where 𝑇_max(‖𝜑₁‖_∞,‖𝜑₂‖_∞,…,‖𝜑_𝑚‖_∞) denotes the eventual blow-up time.

Corollary 2. Under the assumptions of Theorem 1 all solutions of (1)–(3) with positive initial data in 𝐶(Ω) are global. If in addition 𝑏₁,…,𝑏_𝑚,  𝜎>0, then 𝑢 is uniformly bounded in Ω×[0,∞).

Before we prove our results, let us dwell a while on the existing literature concerning Gierer-Meinhardt's type systems.

In 1972, following an ingenious idea of Turing [1], Gierer and Meinhardt [2] proposed a mathematical model for pattern formations of spatial tissue structures of hydra in morphogenesis. It is a system of reaction-diffusion equations of the form: 𝜕_𝑡𝑢−𝑎₁𝑢Δ𝑢=𝜎−𝜇𝑢+^𝑝𝑣^𝑞,𝜕_𝑡𝑣−𝑎₂𝑢Δ𝑣=−𝜈𝑣+^𝑟𝑣^𝑠,𝑥∈Ω,𝑡>0, (10) with Neumann boundary conditions 𝜕𝑢=𝜕𝜂𝜕𝑣𝜕𝜂=0,𝑥∈𝜕Ω,𝑡>0, (11) and initial conditions 𝑢(𝑥,0)=𝜑₁(𝑥)𝑣()>0,𝑥,0=𝜑₂(𝑥)>0,𝑥∈Ω, (12) where Ω⊂ℝ^𝑁 (𝑁=1,2,3 in practice) is a bounded domain with smooth boundary 𝜕Ω,𝑎₁,𝑎₂,𝜇,𝜈,𝜎>0,  and  𝑝, 𝑞, 𝑟 and 𝑠 are non negative windexes with 𝑝>1. Here 𝑢 is the activator, and 𝑣 is the inhibitor.

Global existence of solutions in (0,∞) was proved by Rothe [3], more than ten years after Gierer and Meinhardt's original paper with special choice of the parameters: 𝑝=2, 𝑞=1, 𝑟=2, 𝑠=0, and 𝑁=3. Masuda and Takahashi [4] were able to prove global estimates and bounds of the solution for Gierer and Meinhardt's system in its general form. They proceeded by first proving lower bounds, then 𝐿^𝑝 bounds (for any 𝑝>1), then uniform estimates and bounds in appropriate Sobolev spaces. The key point is represented by the 𝐿^𝑝 bounds, which are derived using in a subtle way the specific structure of the equations.

Li et al. [5] also studied the activator-inhibitor model.

Very recently, Bernasconi [6] considered the larger system: 𝜕_𝑡𝑎(𝑥,𝑡)=𝑑_𝑎𝑎_𝑥𝑥+𝑎²(𝑥,𝑡)ℎ()()𝜕𝑥,𝑡−𝜇𝑎𝑥,𝑡+𝜌,_𝑡ℎ(𝑥,𝑡)=𝑑_ℎℎ_𝑥𝑥(𝑥,𝑡)+𝑎²()()(),𝜕𝑥,𝑡−𝜈ℎ𝑥,𝑡+𝜀𝑠𝑥,𝑡_𝑡𝑠(𝑥,𝑡)=𝑑_𝑠𝑠_𝑥𝑥(𝑥,𝑡)+𝑎(𝑥,𝑡)−𝜅𝑠(𝑥,𝑡), (13) and Meinhardt et al. [7] proposed activator-inhibitor models to describe a theory of biological pattern: 𝜕_𝑡𝑎(𝑥,𝑡)=𝑑_𝑎𝑎_𝑥𝑥+𝑎²(𝑥,𝑡)ℎ()𝑠()()𝜕𝑥,𝑡𝑥,𝑡−𝜇𝑎𝑥,𝑡+𝜌,_𝑡ℎ(𝑥,𝑡)=𝑑_ℎℎ_𝑥𝑥(𝑥,𝑡)+𝑎²()(),𝜕𝑥,𝑡−𝜈ℎ𝑥,𝑡_𝑡𝑠(𝑥,𝑡)=𝑑_𝑠𝑠_𝑥𝑥(𝑥,𝑡)+𝑎(𝑥,𝑡)−𝜅𝑠(𝑥,𝑡), (14) which is Gierer and Meinhardt's system supplemented with a third equation, where 𝑎(𝑥,𝑡) is the activator, ℎ(𝑥,𝑡) is the inhibitor, and 𝑠(𝑥,𝑡) is a source that acts as an inhomogeneous inhibitor.

Our paper generalizes the system in [5] to 𝑚-components.

2. Preliminary Observations and Notations

The usual norms in the spaces 𝐿^𝑝(Ω), 𝐿^∞(Ω), and 𝐶(Ω) are denoted, respectively, by the following: ‖𝑢‖^𝑝_𝑝=1|Ω|_Ω|𝑢(𝑥)|^𝑝‖𝑢‖𝑑𝑥,_∞=esssup_𝑥∈Ω|𝑢(𝑥,‖𝑢‖)|_𝐶(Ω)=max_𝑥∈Ω|𝑢(𝑥)|. (15)

It is well known that to prove global existence of solutions to (1)–(3), it suffices to derive a uniform estimate of ‖𝑓_𝑖(𝑢₁,𝑢₂,…,𝑢_𝑚)‖_𝑝,  𝑖=1,…,𝑚 on [0;𝑇_max) in the space 𝐿^𝑝(Ω) for some 𝑝>𝑛/2 (see Henry [8]).

Since the functions 𝑓_𝑖 are continuously differentiable on ℝ^𝑚₊ for all 𝑖=1,…,𝑚, then for any initial data in 𝐶(Ω), the system (1)–(3) admits a unique, classical solution (𝑢₁,𝑢₂,…,𝑢_𝑚) on (0,𝑇_max)×Ω with the alternative(i)either 𝑇_max=∞;(ii)or 𝑇_max<∞, and lim_{𝑡↗𝑇max}∑^𝑚_𝑖=1‖𝑢_𝑖(𝑡,⋅)‖_∞=∞.

Using the maximum principle, one derives the lower bounds of the components of the solution 𝑢 of (1)–(3): 𝑢_𝑖(𝑡,𝑥)≥𝑒^{−𝑏𝑖𝑡}min𝜑_𝑖(𝑥)>0,𝑖=1,…,𝑚. (16)

Our aim is to construct a Lyapunov functional that allows us to obtain 𝐿^𝑝-bounds on 𝑢_𝑖 leading to global existence.

3. Preparatory Lemmas

For the proof of Theorem 1, we need some preparatory lemmas whose proofs will be in the appendix.

Lemma 3. Assume that the constants 𝑞_𝑖𝑗 satisfy 𝑞₁₁−1𝑞_𝑘1𝑞<min1,^1𝑘𝑞_𝑘𝑘,𝑞+1𝑞^1𝑗_𝑘𝑗,𝑗=2,…,𝑚,𝑗≠𝑘. (17) Then for all ℎ_𝑖−1,𝛼_𝑖>0, 𝑗,𝑖=1,…,𝑚, there exist 𝐶=𝐶(ℎ_𝑖−1,𝛼_𝑖)>0 and 𝜃=𝜃(𝛼₁)∈(0,1), such that 𝛼₁𝑢^{𝑞11−1+𝛼1}₁∏^𝑚_𝑗=2𝑢^{𝑞1𝑗+𝛼𝑗}_𝑗≤𝛼_𝑘𝑢^{𝑞𝑘1+𝛼1}₁𝑢^{𝑞𝑘𝑘+1+𝛼𝑘}_𝑘∏^𝑚_{𝑗=2,𝑗≠𝑘}𝑢^{𝑞𝑘𝑗+𝛼𝑗}_𝑗𝑢+𝐶^𝛼11∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^𝜃, (18)𝑢₁≥0,  𝑢_𝑖≥ℎ_𝑖−1,  𝑖=1,…,𝑚,  𝑘∈{2,…,𝑚}.

Lemma 4 (see [9]). Let 𝐴=(𝑎_𝑖𝑗)_{1≤𝑖,𝑗≤𝑚}. Then one has: 𝐾^𝑚_𝑚=det[𝑚]⋅^{𝑘=𝑚−2}_𝑘=1(det[𝑘])^{2(𝑚−𝑘−2)}𝐾,𝑚>2,₂₂=det[2], (19) where 𝐾^𝑙_𝑚=𝐾^𝑙−1_𝑙−1⋅𝐾_𝑚^𝑙−1−𝐻_𝑚^𝑙−12,𝑙=3,…,𝑚,𝐻^𝑙_𝑚=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠(𝑙+1),…,𝑚,𝑗≠𝑙,..,(𝑚−1)}⋅^{𝑘=𝑙−2}_𝑘=1(det[𝑘])^{2(𝑙−𝑘−2)},𝑙=3,…,𝑚−1,𝐾²_𝑚=𝑎₁₁𝑎_𝑚𝑚−𝑎_1𝑚²,𝐻²_𝑚=𝑎₁₁𝑎_2𝑚−𝑎₁₂𝑎_1𝑚. (20)(21)(22)(23)

Lemma 5. Let 𝛼₁>2max{1,∑^𝑚_𝑖=1𝑏_𝑖/𝑏₁}. One has 𝐾^𝑙_𝑙>𝑆^𝑙_𝑙,𝑙=2,…,𝑚, (24) where 𝐾^𝑟_𝑙=𝐾^𝑟−1_𝑟−1⋅𝐾_𝑙^𝑟−1−𝐻_𝑙^𝑟−12𝐻,𝑟=3,…,𝑙,_𝑟𝑙=det_{1≤𝑖,𝑗≤𝑙}𝑎_{𝑖,𝑗𝑖≠𝑙,…𝑟+1𝑗≠𝑙−1,..𝑟}⋅^{𝑘=𝑟−2}_𝑘=1(det[𝑘])^{2(𝑟−𝑘−2)}𝐾,𝑟=3,…,𝑙−1,_2𝑙=𝛼²₁𝛼²_𝑙𝑎₁𝑎_𝑙𝛼₁−1𝛼₁𝛼_𝑙+1𝛼_𝑙−𝐴²_1𝑙,𝐻²_𝑙=𝛼²₁𝛼₂𝛼_𝑙𝑎₁√𝑎₂𝑎_𝑙𝛼₁−1𝛼₁𝐴_2𝑙−𝐴₁₂𝐴_1𝑙. (25)

Lemma 6 (see Masuda and Takahashi [4]). Let 𝜇,𝑇>0 and let 𝑓_𝑗=𝑓_𝑗(𝑡) be a nonnegative integrable function on [0,𝑇) and 0<𝜃_𝑗<1 (𝑗=1,…,𝐽). Let 𝑊=𝑊(𝑡) be a positive function on [0,𝑇) satisfying the differential inequality 𝑑𝑊(𝑡)(𝑡)+𝑑𝑡≤−𝜇𝑊^𝐽_𝑗=1𝑓_𝑗(𝑡)𝑊^𝜃𝑗(𝑡),0≤𝑡<𝑇. (26) Then, one has 𝑊(𝑡)≤𝜅,0≤𝑡<𝑇, (27) where 𝜅 is the maximal root of the algebraic equation: 𝑥−^𝐽_𝑗=1sup_0<𝑡<𝑇^𝑡₀𝑒^{−𝜇(𝑡−𝜉)}𝑓_𝑗(𝜉)𝑑𝜉𝑥^𝜃𝑗=𝑊(0). (28)

4. Proofs

Proof of Theorem 1. Since 𝑢₁ satisfies 𝜕_𝑡𝑢₁−𝑎₁Δ𝑢₁>0 on {𝑢₁<𝜎/𝑏₁}, the maximum principle implies 𝑢₁≥𝛿∶=min(𝜎/𝑏₁,min𝑢₀(𝑥))>0.
Differentiating 𝐿(𝑡) with respect to 𝑡 yields 𝐿^′(𝑡)=_Ω𝑑𝑢𝑑𝑡∏^𝛼11𝑚_𝑗=2𝑢^𝛼𝑗_𝑗=𝑑𝑥_Ω𝛼₁𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗𝜕_𝑡𝑢₁−^𝑚_𝑖=2𝛼_𝑖𝑢^𝛼1₁𝑢^𝛼𝑖_𝑖⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^𝛼𝑗_𝑗𝜕_𝑡𝑢_𝑖𝑑𝑥. (29) Replacing 𝜕_𝑡𝑢_𝑖, 𝑖=1,…,𝑚, by its expression from (1), we get 𝐿^′(𝑡)=_Ω𝑎₁𝛼₁𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗Δ𝑢₁−^𝑚_𝑖=2𝛼_𝑖𝑎_𝑖𝑢^𝛼1₁𝑢^𝛼𝑖_𝑖⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^𝛼𝑗_𝑗Δ𝑢_𝑖−𝑏₁𝛼𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗+𝛼₁𝑢^{𝑝11−1+𝛼1}₁∏^𝑚_𝑗=2𝑢^{𝑝1𝑗+𝛼𝑗}_𝑗−^𝑚_𝑖=2𝛼_𝑖𝑢^{𝑝𝑖1+𝛼1}₁𝑢^{𝑝𝑖𝑖+1+𝛼𝑖}_𝑘∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^{𝑝𝑖,𝑗+𝛼𝑗}_𝑗+𝜎𝛼₁𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗𝑑𝑥∶=𝐼+𝐽, (30) where we have set 𝐼=𝑎₁𝛼_1Ω𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗Δ𝑢₁−𝑑𝑥_𝑚𝑖=2𝛼_𝑖𝑎_𝑖Ω𝑢^𝛼1₁𝑢^𝛼𝑖_𝑖⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^𝛼𝑗_𝑗Δ𝑢_𝑖𝑑𝑥,𝐽=−𝑏₁𝛼₁+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝐿(𝑡)+𝛼_1Ω𝑢^{𝑝11−1+𝛼1}₁∏^𝑚_𝑗=2𝑢^{𝑝1𝑗+𝛼𝑗}_𝑗−𝑑𝑥_𝑚𝑖=2𝛼_𝑖Ω𝑢^{𝑝𝑖1+𝛼1}₁𝑢^{𝑝𝑖𝑖+1+𝛼𝑖}_𝑘∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^{𝑝𝑖𝑗+𝛼𝑗}_𝑗𝑑𝑥+𝜎𝛼_1Ω𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗𝑑𝑥. (31)(32)Estimation of 𝐼. We are going to show that 𝐼≤0.
Using Green's formula, we obtain 𝐼=_Ω𝑎₁𝛼₁−𝛼₁𝑢−1^𝛼11−2∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗∇𝑢₁²+^𝑚_𝑖=2𝛼_𝑖𝑢^𝛼1₁⁻¹𝑢^𝛼𝑖_𝑖⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^𝛼𝑗_𝑗∇𝑢₁∇𝑢_𝑖+^𝑚_𝑖=2𝑎_𝑖𝛼_𝑖⋅𝛼₁𝑢^𝛼1₁⁻¹𝑢^𝛼𝑖_𝑖⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^𝛼𝑗_𝑗∇𝑢₁∇𝑢_𝑖−𝛼_𝑖𝑢+1^𝛼11𝑢^𝛼𝑖_𝑖⁺²∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^𝛼𝑗_𝑗∇𝑢_𝑖²−^𝑚_{𝑘=2𝑘≠𝑖}𝛼_𝑘×𝑢^𝛼1₁𝑢^𝛼𝑘_𝑘⁺¹𝑢^𝛼𝑖_𝑖⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑖,𝑗≠𝑘}𝑢^𝛼𝑗_𝑗×∇𝑢_𝑘∇𝑢_𝑖𝑑𝑥,=−_Ω𝑢^𝛼1₁⁻²∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗⁺²(𝑄𝑇)⋅𝑇𝑑𝑥, (33) where 𝑄=(𝑎_𝑖,𝑗)_{1≤𝑖,𝑗≤𝑚} is defined in (8) and 𝑇=^𝑚_𝑗=2𝑢_𝑗∇𝑢₁,…,^𝑚_{𝑗=1𝑗≠𝑖}𝑢_𝑗∇𝑢_𝑖,…,^𝑚−1_𝑗=1𝑢_𝑗∇𝑢_𝑚^𝑡. (34)
The matrix 𝑄 is positive definite if and only if all its associated minor matrices Δ₁,Δ₂,…,Δ_𝑚 are positive. To see this, we have the following.(1)Δ₁=𝑎₁𝛼₁(𝛼₁−1)>0. Using (5), we get det[1]>0.(2) According to Lemma 4, we have det[2]=𝐾²₂=𝛼²₁𝛼²₂𝑎₁𝑎₂𝛼₁−1𝛼₁𝛼₂+1𝛼₂−𝐴²₁₂. (35) Using (6) and (24) for 𝑙=2, we get det[2]>0.(3) Again according to Lemma 4, we have 𝐾³₃=det[3]det[1]. (36) But det[1]>0, thus sign(𝐾³₃)=sign(det[3]). Using (6) and (24) for 𝑙=3, we get det[3]>0.(4) We suppose that det[𝑘]>0, 𝑘=1,2,…,𝑙−1 and prove that det[𝑙]>0; thus [𝑘]()⟹det>0,𝑘=1,…,𝑙−1_{𝑘=𝑙−2𝑘=1}(det[𝑘])^{2(𝑙−𝑘−2)}>0. (37)
From Lemma 4, 𝐾^𝑙_𝑙=det[𝑙]⋅^{𝑘=𝑙−2}_𝑘=1(det[𝑘])^{2(𝑙−𝑘−2)}. (38)
This along with (37) yields sign𝐾^𝑙_𝑙=sign(det[𝑙]). (39) But from (6) and (24) 𝐾^𝑙_𝑙>0; thus det[𝑙]>0.
Consequently, we have 𝐼≤0.
Estimation of 𝐽. We are going to estimate 𝐽 by a function of 𝐿(𝑡).
According to the maximum principle, there exists 𝐶₀ depending on 𝜑_𝑖(𝑥),  𝑖=1,…,𝑚, such that 𝑢_𝑖≥𝐶₀>0,  𝑖=2,…,𝑚. We then have 𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗=𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1𝑚}_𝑗=21𝑢_𝑗^{𝛼𝑗/𝛼1}≤𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1}1𝐶₀^{∑𝑚𝑗=2𝛼𝑗/𝛼1}, (40) whereupon 𝑢^𝛼1₁⁻¹∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗≤𝐶₂𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1},where𝐶₂=1𝐶₀^{∑𝑚𝑗=2𝛼𝑗/𝛼1}. (41) We have 𝐽≤−𝑏₁𝛼₁+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝐿(𝑡)+𝛼_1Ω𝑢^{𝑞11−1+𝛼1}₁∏^𝑚_𝑗=2𝑢^{𝑞1𝑗+𝛼𝑗}_𝑗−𝑑𝑥_𝑚𝑖=2𝛼_𝑖Ω𝑢^{𝑞𝑖1+𝛼1}₁𝑢^{𝑞𝑖𝑖+1+𝛼𝑖}_𝑘∏^𝑚_{𝑗=2,𝑗≠𝑖}𝑢^{𝑞𝑖𝑗+𝛼𝑗}_𝑗𝑑𝑥+𝜎𝛼_1Ω𝐶₂𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1}𝑑𝑥. (42) Using Lemma 3, we obtain 𝐽≤−𝑏₁𝛼₁+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝐿(𝑡)+_Ω𝐶𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^𝜃𝑑𝑥+𝜎𝛼_1Ω𝐶₂𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1}𝑑𝑥. (43) Applying Hölder's inequality, we obtain _Ω𝐶𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^𝜃≤𝑑𝑥_Ω𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗𝑑𝑥^𝜃𝐶(meas(Ω))^1−𝜃. (44) So _Ω𝐶𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^𝜃𝑑𝑥≤𝐶₃𝐿^𝜃(𝑡),𝐶₃=𝐶(meas(Ω))^1−𝜃. (45) Also, we have _Ω𝐶₂𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1}≤𝑑𝑥_Ω𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗𝑑𝑥^{(𝛼1−1)/𝛼1}⋅_Ω𝐶₂^𝛼1𝑑𝑥^1/𝛼1. (46) So _Ω𝐶₂𝑢^𝛼1₁∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^{(𝛼1−1)/𝛼1}𝑑𝑥≤𝐶₄𝐿(𝛼₁−1)𝛼₁(𝑡),where𝐶₄=𝐶₂(meas(Ω))^1/𝛼1. (47) We then get 𝐽≤−𝑏₁𝛼₁+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝐿(𝑡)+𝐶₃𝐿^𝜃(𝑡)+𝛼₁𝜎𝐶₄𝐿^{(𝛼1−1)/𝛼1}(𝑡), (48) which implies 𝐽≤−𝑏₁𝛼₁+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝐿(𝑡)+𝐶₅𝐿^𝜃(𝑡)+𝛼₁𝜎𝐿^{(𝛼1−1)/𝛼1}(𝑡). (49) This yields the differential inequality: 𝐿^′(𝑡)≤−𝑏₁𝛼₁+^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖𝐿(𝑡)+𝐶₅𝐿^𝜃(𝑡)+𝛼₁𝜎𝐿^{(𝛼1−1)/𝛼1}(𝑡). (50) Thus under conditions (5), (6), and (8), we obtain −𝑏₁𝛼₁+∑^𝑚_𝑖=2𝑏_𝑖𝛼_𝑖<0; using Lemma 6 we deduce that 𝐿(𝑡) is bounded on (0,𝑇_max); that is, 𝐿(𝑡)≤𝛾₁, where 𝛾₁ depends on 𝜑_𝑖(𝑥), 𝑖=1,…,𝑚.

Proof of Corollary 2 (𝐿^∞-bounds). By Theorem 1, we have 𝑢^𝑝𝑖1₁/∏^𝑚_𝑗=2𝑢^𝑝𝑖𝑗_𝑗∈𝐿^∞((0,𝑇_max),𝐿^𝑟(Ω)),  𝑖=2,…,𝑚 for all 𝑟>𝑁/2. By a simple argument relying on the variation-of-constants formula and the 𝐿^𝑝-𝐿^𝑞-estimate (Proposition 48.4 see [10]), we deduce that 𝑢 is uniformly bounded. Consequently, 𝑇_max=∞.

Appendix

The purpose of this appendix is to prove the lemmas of Section 3 which have been used in the proof of Theorem 1.

Proof of Lemma 3. Inequality (18) is equivalent to 𝛼₁𝑢^𝑞11₁⁻¹∏^𝑚_𝑗=2𝑢^𝑞1𝑗_𝑗≤𝛼_𝑘𝑢^𝑞𝑘1₁𝑢^𝑞𝑘𝑘_𝑘⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑘}𝑢^𝑞𝑘𝑗_𝑗𝑢+𝐶^𝛼11∏^𝑚_𝑗=2𝑢^𝛼𝑗_𝑗^𝜃−1. (A.1) Let us set 𝜁=(𝛼_𝑘𝑢^𝑞𝑘1₁)/(𝑢^𝑞𝑘𝑘_𝑘⁺¹∏^𝑚_{𝑗=2,𝑗≠𝑘}𝑢^𝑞𝑘𝑗_𝑗).
Now, we write 𝛼₁𝑢^𝑞11₁⁻¹∏^𝑚_𝑗=2𝑢^𝑞1𝑗_𝑗=𝛼₁𝛼_𝑘^{−(𝑞11−1)/𝑞𝑘1}(𝜁)^{(𝑞11−1)/𝑞𝑘1}⋅^𝑚_{𝑗=2𝑗≠𝑘}𝑢_𝑗^{𝑞𝑘𝑗(𝑞11−1)/𝑞𝑘1−𝑞1𝑗}⋅𝑢_𝑘^{(𝑞𝑘𝑘+1)(𝑞11−1)/𝑞𝑘1−𝑞1𝑘}. (A.2) For each 𝜖 such that 0<𝜖<min{1,𝑞_1𝑘/(𝑞_𝑘𝑘+1),𝑞_1𝑗/𝑞_𝑘𝑗,𝑗=2,…,𝑚,and𝑗≠𝑘}−(𝑞₁₁−1)/𝑞_𝑘1, 𝛼₁𝑢^𝑞11₁⁻¹∏^𝑚_𝑗=2𝑢^𝑞1𝑗_𝑗=𝛼₁𝛼_𝑘^{−(𝑞11−1)/𝑞𝑘1}(𝜁)^{(𝑞11−1)/𝑞𝑘1+𝜖}×(𝜁)^𝑚−𝜖_{𝑗=2𝑗≠𝑘}𝑢_𝑗^{𝑞𝑘𝑗(𝑞11−1)/𝑞𝑘1−𝑞1𝑗}×𝑢_𝑘^{(𝑞𝑘𝑘+1)(𝑞11−1)/𝑞𝑘1−𝑞1𝑘}=𝛼₁𝛼_𝑘^{−(𝑞11−1)/𝑞𝑘1−𝜖}(𝜁)^{(𝑞11−1)/𝑞𝑘1+𝜖}×1𝑢^𝛼1₁^{𝑞𝑘1𝜖/𝛼1}×^𝑚_{𝑗=2𝑗≠𝑘}𝑢_𝑗^{𝑞𝑘,𝑗(𝑞11−1)/𝑞𝑘,1−𝑞1𝑗+𝜖𝑞𝑘𝑗}×𝑢_𝑘^{(𝑞𝑘𝑘+1)(𝑞11−1)/𝑞𝑘,1−𝑞1𝑘+𝜖(𝑞𝑘𝑘+1)}≤𝛼₁𝛼_𝑘^{−(𝑞11−1)/𝑞𝑘1−𝜖}(𝜁)^{(𝑞11−1)/𝑞𝑘1+𝜖}×1𝑢^𝛼1₁^{𝑞𝑘1𝜖/𝛼1}×^𝑚_{𝑗=2𝑗≠𝑘}ℎ_𝑗^{𝑞𝑘𝑗(𝑞11−1)/𝑞𝑘1−𝑞1𝑗+𝜖𝑞𝑘𝑗}×ℎ_𝑘^{(𝑞𝑘𝑘+1)(𝑞11−1)/𝑞𝑘1−𝑞1𝑘+𝜖(𝑞𝑘𝑘𝑚+1)}_𝑗=2𝑢_𝑗ℎ_𝑗^{𝛼𝑗𝑞𝑘1𝜖/𝛼1}≤𝐶₁(𝜁)^{(𝑞11−1)/𝑞𝑘1+𝜖}∏^𝑚_𝑗=2𝑢_𝑗^𝛼𝑗𝑢^𝛼1₁^{𝑞𝑘1𝜖/𝛼1}, (A.3) where 𝐶₁=𝛼₁𝛼_𝑘^{−(𝑞11−1)/𝑞𝑘1−𝜖}×^𝑚_{𝑗=2𝑗≠𝑘}ℎ_𝑗^{𝑞𝑘𝑗(𝑞11−1)/𝑞𝑘1−𝑞1,𝑗+𝜖𝑞𝑘,𝑗−𝛼𝑗𝑞𝑘1𝜖/𝛼1}×ℎ_𝑘^{(𝑞𝑘𝑘+1)(𝑞11−1)/𝑞𝑘1−𝑞1𝑘+𝜖(𝑞𝑘𝑘+1)−𝛼𝑘𝑞𝑘1𝜖/𝛼1}. (A.4) Using Young's inequality for (A.3) with 𝐶=𝐶^{1+(𝑞11−1+𝑞𝑘1𝜖)/(𝑞𝑘1−(𝑞11−1)−𝑞𝑘1}₁^𝜖),𝑞𝜃=1−^𝑘1𝜖𝛼₁1−𝑞₁₁−1/𝑞_𝑘1,−𝜖 (A.5) where 𝜖 is sufficiently small, we get inequality (18).

Proof of Lemma 4. We prove this lemma by induction.
For 𝑚=2, we have 𝐾²₂=det[2].
We consider the case 𝑚=3.
By using the well-known Dodgson condensation [11] for the symmetric 3-square matrix: det[1]det[3]=det[2]det_{1≤𝑖,𝑗≤3}𝑎_{𝑖,𝑗𝑖≠2,𝑗≠2}−det_{1≤𝑖,𝑗≤3}𝑎_{𝑖,𝑗𝑖≠3,𝑗≠2}². (A.6) But det[2]=𝐾²₂,det_{1≤𝑖,𝑗≤3}𝑎_{𝑖,𝑗𝑖≠2𝑗≠2}=𝑎₁₁𝑎₃₃−𝑎₁₃²=𝐾²₃,det_{1≤𝑖,𝑗≤3}𝑎_{𝑖,𝑗𝑖≠2𝑗≠3}=𝑎₁₁𝑎₂₃−𝑎₁₂𝑎₁₃=𝐻²₃. (A.7) So det[1]det[3]=𝐾²₂⋅𝐾²₃−𝐻²₃². (A.8) Hence by using formula (20), formula (19) is correct for 𝑚=3.
When 𝑚≥4, we suppose that formula (19) is correct for (𝑚−1),𝑚−2,𝑚−3,…,4, and we prove it for 𝑚.
It is sufficient to prove that 𝐾_𝑚^𝑚−1=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠𝑚−1𝑗≠𝑚−1}⋅^{𝑘=𝑚−3}_𝑘=1(det[𝑘])^{2(𝑚−𝑘−3)}. (A.9) By putting 𝑙=𝑚−1 in formula (21), we get 𝐻_𝑚^𝑚−1=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠𝑚𝑗≠𝑚−1}⋅^{𝑘=𝑚−3}_𝑘=1(det[𝑘])^{2(𝑚−𝑘−3)}. (A.10) From the mathematical induction proof, we have 𝐾₍₎^(𝑚−1)_𝑚−1[]⋅=det𝑚−1_{𝑘=𝑚−3𝑘=1}(det[𝑘])^{2(𝑚−𝑘−3)}. (A.11) By putting 𝑙=𝑚 in formula (20), we get 𝐾^𝑚_𝑚=𝐾^𝑚−1_𝑚−1⋅𝐾_𝑚^𝑚−1−𝐻_𝑚^𝑚−12. (A.12) By replacing (A.9), (A.10), and (A.11) in (A.12), we obtain 𝐾^𝑚_𝑚=^{𝑘=𝑚−3}_𝑘=1(det[𝑘])^{2(𝑚−𝑘−2)}[][𝑚][𝑚]⋅⋅det𝑚−2⋅det=det^{𝑘=𝑚−2}_𝑘=1(det[𝑘])^{2(𝑚−𝑘−2)}, (A.13) and thus formula (19) is correct for 𝑚.
Now, we prove formula (A.9); we may generalize formula (A.9) as follows: 𝐾^𝑙_𝑚=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖𝑗𝑖≠𝑚−1,…𝑙𝑗≠𝑚−1,…𝑙}⋅^{𝑘=𝑙−2}_𝑘=1(det[𝑘])^{2((𝑙−2)−𝑘)},𝑙=3,…,𝑚−1. (A.14) Also, we prove formula (A.14) by induction. It is a second inductive proof included in the first one.
It is evident for 𝑙=2.
For 𝑙=3, formula (20) will be: 𝐾³_𝑚=𝐾²₂⋅𝐾²_𝑚−𝐻²_𝑚². (A.15) Since we already know that 𝐾²₂[2],𝐾=det_2𝑚=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠𝑚−1,…2𝑗≠𝑚−1,…2},𝐻²_𝑚=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠𝑚−1,…,2𝑗≠𝑚,…,3}, (A.16) simple substitution of these three formulas in the formula (A.15) followed by the application of the modified well-known Dodgson condensation which has been modified in [11] will lead to formula (A.14) for 𝑙=3. directly.
When 𝑙≥4, we suppose that formula (A.14) is correct for 𝑙−1, and we prove it for 𝑙.
Formula (20) for 𝑙−1 reads 𝐾^𝑙_𝑚=𝐾^𝑙−1_𝑙−1⋅𝐾_𝑚^𝑙−1−𝐻_𝑚^𝑙−12. (A.17)
According to the first induction, we have 𝐾₍₎^(𝑙−1)_𝑙−1=det[𝑙−1]^{𝑘=𝑙−3}_𝑘=1(det[𝑘])^{2(𝑙−𝑘−3)}. (A.18) According to the second induction, we have 𝐾_𝑚^𝑙−1=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠𝑚−1,…,𝑙−1𝑗≠𝑚−1,…,𝑙−1}⋅^{()𝑘=𝑙−3}_𝑘=1(det[𝑘])^{2((𝑙−3)−𝑘)}. (A.19) According to formula (21), we have: 𝐻_𝑚^𝑙−1=det_{1≤𝑖,𝑗≤𝑚}𝑎_{𝑖,𝑗𝑖≠𝑚,…,𝑙𝑗≠𝑚−1,..,𝑙−1}⋅^{𝑘=𝑙−3}_𝑘=1(det[𝑘])^{2((𝑙−3)−𝑘)}. (A.20) By replacing (A.18), (A.19), and (A.20) in (A.17) and by using the well-known Dodgson condensation, we obtain formula (A.14) for 𝑙. Therefore, the second inductive proof is finished and consequently the first one.

Proof of Lemma 5. We prove this lemma by induction: 𝐾^𝑙_𝑙>𝑆^𝑙_𝑙,𝑙=2,…,𝑚. (A.21)
For 𝑙=2, we have 𝐾²₂=𝛼²₁𝛼²₂𝑎₁𝑎₂𝛼₁−1𝛼₁𝛼₂+1𝛼₂−𝐴²₁₂>𝛼²₁𝛼²₂𝑎₁𝑎₂12𝛼₂−𝐴²₁₂=𝑆²₂. (A.22) Because 𝛼₁>2,then𝛼₁−1𝛼₁𝛼₂+1𝛼₂>12𝛼₂. (A.23)
Assuming 𝑙≥3, we suppose (24) is true for (𝑙−1),𝑙−2,𝑙−3,…,3, and we prove it for 𝑙. Hence, we aim to prove 𝐾²₂>𝑆²₂,𝐾³₃>𝑆³₃,𝐾⁴₄>𝑆⁴₄𝐾,…,_{𝑙−1𝑙−1}>𝑆^𝑙−1_𝑙−1⟹𝐾^𝑙_𝑙>𝑆^𝑙_𝑙. (A.24)
Recall that 𝐾^𝑙_𝑙=𝐾^𝑙−1_𝑙−1𝐾_𝑙^𝑙−1−𝐻_𝑙^𝑙−12. (A.25) It is then sufficient to prove 𝐾_𝑙^𝑙−1>𝑆_𝑙^𝑙−1, (A.26) which will satisfy the inequality 𝐾^𝑙_𝑙=𝐾^𝑙−1_𝑙−1𝐾_𝑙^𝑙−1−𝐻_𝑙^𝑙−12>𝑆^𝑙−1_𝑙−1𝑆_𝑙^𝑙−1−𝐻_𝑙^𝑙−12=𝑆^𝑙_𝑙. (A.27)
In order to prove (A.26), we first generalize it in the form 𝐾^𝑟_𝑙>𝑆^𝑟_𝑙,𝑟=2,…,𝑙−1. (A.28) This can be proven by mathematical induction. It is a secondary inductive proof inside the primary one. For 𝑟=2, it is evident that 𝐾²_𝑙>𝑆²_𝑙. (A.29) For 𝑟=3, the formula 𝐾³_𝑙=𝐾²₂𝐾²_𝑙−𝐻²_𝑙²>𝑆²₂𝑆²_𝑙−𝐻²_𝑙²=𝑆³_𝑙 (A.30) is evident too.
When 𝑟≥4, we suppose formula (A.28) true for 𝑙−2: 𝐾_𝑙^𝑙−2>𝑆_𝑙^𝑙−2 (A.31) and we prove it for 𝑙−1: 𝐾_𝑙^𝑙−1>𝑆_𝑙^𝑙−1. (A.32)
We have 𝐾_𝑙^𝑙−1=𝐾^𝑙−2_𝑙−2𝐾_𝑙^𝑙−2−𝐻_𝑙^𝑙−22>𝑆^𝑙−2_𝑙−2𝑆_𝑙^𝑙−2−𝐻_𝑙^𝑙−22=𝑆_𝑙^𝑙−1. (A.33)
Then 𝐾_𝑙^𝑙−1>𝑆_𝑙^𝑙−1. (A.34)
Accordingly, we have 𝐾^𝑙_𝑙>𝑆^𝑙_𝑙. (A.35) This finishes the proof.

Acknowledgment

The authors would like to express their deepest gratitude for Professor KIRANE (Université de la Rochelle, France) for all his valuable input and guidance throughout the research and authoring of this paper.