Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 873683 | 22 pages | https://doi.org/10.1155/2009/873683

Existence of Weak Solutions to a Class of Degenerate Semiconductor Equations Modeling Avalanche Generation

Academic Editor: Shijun Liao
Received22 Feb 2009
Accepted03 Jun 2009
Published30 Jul 2009

Abstract

We consider the drift-diffusion model with avalanche generation for evolution in time of electron and hole densities 𝑛, 𝑝 coupled with the electrostatic potential πœ“ in a semiconductor device. We also assume that the diffusion term is degenerate. The existence of local weak solutions to this Dirichlet-Neumann mixed boundary value problem is obtained.

1. Introduction

In this paper, we consider the following degenerate semiconductor equations modeling avalanche generation: π‘›βˆ’βˆ‡β‹…(βˆ‡πœ“)=π‘βˆ’π‘›+𝐢(π‘₯),(1.1)π‘‘βˆ’βˆ‡β‹…π½π‘›=𝑅(𝑛,𝑝)+𝑔,𝐽𝑛=βˆ‡(𝑛𝛾)βˆ’πœ‡1π‘π‘›βˆ‡πœ“,(1.2)𝑑+βˆ‡β‹…π½π‘=𝑅(𝑛,𝑝)+𝑔,βˆ’π½π‘=βˆ‡(𝑝𝛾)+πœ‡2π‘βˆ‡πœ“(1.3) with initial and boundary conditions ξ€·(πœ“,𝑛,𝑝)=πœ“,𝑛,𝑝,(π‘₯,𝑑)βˆˆΞ£π·β‰‘Ξ“π·Γ—ξ‚΅(0,𝑇),(1.4)πœ•πœ“,πœ•πœ‚πœ•π‘›,πœ•πœ‚πœ•π‘ξ‚Άπœ•πœ‚=(0,0,0),(π‘₯,𝑑)βˆˆΞ£π‘β‰‘Ξ“π‘ξ€·π‘›Γ—(0,𝑇),(1.5)(𝑛,𝑝)=0,𝑝0ξ€Έ,π‘₯∈Ω,𝑑=0.(1.6) Here the unknowns πœ“,𝑛, and 𝑝 denote the electrostatic potential, the electron density, and the hole density, respectively. The boundary πœ•Ξ© consists of two disjoint subsets Γ𝐷 and Γ𝑁. The carrier densities and the potential are fixed at Γ𝐷 (Ohmic contacts), whereas Γ𝑁 models the union of insulating boundary segments. 𝐽𝑛 represents the electron current, and 𝐽𝑝 is the analogously defined physical quantity of the positively charged holes. Function 𝐢(π‘₯) denotes the doping profile (fixed charged background ions) characterizing the semiconductor under consideration, while the term 𝑔=𝛼1(βˆ‡πœ“)|𝐽𝑛|+𝛼2(βˆ‡πœ“)|𝐽𝑝| models the effect of impact ionization (avalanche generation of charged particles) (cf. [1, 2] for details). 𝑅(𝑛,𝑝)=π‘Ÿ(𝑛,𝑝)(1βˆ’π‘›π‘) is the net recombination-generation rate, where π‘Ÿ characterizes the mechanism of particle transition. The constant 𝛾 is the adiabatic or isothermal (if 𝛾=1) exponent. The regime 0<𝛾<1 describes a fast diffusion process in the electron (hole) density, whereas 1<𝛾≀5 is related to slow diffusion.

The standard drift-diffusion model corresponding to 𝛾=1 has been mathematically and numerically investigated in many papers (see [3–6]). Existence and uniqueness of weak solutions have been shown. The standard model can be derived from Boltzmann's equation once assumed that the semiconductor device is in the low injection regime, that is, for small absolute values of the applied voltage. In [7] JΓΌngel showed that in the high-injection regime diffusion terms are no longer linear. A useful choice for 𝛾 is 𝛾=5/3. In this case, the parabolic equations (1.2) and (1.3) become of degenerate type, and existence of solutions does not follow from standard theory. Recently, many authors [8–10] have studied the existence and uniqueness of weak solutions of this type of degenerate semiconductor equations without avalanche generation term. In [9], the degenerate semiconductor equations based on Fermi-Dirac statistics were introduced by JΓΌngel for the first time. The existence and uniqueness results are shown under the assumption that the solution πœ“ of Poisson equation with Dirichlet-Neumann mixed boundary conditions had the regularity πœ“βˆˆπ‘Š2,π‘Ÿ(Ξ©)(π‘Ÿ>𝑁), this amounts to a geometric condition on Ξ©, for example Ω∈𝐢1,1 and Ξ“π·βˆ©Ξ“π‘=βˆ…. ([11, Theorem 3.29]). Then Guan and Wu [8] obtain similar results without the assumption above.

There are some papers concerning the semiconductor equations modeling avalanche generation. For instance, the existence of weak solutions of nondegenerate stationary problem has been investigated in [12, 13]. When 𝛾=1, that is, the diffusion term is not degenerate, the authors [14] obtained the existence of local weak solutions of problem (1.1)–(1.6).

Our main goal in this paper is to study the existence of weak solutions of problem (1.1)–(1.6). In contrast to the above works, degeneration of diffusion term we are going to study introduces significant new technical difficulties to estimate the avalanche term.

We make the following assumptions:

(H1)Ξ©βŠ‚π‘…π‘(𝑁=1,2,3) is bounded and πœ•Ξ©βˆˆπΆ0,1, whose outward normal vector is πœ‚ and πœ•Ξ©=Γ𝐷βˆͺΓ𝑁,Ξ“π·βˆ©Ξ“π‘=βˆ…, measπ‘βˆ’1(Γ𝐷)>0;(H2)𝐢(π‘₯)∈𝐿∞(Ξ©);(H3)π‘Ÿ(𝑛,𝑝) is a locally Lipschitz continuous function defined for (𝑛,𝑝) and 0β‰€π‘Ÿ(𝑛,𝑝)β‰€π‘Ÿ<∞;(H4)π›Όπ‘–βˆˆπΆ(𝑅𝑁), 0≀𝛼𝑖(πœ‰)≀𝛼𝑖0=  const.<+∞, forallπœ‰βˆˆπ‘…π‘(𝑖=1,2);(H5)𝑛,π‘βˆˆπ‘Š1,∞(Ξ©)∩𝐿∞(Ξ©), πœ“βˆˆπ»1(Ξ©)∩𝐿∞(Ξ©), and 𝑛, 𝑝β‰₯0 in Ξ©;(H6)𝑛0,𝑝0∈𝐿∞(Ξ©) and 𝑛0,𝑝0β‰₯0 a.e. in Ξ©.

Let ξ€½π‘Œ=πœ”βˆˆπ»1(Ξ©)|πœ”|Γ𝐷=0.(1.7)

Definition 1.1. (πœ“,𝑛,𝑝) is called the weak solution to the problem (1.1)–(1.6) if π‘›π›Ύβˆˆπ‘›π›Ύ+𝐿2(0,𝑇;π‘Œ), π‘π›Ύβˆˆπ‘π›Ύ+𝐿2(0,𝑇;π‘Œ), πœ“βˆˆπœ“+𝐿2(0,𝑇;π‘Œ), 𝑛𝑑,π‘π‘‘βˆˆπΏ2(0,𝑇;π‘Œβˆ—),π‘›βˆ£π‘‘=0=𝑛0,π‘βˆ£π‘‘=0=𝑝0, and there hold ξ€œΞ©ξ€œβˆ‡πœ“β‹…βˆ‡πœdπ‘₯=Ξ©(ξ€œπ‘βˆ’π‘›+𝐢(π‘₯))𝜁dπ‘₯,βˆ€π‘‘βˆˆ(0,𝑇),βˆ€πœβˆˆπ‘Œπ‘‡0βŸ¨π‘›π‘‘,πœβŸ©π‘Œβˆ—,π‘Œ+ξ€œπ‘‡0ξ€œΞ©ξ€·βˆ‡(𝑛𝛾)βˆ’πœ‡1ξ€Έ=ξ€œπ‘›βˆ‡πœ“β‹…βˆ‡πœdπ‘₯d𝑑𝑇0ξ€œΞ©[π‘Ÿ](𝑛,𝑝)(1βˆ’π‘›π‘)+π‘”πœdπ‘₯d𝑑,βˆ€πœβˆˆπΏ2ξ€œ(0,𝑇;π‘Œ),𝑇0βŸ¨π‘π‘‘,πœβŸ©π‘Œβˆ—,π‘Œ+ξ€œπ‘‡0ξ€œΞ©ξ€·βˆ‡(𝑝𝛾)+πœ‡2ξ€Έ=ξ€œπ‘βˆ‡πœ“β‹…βˆ‡πœdπ‘₯d𝑑𝑇0ξ€œΞ©[]π‘Ÿ(𝑛,𝑝)(1βˆ’π‘›π‘)+π‘”πœdπ‘₯d𝑑,βˆ€πœβˆˆπΏ2(0,𝑇;π‘Œ).(1.8) Our main result in this paper is as follows.

Theorem 1.2. Under hypotheses (H1)–(H6), there exists at least one local weak solution to the problem (1.1)–(1.6).

2. Approximate Problem

For simplicity, we assume that πœ‡π‘–=1,𝑖=1,2. As [14], we first construct the following bound approximate sequence π‘”πœ of avalanche generation term 𝑔: π‘”πœξ€·π›Ό(𝑛,𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“)=1||𝐽(βˆ‡πœ“)𝑛||+𝛼2||𝐽(βˆ‡πœ“)𝑝||ξ€Έβ‹…ξ€Ί1+𝜏(𝛼1||𝐽(βˆ‡πœ“)𝑛||+𝛼2||𝐽(βˆ‡πœ“)𝑝||)ξ€»βˆ’1,(2.1) here 0<𝜏<1. Obviously, 0<π‘”πœ<1/𝜏 .

Now we introduce the following approximate problem with the initial and boundary conditions (1.4)–(1.6):

π‘›βˆ’βˆ‡β‹…(βˆ‡πœ“)=π‘βˆ’π‘›+𝐢(π‘₯),(2.2)π‘‘βˆ’βˆ‡β‹…(βˆ‡π‘›π›Ύβˆ’π‘›βˆ‡πœ“)=π‘Ÿ(𝑛,𝑝)(1βˆ’π‘›π‘)+π‘”πœπ‘(𝑛,𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“),(2.3)π‘‘βˆ’βˆ‡β‹…(βˆ‡π‘π›Ύ+π‘βˆ‡πœ“)=π‘Ÿ(𝑛,𝑝)(1βˆ’π‘›π‘)+π‘”πœ(𝑛,𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“).(2.4)

This section is devoted to the proof of global existence of weak solutions to the above approximate problem (2.2)–(2.4), (1.4)–(1.6). We will prove the following existence theorem.

Theorem 2.1. Under hypotheses (H1)–(H6), there exists at least one global weak solution to the problem (2.2)–(2.4), (1.4)–(1.6).

The proof is based on Schauder's fixed pointed theorem. The main difficulty in the proof is that problem (2.2)–(2.4), (1.4)–(1.6) is degenerated at points where 𝑛,𝑝=0. This difficulty leads us to consider the following auxiliary regularized problem with the initial and boundary conditions (1.4)–(1.6): βˆ’βˆ‡β‹…(βˆ‡πœ“)=π‘π‘˜βˆ’π‘›π‘˜π‘›+𝐢(π‘₯),(2.5)π‘‘βˆ’βˆ‡β‹…ξ‚€ξ‚€π›Ύπ‘›π‘˜π›Ύβˆ’1+πœ€βˆ‡π‘›βˆ’π‘›π‘˜ξ‚ξ€·βˆ‡πœ“=π‘Ÿ(𝑛,𝑝)1βˆ’π‘›π‘π‘˜ξ€Έπ‘+β„Ž(𝑛,𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“),(2.6)π‘‘βˆ’βˆ‡β‹…ξ‚€ξ‚€π›Ύπ‘π‘˜π›Ύβˆ’1+πœ€βˆ‡π‘+π‘π‘˜ξ‚ξ€·βˆ‡πœ“=π‘Ÿ(𝑛,𝑝)1βˆ’π‘›π‘˜π‘ξ€Έ+β„Ž(𝑛,𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“),(2.7) where π‘ π‘˜=min{max{0,𝑠},π‘˜} and β„Žξ€·πœ‰,πœ‚,πœ‰ξ…ž,πœ‚ξ…žξ€Έ=𝛼,𝜁1||(𝜁)π›Ύπœ‰π‘˜π›Ύβˆ’1πœ‰ξ…žβˆ’πœ‰π‘˜πœ||+𝛼2||(𝜁)π›Ύπœ‚π‘˜π›Ύβˆ’1πœ‚ξ…ž+πœ‚π‘˜πœ||×𝛼1+𝜏1||(𝜁)π›Ύπœ‰π‘˜π›Ύβˆ’1πœ‰ξ…žβˆ’πœ‰π‘˜πœ||+𝛼2||(𝜁)π›Ύπœ‚π‘˜π›Ύβˆ’1πœ‚ξ…ž+πœ‚π‘˜πœ||ξ‚ξ‚„βˆ’1(2.8) for any πœ‰,πœ‚βˆˆπ‘… and πœ‰ξ…ž,πœ‚ξ…ž,πœβˆˆπ‘…π‘.

Let 𝒦=π‘₯∈𝐿2ξ€·π‘„π‘‡ξ€Έβˆ£β€–π‘₯‖𝐿2𝑄𝑇≀𝑅,(2.9) and ̃𝑛,Μƒπ‘βˆˆπ’¦. It is obvious that 𝒦 is a closed convex set and weakly compact in 𝐿2(𝑄𝑇).

The theory of linear elliptic boundary value problems [15] gives a unique πœ“ such that βˆ’βˆ‡β‹…(βˆ‡πœ“)=Μƒπ‘π‘˜βˆ’Μƒπ‘›π‘˜+𝐢(π‘₯),(2.10)πœ“βˆ£Ξ£π·=πœ“,πœ•πœ“||||πœ•πœ‚Ξ£π‘=0,(2.11)β€–βˆ‡πœ“β€–πΏ2(𝑄𝑇)+β€–πœ“β€–πΏβˆž(𝑄𝑇)≀𝐢,(2.12) where 𝐢 is dependent on π‘˜,𝑄𝑇 and the 𝐿∞ norms for 𝐢(π‘₯) and πœ“, but not on 𝑅.

Next, for the unique weak solution πœ“ to problem (2.10)–(2.11), we consider the following problem: π‘›π‘‘βˆ’βˆ‡β‹…ξ‚€ξ‚€π›ΎΜƒπ‘›π‘˜π›Ύβˆ’1+πœ€βˆ‡π‘›βˆ’Μƒπ‘›π‘˜ξ‚ξ€·βˆ‡πœ“=π‘Ÿ(̃𝑛,̃𝑝)1βˆ’π‘›Μƒπ‘π‘˜ξ€Έπ‘+β„Ž(̃𝑛,̃𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“),(2.13)π‘‘βˆ’βˆ‡β‹…ξ‚€ξ‚€π›ΎΜƒπ‘π‘˜π›Ύβˆ’1+πœ€βˆ‡π‘+Μƒπ‘π‘˜ξ‚ξ€·βˆ‡πœ“=π‘Ÿ(̃𝑛,̃𝑝)1βˆ’Μƒπ‘›π‘˜π‘ξ€Έ+β„Ž(̃𝑛,̃𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“),(2.14)(𝑛,𝑝)∣Σ𝐷=𝑛,𝑝,ξ‚΅πœ•π‘›,πœ•πœ‚πœ•π‘ξ‚Ά||||πœ•πœ‚Ξ£π‘ξ€·π‘›=(0,0),(2.15)(𝑛,𝑝)=0,𝑝0ξ€Έ,π‘₯∈Ω,𝑑=0.(2.16)

Lemma 2.2. Under hypotheses (H1)–(H6), there exists one global and unique weak solution to the problem (2.13)–(2.16).
Further there are bounds on ‖𝑛‖𝐿2(0,𝑇;𝐻1(Ξ©)) and ‖𝑛𝑑‖𝐿2(0,𝑇;π‘Œβˆ—) which depend on πœ€,π‘˜,𝜏,𝑄𝑇, and the known data, but not on 𝑅. Similar estimates also hold for 𝑝.

Proof. We begin by choosing a constant 𝜌 such that ξ€½π›ΌπœŒβ‰₯max10,𝛼20ξ€Ύ2𝛾2π‘˜2π›Ύβˆ’2𝛼2min11,𝛼21ξ€Ύ(2.17) and observe that (𝑛,𝑝) satisfies (2.13)–(2.16) if and only if (π‘ˆ,𝑉)=(π‘’βˆ’πœŒπ‘‘π‘›,π‘’βˆ’πœŒπ‘‘π‘) satisfies π‘ˆπ‘‘ξ€·π›Όβˆ’βˆ‡β‹…11βˆ‡π‘ˆβˆ’π‘’βˆ’πœŒπ‘‘Μƒπ‘›π‘˜ξ€Έβˆ‡πœ“+𝛼12π‘ˆ=π‘’βˆ’πœŒπ‘‘π»ξ€·π‘’πœŒπ‘‘βˆ‡π‘ˆ,π‘’πœŒπ‘‘ξ€Έβˆ‡π‘‰+π‘’βˆ’πœŒπ‘‘π‘Ÿπ‘‰(̃𝑛,̃𝑝),(2.18)π‘‘ξ€·π›Όβˆ’βˆ‡β‹…21βˆ‡π‘‰+π‘’βˆ’πœŒπ‘‘Μƒπ‘π‘˜ξ€Έβˆ‡πœ“+𝛼22𝑉=π‘’βˆ’πœŒπ‘‘π»ξ€·π‘’πœŒπ‘‘βˆ‡π‘ˆ,π‘’πœŒπ‘‘ξ€Έβˆ‡π‘‰+π‘’βˆ’πœŒπ‘‘π‘Ÿ(̃𝑛,̃𝑝),(2.19)(π‘ˆ,𝑉)∣Σ𝐷=ξ€·π‘’βˆ’πœŒπ‘‘π‘›,π‘’βˆ’πœŒπ‘‘π‘ξ€Έ,ξ‚΅πœ•π‘ˆ,πœ•πœ‚πœ•π‘‰ξ‚Ά||||πœ•πœ‚Ξ£π‘=𝑛(0,0),(2.20)(π‘ˆ,𝑉)=0,𝑝0ξ€Έ,π‘₯∈Ω,𝑑=0,(2.21) or if and only if (𝑒,𝑣)=(π‘ˆβˆ’π‘’βˆ’πœŒπ‘‘π‘›,π‘‰βˆ’π‘’βˆ’πœŒπ‘‘π‘) satisfies π‘’π‘‘ξ€·π›Όβˆ’βˆ‡β‹…11ξ€Έβˆ‡π‘’+𝛼12𝑒=π‘’βˆ’πœŒπ‘‘π»ξ€·π‘’πœŒπ‘‘βˆ‡π‘’+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡π‘£+βˆ‡π‘ξ€Έ+𝐹1𝑣,(2.22)π‘‘ξ€·π›Όβˆ’βˆ‡β‹…21ξ€Έβˆ‡π‘£+𝛼22𝑣=π‘’βˆ’πœŒπ‘‘π»ξ€·π‘’πœŒπ‘‘βˆ‡π‘’+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡π‘£+βˆ‡π‘ξ€Έ+𝐹2,(2.23)(𝑒,𝑣)∣Σ𝐷=ξ‚΅(0,0),πœ•π‘’,πœ•πœ‚πœ•π‘£ξ‚Ά||||πœ•πœ‚Ξ£π‘=ξ‚΅βˆ’π‘’βˆ’πœŒπ‘‘πœ•π‘›πœ•πœ‚,βˆ’π‘’βˆ’πœŒπ‘‘πœ•π‘ξ‚Άξ€·π‘›πœ•πœ‚,(2.24)(𝑒,𝑣)=0βˆ’π‘›,𝑝0βˆ’π‘ξ€Έ,π‘₯∈Ω,𝑑=0.(2.25) where 𝛼𝑖𝑗2Γ—2=ξƒ©π›ΎΜƒπ‘›π‘˜π›Ύβˆ’1+πœ€πœŒ+π‘Ÿ(̃𝑛,̃𝑝)Μƒπ‘π‘˜π›ΎΜƒπ‘π‘˜π›Ύβˆ’1+πœ€πœŒ+π‘Ÿ(̃𝑛,̃𝑝)Μƒπ‘›π‘˜ξƒͺ,𝐻𝐹(βˆ‡π‘’,βˆ‡π‘£)=β„Ž(̃𝑛,̃𝑝,βˆ‡π‘’,βˆ‡π‘£,βˆ‡πœ“),1𝐹2ξ‚Ά=ξƒ©π‘’βˆ’πœŒπ‘‘ξ€·ξ€·π›Όβˆ‡β‹…11βˆ‡π‘›βˆ’Μƒπ‘›π‘˜ξ€Έ+ξ€·βˆ‡πœ“πœŒβˆ’π›Ό12𝑒𝑛+π‘Ÿ(̃𝑛,̃𝑝)βˆ’πœŒπ‘‘ξ€·ξ€·π›Όβˆ‡β‹…21βˆ‡π‘+Μƒπ‘π‘˜ξ€Έ+ξ€·βˆ‡πœ“πœŒβˆ’π›Ό22ξ€Έξ€Έξƒͺ.𝑝+π‘Ÿ(̃𝑛,̃𝑝)(2.26) Clearly, (𝐹1,𝐹2)∈(𝐿2(0,𝑇;π‘Œβˆ—))2.
First, we prove the uniqueness of weak solution to the problem (2.13)–(2.16) which is equivalent to (2.18)–(2.21). Let (π‘ˆπ‘–,𝑉𝑖),𝑖=1,2 be two weak solutions to the problem (2.18)–(2.21), then (𝑁,𝑃)=(π‘ˆ1βˆ’π‘ˆ2,𝑉1βˆ’π‘‰2) satisfies
π‘π‘‘ξ€·π›Όβˆ’βˆ‡β‹…11ξ€Έβˆ‡π‘+𝛼12𝑁=π‘’βˆ’πœŒπ‘‘πΊξ€·βˆ‡π‘ˆ1,βˆ‡π‘ˆ2,βˆ‡π‘‰1,βˆ‡π‘‰2𝑃,(2.27)π‘‘ξ€·π›Όβˆ’βˆ‡β‹…21ξ€Έβˆ‡π‘ƒ+𝛼22𝑃=π‘’βˆ’πœŒπ‘‘πΊξ€·βˆ‡π‘ˆ1,βˆ‡π‘ˆ2,βˆ‡π‘‰1,βˆ‡π‘‰2ξ€Έ,(2.28)(𝑁,𝑃)∣Σ𝐷=ξ‚΅(0,0),πœ•π‘,πœ•πœ‚πœ•π‘ƒξ‚Ά||||πœ•πœ‚Ξ£π‘=(0,0),(2.29)(π‘ˆ,𝑉)=(0,0),π‘₯∈Ω,𝑑=0,(2.30) here πΊξ€·βˆ‡π‘ˆ1,βˆ‡π‘ˆ2,βˆ‡π‘‰1,βˆ‡π‘‰2𝑒=π»πœŒπ‘‘βˆ‡π‘ˆ1,π‘’πœŒπ‘‘βˆ‡π‘‰1ξ€Έξ€·π‘’βˆ’π»πœŒπ‘‘βˆ‡π‘ˆ2,π‘’πœŒπ‘‘βˆ‡π‘‰2ξ€Έ(2.31) such that πΊξ€·βˆ‡π‘ˆ1,βˆ‡π‘ˆ2,βˆ‡π‘‰1,βˆ‡π‘‰2≀𝛼1||||(βˆ‡πœ“)π›ΎΜƒπ‘›π‘˜π›Ύβˆ’1π‘’πœŒπ‘‘βˆ‡π‘ˆ1βˆ’Μƒπ‘›π‘˜||βˆ’||βˆ‡πœ“π›ΎΜƒπ‘›π‘˜π›Ύβˆ’1π‘’πœŒπ‘‘βˆ‡π‘ˆ2βˆ’Μƒπ‘›π‘˜||||βˆ‡πœ“+𝛼2||||(βˆ‡πœ“)π›ΎΜƒπ‘π‘˜π›Ύβˆ’1π‘’πœŒπ‘‘βˆ‡π‘‰1+Μƒπ‘π‘˜||βˆ’||βˆ‡πœ“π›ΎΜƒπ‘π‘˜π›Ύβˆ’1π‘’πœŒπ‘‘βˆ‡π‘‰2+Μƒπ‘π‘˜||||ξ€½π›Όβˆ‡πœ“β‰€max10,𝛼20ξ€Ύπ›Ύπ‘˜π›Ύβˆ’1π‘’πœŒπ‘‘ξ€·||||+||||ξ€Έ.βˆ‡π‘βˆ‡π‘ƒ(2.32) Take 𝑁,𝑃 as test functions in (2.27), (2.28), respectively. By (2.32) and HΓΆlder inequality, 12dξ€œd𝑑Ω||||𝑁(𝑑)2+||||𝑃(𝑑)2𝛼+min11,𝛼21ξ€Ύξ€œΞ©ξ‚€||||βˆ‡π‘2+||||βˆ‡π‘ƒ2ξ‚ξ€œ+πœŒΞ©ξ€·π‘2+𝑃2𝛼≀max10,𝛼20ξ€Ύπ›Ύπ‘˜π›Ύβˆ’1ξ€œΞ©ξ€·||||+||||||𝑁||+||𝑃||≀1βˆ‡π‘βˆ‡π‘ƒξ€Έξ€·2𝛼min11,𝛼21ξ€Ύξ€œΞ©ξ‚€||||βˆ‡π‘2+||||βˆ‡π‘ƒ2+𝛼max10,𝛼20ξ€Ύ2𝛾2π‘˜2π›Ύβˆ’2𝛼2min11,𝛼21ξ€Ύξ€œΞ©ξ€·π‘2+𝑃2ξ€Έ.(2.33) Thus the uniqueness is established by Gronwall's inequality.
We are now in a position to prove the existence result. Define π‘Š=𝐿2(0,𝑇;π‘Œ), 𝒱=π‘ŠΓ—π‘Š. Clearly, 𝒱 is a Hilbert space with respect to the scalar product
ξ€œ((𝑒,𝑣),(πœ‰,πœ‚))=𝑇0ξ€œΞ©(βˆ‡π‘’β‹…βˆ‡πœ‰+βˆ‡π‘£β‹…βˆ‡πœ‚).(2.34) Set, for (𝑒,𝑣),(πœ‰,πœ‚)βˆˆπ’±, ξ€œβŸ¨π’œ(𝑒,𝑣),(πœ‰,πœ‚)⟩=𝑇0ξ€œΞ©ξ€·π›Ό11βˆ‡π‘’β‹…βˆ‡πœ‰+𝛼21ξ€Έ+ξ€œβˆ‡π‘£β‹…βˆ‡πœ‚π‘‡0ξ€œΞ©ξ€·π›Ό12π‘’πœ‰+𝛼22ξ€Έ+ξ€œπ‘£πœ‚π‘‡0ξ€œΞ©π‘’βˆ’πœŒπ‘‘π»ξ€·π‘’πœŒπ‘‘βˆ‡π‘’+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡π‘£+βˆ‡π‘ξ€Έ(πœ‰+πœ‚).(2.35) The operator π’œβˆΆπ’±β†’π’±βˆ— is well defined and bounded (because 0≀𝐻≀1/𝜏). To prove the existence result by using [16, Theorem 30.A], it suffices to verify that the operator π’œ is hemicontinuous, monotone, and coercive.
Note that
π‘’βˆ’πœŒπ‘‘||π»ξ€·π‘’πœŒπ‘‘βˆ‡(𝑒+πœ†πœ‘)+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡(𝑣+πœ†πœ™)+βˆ‡π‘ξ€Έξ€·π‘’βˆ’π»πœŒπ‘‘βˆ‡π‘’+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡π‘£+βˆ‡π‘ξ€Έ||𝛼≀max10,𝛼20ξ€Ύπœ†π›Ύπ‘˜π›Ύβˆ’1ξ€·||||+||||ξ€Έβˆ‡πœ‘βˆ‡πœ™(2.36) for any (𝑒,𝑣),(πœ‘,πœ™)βˆˆπ’±. The hemicontinuity of π’œ is easily obtained by the standard method.
For the monotone, we first notice that
ξ€œπ‘‡0ξ€œ0π‘₯0200π‘‘Ξ©ξ€Ίπ»ξ€·π‘’πœŒπ‘‘βˆ‡π‘’1+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡π‘£1+βˆ‡π‘ξ€Έξ€·π‘’βˆ’π»πœŒπ‘‘βˆ‡π‘’2+βˆ‡π‘›,π‘’πœŒπ‘‘βˆ‡π‘£2+βˆ‡π‘π‘’ξ€Έξ€»ξ€Ίξ€·1βˆ’π‘’2ξ€Έ+𝑣1βˆ’π‘£2𝛼β‰₯βˆ’max10,𝛼20ξ€Ύπ›Ύπ‘˜π›Ύβˆ’1ξ€œπ‘‡0ξ€œΞ©ξ€·||βˆ‡ξ€·π‘’1βˆ’π‘’2ξ€Έ||+||βˆ‡ξ€·π‘£1βˆ’π‘£2ξ€Έ||||𝑒1βˆ’π‘’2||+||𝑣1βˆ’π‘£2||𝛼β‰₯βˆ’min11,𝛼21ξ€Ύ2ξ€œπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑒1βˆ’π‘’2)||2+||βˆ‡(𝑣1βˆ’π‘£2)||2ξ‚βˆ’ξ€½π›Όmax10,𝛼20ξ€Ύ2𝛾2π‘˜2π›Ύβˆ’2𝛼2min11,𝛼21ξ€Ύξ€œπ‘‡0ξ€œΞ©ξ‚ƒξ€·π‘’1βˆ’π‘’2ξ€Έ2+𝑣1βˆ’π‘£2ξ€Έ2ξ‚„.(2.37) Hence ξ«π’œξ€·π‘’1,𝑣1ξ€Έξ€·π‘’βˆ’π’œ2,𝑣2ξ€Έ,𝑒1βˆ’π‘’2,𝑣1βˆ’π‘£2β‰₯𝛼min11,𝛼21ξ€Ύ2ξ€œπ‘‡0ξ€œ0π‘₯0200𝑑Ω||βˆ‡(𝑒1βˆ’π‘’2)||2+||βˆ‡(𝑣1βˆ’π‘£2)||2+ξƒ©ξ€½π›ΌπœŒβˆ’max10,𝛼20ξ€Ύ2𝛾2π‘˜2π›Ύβˆ’2𝛼2min11,𝛼21ξ€Ύξƒͺξ€œπ‘‡0ξ€œΞ©ξ‚ƒξ€·π‘’1βˆ’π‘’2ξ€Έ2+𝑣1βˆ’π‘£2ξ€Έ2ξ‚„.(2.38) By the choice of 𝜌, we can easily obtain the monotone of π’œ. Moreover, from (2.38) we also know that the operator π’œ is coercive.
Therefore, there exists a unique (𝑒,𝑣)βˆˆπ’± with (𝑒𝑑,𝑣𝑑)βˆˆπ’±βˆ— such that (𝑒0,𝑣0)=(𝑛0βˆ’π‘›,𝑝0βˆ’π‘) in Ξ© and
𝑒𝑑,𝑣𝑑=𝐹+π’œ(𝑒,𝑣),(πœ‰,πœ‚)1,𝐹2ξ€Έ,(πœ‰,πœ‚),βˆ€(πœ‰,πœ‚)βˆˆπ’±.(2.39) Especially, βŸ¨ξ€·π‘’π‘‘,𝑣𝑑𝐹+π’œ(𝑒,𝑣),(πœ‰,0)⟩=⟨1,𝐹2ξ€Έ,βŸ¨ξ€·π‘’(πœ‰,0)⟩,𝑑,𝑣𝑑𝐹+π’œ(𝑒,𝑣),(0,πœ‚)⟩=⟨1,𝐹2ξ€Έ,(0,πœ‚)⟩.(2.40) That is, (𝑒,𝑣) is a weak solution to the problem (2.22)–(2.25).
Finally, noting that 0β‰€β„Žβ‰€1/𝜏 we can easily establish the bounds on β€–βˆ‡π‘’β€–πΏ2(0,𝑇;𝐻1(Ξ©)) and ‖𝑒𝑑‖𝐿2(0,𝑇;π‘Œβˆ—) by the standard energy estimate.

Lemma 2.3. Under hypotheses (H1)–(H6), there exists at least one global weak solution to the problem (2.5)–(2.7), (1.4)–(1.6).

Proof. We define the mapping 𝑆 as π‘†βˆΆπ’¦2βŸΆξ€·πΏ2𝑄𝑇2,(̃𝑛,̃𝑝)⟼(𝑛,𝑝)(2.41) with (𝑛,𝑝) solution of (2.13)–(2.16). From Lemma 2.2, we know that 𝑆 is well defined and compact. Indeed, (𝑛,𝑝) lies in a bounded sunsets of (𝐿2(0,𝑇;𝐻1(Ξ©)))2, and (𝑛𝑑,𝑝𝑑) lies in a bounded subset of (𝐿2(0,𝑇;π‘Œβˆ—))2. Since the injection 𝐻1(Ξ©)β†ͺ𝐿2(Ξ©) is compact, we conclude from Aubin's lemma that 𝑆 is relatively compact in (𝐿2(𝑄𝑇))2. And for given 𝜏 and π‘˜, 𝑆(𝒦2)β†ͺ𝒦2 holds if we choose 𝑅 large enough.
To apply Schauder's fixed point theorem, we still need to prove that the mapping 𝑆 is continuous. Consider any sequence (̃𝑛𝑗,̃𝑝𝑗)βŠ‚π’¦2β†’(̃𝑛,̃𝑝) strongly in (𝐿2(𝑄𝑇))2 and let 𝑆(̃𝑛𝑗,̃𝑝𝑗)=(𝑛𝑗,𝑝𝑗). Since 𝑆(𝒦2) is relatively compact in (𝐿2(𝑄𝑇))2 and bounded in (𝐿2(0,𝑇;𝐻1(Ξ©)))2, we can extract subsequences such that
𝑛𝑗𝑑,ξ€·π‘π‘—ξ€Έπ‘‘ξ€ΈβŸΆξ€·π‘›π‘‘,𝑝𝑑weaklyin𝐿2ξ€·0,𝑇;π‘Œβˆ—ξ€Έ,π‘›π‘—βŸΆπ‘›,π‘π‘—βŸΆπ‘stronglyin𝐿2𝑄𝑇,π‘›π‘—βŸΆπ‘›,π‘π‘—βŸΆπ‘weaklyin𝐿2ξ€·0,𝑇;𝐻1ξ€Έ,𝑛(Ξ©)π‘—βŸΆπ‘›,π‘π‘—βŸΆπ‘a.e.in𝑄𝑇.(2.42) We only have to show 𝑆(̃𝑛,̃𝑝)=(𝑛,𝑝). To do this, we only need to prove ξ€œπ‘‡0ξ€œ0π‘₯0200π‘‘Ξ©β„Žξ€·Μƒπ‘›π‘—,̃𝑝𝑗,βˆ‡π‘›π‘—,βˆ‡π‘π‘—,βˆ‡πœ“π‘—ξ€Έξ€œβ‹…πœβŸΆπ‘‡0ξ€œΞ©β„Ž(̃𝑛,̃𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“)β‹…πœ,(2.43) where 𝜁∈𝐿2(0,𝑇;π‘Œ) is a test function, and πœ“π‘—,πœ“ are solutions of (2.10)–(2.11) corresponding to (̃𝑛𝑗,̃𝑝𝑗),(̃𝑛,̃𝑝), respectively. The reminder of convergence proof is standard (details see [8] or [9]). Use π‘›π‘—βˆ’π‘›βˆˆπΏ2(0,𝑇;π‘Œ) as test function in a modification of (2.13) in which the functions ̃𝑛,̃𝑝 have been replaced by ̃𝑛𝑗,̃𝑝𝑗, respectively. Then we have πœ€ξ€œπ‘‡0ξ€œΞ©||βˆ‡(𝑛𝑗||βˆ’π‘›)2ξ€œβ‰€βˆ’π‘‡0𝑛𝑑,π‘›π‘—ξ¬βˆ’π‘›π‘Œβˆ—,π‘Œξ€œβˆ’πœ€π‘‡0ξ€œΞ©ξ€·π‘›βˆ‡π‘›β‹…βˆ‡π‘—ξ€Έ+ξ€œβˆ’π‘›π‘‡0ξ€œΞ©ξ€·Μƒπ‘›π‘—ξ€Έπ‘˜βˆ‡πœ“π‘—ξ€·π‘›β‹…βˆ‡π‘—ξ€Έ+ξ€œβˆ’π‘›π‘‡0ξ€œΞ©π‘Ÿξ€·Μƒπ‘›π‘—,̃𝑝𝑗1βˆ’π‘›π‘—ξ€·Μƒπ‘π‘—ξ€Έπ‘˜π‘›ξ€Έξ€·π‘—ξ€Έξ€½π›Όβˆ’π‘›+max01,𝛼02ξ€Ύξ€½maxπ›Ύπ‘˜π›Ύβˆ’1ξ€Ύξ€œ,2π‘˜π‘‡0ξ€œΞ©ξ€·||βˆ‡π‘›π‘—||+||βˆ‡π‘π‘—||+||βˆ‡πœ“π‘—||π‘›ξ€Έξ€·π‘—ξ€Έξ€œβˆ’π‘›β‰€βˆ’π‘‡0𝑛𝑑,π‘›π‘—ξ¬βˆ’π‘›π‘Œβˆ—,π‘Œξ€œβˆ’πœ€π‘‡0ξ€œ0π‘₯0200π‘‘Ξ©ξ€·π‘›βˆ‡π‘›β‹…βˆ‡π‘—ξ€Έ+πœ€βˆ’π‘›4ξ‚€β€–β€–βˆ‡ξ€·π‘›π‘—ξ€Έβ€–β€–βˆ’π‘›2𝐿2𝑄𝑇+β€–β€–βˆ‡ξ€·π‘π‘—ξ€Έβ€–β€–βˆ’π‘2𝐿2ξ€·π‘„π‘‡ξ€Έξ‚ξ€œ+𝐢𝑇0ξ€œΞ©ξ€·||𝑛1+𝑗||ξ€Έ||𝑛𝑗||ξƒ©β€–β€–βˆ‡ξ€·πœ“βˆ’π‘›+𝐢(πœ€)π‘—ξ€Έβ€–β€–βˆ’πœ“2𝐿2(𝑄𝑇)+||||ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›βˆ‡πœ“β‹…βˆ‡π‘—ξ€Έ||||+β€–β€–π‘›βˆ’π‘›π‘—β€–β€–βˆ’π‘›2𝐿2(𝑄𝑇)ξƒͺξ€œ+𝐢(πœ€)𝑇0ξ€œΞ©ξ€·||||+||||+||||ξ€Έ||π‘›βˆ‡π‘›βˆ‡π‘βˆ‡πœ“π‘—||.βˆ’π‘›(2.44) A similar estimate holds for πœ€βˆ«π‘‡0∫Ω|βˆ‡(π‘π‘—βˆ’π‘)|2. Then adding the two inequalities and using βˆ‡πœ“π‘—β†’βˆ‡πœ“ strongly in 𝐿2(𝑄𝑇) (details see [9]) and (2.42), we conclude that β€–βˆ‡(π‘›π‘—βˆ’π‘›)‖𝐿2(𝑄𝑇)+β€–βˆ‡(π‘π‘—βˆ’π‘)‖𝐿2(𝑄𝑇)β†’0. This implies that (βˆ‡π‘›π‘—,βˆ‡π‘π‘—)β†’(βˆ‡π‘›,βˆ‡π‘) a.e. in 𝑄𝑇. Then we can easily prove (2.43) by using Vitali's theorem.
Now existence of a fixed point of 𝑆 follows which is a solution of (2.5)–(2.7), (1.4)–(1.6).

To obtain the existence result of problem (2.2)–(2.4), (1.4)–(1.6), the following 𝐿∞ estimates on 𝑛,𝑝,πœ“ uniformly in πœ€ are necessary.

Lemma 2.4. The solutions of problem (2.5)–(2.7), (1.4)–(1.6) satisfy the estimates 0≀𝑛(π‘₯,𝑑),𝑝(π‘₯,𝑑)≀𝐢,a.e.(π‘₯,𝑑)βˆˆπ‘„π‘‡,(2.45) where 𝐢 is dependent on 𝜏,𝑄𝑇 and the known data, but not on πœ€.

Proof. By taking π‘›βˆ’=min{𝑛,0}∈𝐿2(0,𝑇;π‘Œ) as test function in (2.6), we have 12ξ€œΞ©π‘›βˆ’(𝑑)2ξ€œ+πœ€π‘‡0ξ€œ0π‘₯0200𝑑Ω||βˆ‡π‘›βˆ’||2β‰€ξ€œπ‘‡0ξ€œΞ©π‘›π‘˜βˆ‡πœ“β‹…βˆ‡π‘›βˆ’βˆ’ξ€œπ‘‡0ξ€œΞ©π‘Ÿ(𝑛,𝑝)π‘π‘˜π‘›βˆ’2+ξ€œπ‘‡0ξ€œΞ©(π‘Ÿ(𝑛,𝑝)+β„Ž)π‘›βˆ’.(2.46) By taking into account π‘›π‘˜=0 in {𝑛≀0} and the nonnegativity of π‘Ÿ and β„Ž we obtain 12ξ€œΞ©π‘›βˆ’(𝑑)2≀0,(2.47) and thus 𝑛(π‘₯,𝑑)β‰₯0 a.e. in 𝑄𝑇. Similarly, we have 𝑝(π‘₯,𝑑)β‰₯0 a.e. in 𝑄𝑇.
To obtain the upper bound set
‖‖𝑀=maxπ‘›β€–β€–πΏβˆž(Ξ©),‖‖𝑛0β€–β€–πΏβˆž(Ξ©),β€–β€–π‘β€–β€–πΏβˆž(Ξ©),‖‖𝑝0β€–β€–πΏβˆž(Ξ©)ξ€Ύ,π‘˜β‰₯𝑀(2.48) and use (π‘›π‘˜βˆ’π‘€)+π‘ž as test function in (2.6), then 1ξ€œπ‘ž+1Ξ©ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)ξ€œ+πœ€π‘‡0ξ€œΞ©π‘žξ€·π‘›π‘˜ξ€Έβˆ’π‘€+(π‘žβˆ’1)||βˆ‡ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+||2β‰€ξ€œπ‘‡0ξ€œΞ©ξ‚΅π‘žβˆ‡πœ“β‹…βˆ‡ξ€·π‘›π‘ž+1π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)𝑛+π‘€π‘˜ξ€Έβˆ’π‘€+π‘žξ‚Ά+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘Ÿξ€·(𝑛,𝑝)1βˆ’π‘›π‘π‘˜ξ€Έπ‘›+β„Žξ€Έξ€·π‘˜ξ€Έβˆ’π‘€+π‘žβ‰€ξ€œπ‘‡0ξ€œΞ©ξ€·π‘π‘˜βˆ’π‘›π‘˜ξ€Έξ‚΅π‘ž+𝐢(π‘₯)ξ€·π‘›π‘ž+1π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)𝑛+π‘€π‘˜ξ€Έβˆ’π‘€+π‘žξ‚Ά+ξ€œπ‘‡0ξ€œΞ©ξ‚€1π‘Ÿ+πœξ‚ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+π‘ž.(2.49) Adding the equality (2.49) for 𝑛 and a similar inequality for 𝑝, we get 1ξ€œπ‘ž+1Ξ©ξ‚€ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)+ξ€·π‘π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)ξ‚β‰€π‘žξ€œπ‘ž+1𝑇0ξ€œΞ©ξ€·π‘π‘˜βˆ’π‘›π‘˜ξ€Έξ‚€ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)βˆ’ξ€·π‘π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)ξ‚ξ€œ+𝑀𝑇0ξ€œΞ©ξ€·π‘π‘˜βˆ’π‘›π‘˜ξ€Έξ‚€ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+π‘žβˆ’ξ€·π‘π‘˜ξ€Έβˆ’π‘€+π‘žξ‚+π‘žξ€œπ‘ž+1𝑇0ξ€œΞ©πΆξ‚€ξ€·π‘›(π‘₯)π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)βˆ’ξ€·π‘π‘˜ξ€Έβˆ’π‘€+(π‘ž+1)+ξ€œπ‘‡0ξ€œΞ©ξ‚€1π‘Ÿ+πœξ€·π‘›+𝑀𝐢(π‘₯)ξ‚ξ‚€π‘˜ξ€Έβˆ’π‘€+π‘žβˆ’ξ€·π‘π‘˜ξ€Έβˆ’π‘€+π‘žξ‚.(2.50) Noticing that ξ€·π‘π‘˜βˆ’π‘›π‘˜ξ€Έξ‚€ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+πœƒβˆ’ξ€·π‘π‘˜ξ€Έβˆ’π‘€+πœƒξ‚β‰€0,βˆ€πœƒ>1(2.51) and applying HΓΆlder inequality, we further have β€–β€–ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+β€–β€–πΏπ‘ž+1π‘ž+1(Ξ©)+β€–β€–ξ€·π‘π‘˜ξ€Έβˆ’π‘€+β€–β€–πΏπ‘ž+1π‘ž+1(Ξ©)ξ€œβ‰€πΆ(π‘ž+1)𝑇0ξ‚€β€–β€–ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+β€–β€–πΏπ‘ž+1π‘ž+1(Ξ©)+β€–β€–ξ€·π‘π‘˜ξ€Έβˆ’π‘€+β€–β€–πΏπ‘ž+1π‘ž+1(Ξ©)+𝐢,(2.52) where 𝐢 is independent of πœ€ and π‘ž. Gronwall's inequality then implies that β€–β€–ξ€·π‘›π‘˜ξ€Έβˆ’π‘€+β€–β€–πΏπ‘ž+1(Ξ©)+β€–β€–ξ€·π‘π‘˜ξ€Έβˆ’π‘€+β€–β€–πΏπ‘ž+1(Ξ©)≀𝐢𝑒𝐢𝑑(2.53) for all π‘žβ‰₯1 and π‘˜β‰₯𝑀. Since the right-hand side of this equality does not depend on π‘˜, we can let π‘˜β†’βˆž and then π‘žβ†’βˆž to obtain the desired upper bound.

Thus, taking π‘˜ large enough, we see that (πœ“πœ€,π‘›πœ€,π‘πœ€) solves ξ€·βˆ’βˆ‡β‹…βˆ‡πœ“πœ€ξ€Έ=π‘πœ€βˆ’π‘›πœ€ξ€·π‘›+𝐢(π‘₯),(2.54)πœ€ξ€Έπ‘‘βˆ’βˆ‡β‹…ξ‚€ξ‚€π›Ύπ‘›πœ€π›Ύβˆ’1+πœ€βˆ‡π‘›πœ€βˆ’π‘›πœ€βˆ‡πœ“πœ€ξ‚ξ€·π‘›=π‘Ÿπœ€,π‘πœ€ξ€Έξ€·1βˆ’π‘›πœ€π‘πœ€ξ€Έ+π‘”πœξ€·π‘›πœ€,π‘πœ€,βˆ‡π‘›πœ€,βˆ‡π‘πœ€,βˆ‡πœ“πœ€ξ€Έξ€·π‘,(2.55)πœ€ξ€Έπ‘‘βˆ’βˆ‡β‹…ξ‚€ξ‚€π›Ύπ‘πœ€π›Ύβˆ’1+πœ€βˆ‡π‘πœ€+π‘πœ€ξ‚ξ€·π‘›βˆ‡πœ“=π‘Ÿπœ€,π‘πœ€ξ€Έξ€·1βˆ’π‘›πœ€π‘πœ€ξ€Έ+π‘”πœξ€·π‘›πœ€,π‘πœ€,βˆ‡π‘›πœ€,βˆ‡π‘πœ€,βˆ‡πœ“πœ€ξ€Έ(2.56) subject to the initial and boundary conditions (1.4)–(1.6).

Proof of Theorem 2.1. Noticing that the function π‘”πœ is bound, we can obtain the following convergence properties by using the same method as the proof in [8, Theorem 1.1] and Lemma 3.2: ξ‚€π‘›πœ€(π›Ύβˆ’1)/2,π‘πœ€(π›Ύβˆ’1)/2ξ‚βŸΆξ€·π‘›(π›Ύβˆ’1)/2,𝑝(π›Ύβˆ’1)/2ξ€Έstronglyin𝐿2𝑄𝑇,ξ€·π‘›πœ€,π‘πœ€ξ€ΈβŸΆ(𝑛,𝑝),a.e.in𝑄𝑇,ξ€·βˆ‡π‘›π›Ύπœ€,βˆ‡π‘π›Ύπœ€ξ€ΈβŸΆ(βˆ‡π‘›π›Ύ,βˆ‡π‘π›Ύ)weaklyin𝐿2ξ€·0,𝑇;𝐻1ξ€Έ,𝑛(Ξ©)ξ€·ξ€·πœ€ξ€Έπ‘‘,ξ€·π‘πœ€ξ€Έπ‘‘,ξ€ΈβŸΆξ€·π‘›π‘‘,𝑝𝑑weaklyin𝐿2ξ€·0,𝑇;π‘Œβˆ—0ξ€Έ,πœ“πœ€βŸΆπœ“weaklyinπΏβˆžξ€·0,𝑇;𝐻1ξ€Έ,𝑛(Ξ©)πœ€βˆ‡πœ“πœ€,π‘πœ€βˆ‡πœ“πœ€ξ€ΈβŸΆ(π‘›βˆ‡πœ“,π‘βˆ‡πœ“)weaklyin𝐿2𝑄𝑇.(2.57) In addition, a standard elliptic estimate gives β€–β€–βˆ‡ξ€·πœ“πœ€ξ€Έβ€–β€–βˆ’πœ“πΏ2(𝑄𝑇)β‰€β€–β€–ξ€·π‘πœ€ξ€Έβˆ’ξ€·π‘›βˆ’π‘πœ€ξ€Έβ€–β€–βˆ’π‘›πΏ2(𝑄𝑇),(2.58) from which we conclude βˆ‡πœ“πœ€βŸΆβˆ‡πœ“stronglyin𝐿2𝑄𝑇,(2.59) and furthermore βˆ‡πœ“πœ€βŸΆβˆ‡πœ“a.e.in𝑄𝑇.(2.60) Next, using (π‘›π›Ύπœ€βˆ’π‘›π›Ύ) as test function in (2.55), we get ξ€œπ‘‡0ξ€œΞ©||βˆ‡(π‘›π›Ύπœ€βˆ’π‘›π›Ύ)||2ξ€œβ‰€βˆ’π‘‡0βŸ¨π‘›π‘‘,π‘›π›Ύπœ€βˆ’π‘›π›ΎβŸ©π‘Œβˆ—,π‘Œξ€œ+πœ€π‘‡0ξ€œΞ©βˆ‡π‘›πœ€ξ€·π‘›β‹…βˆ‡π›Ύπœ€βˆ’π‘›π›Ύξ€Έ+ξ€œπ‘‡0ξ€œΞ©βˆ‡π‘›π›Ύξ€·π‘›β‹…βˆ‡π›Ύπœ€βˆ’π‘›π›Ύξ€Έ+ξ€œπ‘‡0ξ€œΞ©π‘›πœ€βˆ‡πœ“πœ€ξ€·π‘›β‹…βˆ‡π›Ύπœ€βˆ’π‘›π›Ύξ€Έξ€½π›Ό+max10,𝛼20ξ€Ύξ€œπ‘‡0ξ€œΞ©ξ€·||βˆ‡π‘›π›Ύπœ€||+||βˆ‡π‘π›Ύπœ€||+ξ€·π‘›πœ€+π‘πœ€ξ€Έ||βˆ‡πœ“πœ€||ξ€Έ||π‘›π›Ύπœ€βˆ’π‘›π›Ύ||⟢0(2.61) as πœ€β†’0, where we have used πœ€ξ€œπ‘‡0ξ€œΞ©βˆ‡π‘›πœ€ξ€·π‘›β‹…βˆ‡π›Ύπœ€βˆ’π‘›π›Ύξ€Έβ‰€πœ€1/2ξ‚΅ξ€œπ‘‡0ξ€œΞ©πœ€||βˆ‡π‘›πœ€||2ξ‚Ά1/2ξ‚΅ξ€œπ‘‡0ξ€œΞ©||βˆ‡ξ€·π‘›π›Ύπœ€βˆ’π‘›π›Ύξ€Έ||2ξ‚Ά1/2β‰€πœ€1/2ξ€œπΆβŸΆ0,𝑇0ξ€œΞ©π‘›πœ€βˆ‡πœ“πœ€ξ€·π‘›β‹…βˆ‡π›Ύπœ€βˆ’π‘›π›Ύξ€Έ=1ξ€œπ›Ύ+1𝑇0ξ€œΞ©ξ€·π‘πœ€βˆ’π‘›πœ€ξ€Έξ‚€π‘›+𝐢(π‘₯)πœ€π›Ύ+1βˆ’π‘›π›Ύ+1ξ‚βˆ’ξ€œπ‘‡0ξ€œΞ©π‘›πœ€βˆ‡πœ“πœ€β‹…βˆ‡π‘›π›Ύ+1ξ€œπ›Ύ+1𝑇0ξ€œΞ©βˆ‡πœ“πœ€β‹…βˆ‡π‘›π›Ύ+1⟢0,(2.62) and (2.57)–(2.60). The same argument shows that β€–βˆ‡π‘π›Ύπœ€βˆ’βˆ‡π‘π›Ύβ€–πΏ2(𝑄𝑇)β†’0 as πœ€β†’0.
Thus, there exists a subsequence (not relabeled) such that βˆ‡π‘›π›Ύπœ€β†’βˆ‡π‘›π›Ύ,βˆ‡π‘π›Ύπœ€β†’βˆ‡π‘π›Ύ almost everywhere in 𝑄𝑇 as πœ€β†’0. Then it follows from Vitali's theorem that
π‘”πœξ€·π‘›πœ€,π‘πœ€,βˆ‡π‘›πœ€,βˆ‡π‘πœ€,βˆ‡πœ“πœ€ξ€ΈβŸΆπ‘”πœ(𝑛,𝑝,βˆ‡π‘›,βˆ‡π‘,βˆ‡πœ“)stronglyin𝐿2𝑄𝑇.(2.63)
Now we can conclude that (πœ“,𝑛,𝑝) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6) from the above convergence by standard method and then complete the proof of Theorem 2.1.

3. Proof of the Main Result

In the last section, we prove that there is at least one global weak solution (πœ“πœ,π‘›πœ,π‘πœ) to the problem (2.2)–(2.4), (1.4)–(1.6) for every given 𝜏. In the following what we need to do is to prove that the limit of (π‘›πœ,π‘πœ,πœ“πœ) is a solution of (1.1)–(1.6). To this end, we first give some uniform estimates for the problem (2.2)–(2.4), (1.4)–(1.6). For simplicity, we drop the subscript 𝜏 of (π‘›πœ,π‘πœ,πœ“πœ) and set 𝛼𝑖=1,𝑖=1,2.

Lemma 3.1. Forallπ‘’βˆˆπ»1(Ξ©), there holds ‖𝑒‖𝐿𝑠(Ξ©)≀𝐢‖𝑒‖𝛼𝐿2(Ξ©)‖𝑒‖𝐻1βˆ’π›Ό1(Ξ©),(3.1) where 𝛼,𝑠 satisfy ξ‚€10<1βˆ’π›Ό=𝑁2βˆ’1𝑠<1,1<𝑠<2π‘π‘βˆ’2.(3.2)

This is the well-known Gagliardo-Nirenberg Inequality [15].

Lemma 3.2. If (πœ“,𝑛,𝑝) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6), the following estimate holds: ξ€œπ‘‡0ξ€œ0π‘₯0200𝑑Ω(𝑛𝛾+𝑝𝛾)||||βˆ‡πœ“2ξ€œβ‰€π›Ώπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚ξƒ¬ξ€œ+𝐢(𝛿)𝑇0ξ‚΅ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έξ‚Άdπ‘₯2ξƒ­+1,βˆ€π›Ύβ‰₯1,(3.3) where 𝛿 is a sufficiently small constant.

Proof. First of all, the following 𝐿∞ estimate of πœ“β€–πœ“(𝑑)β€–πΏβˆž(Ξ©)ξƒ―ξ‚΅ξ€œβ‰€πΆ1+Ξ©(𝑛2(𝑑)+𝑝2ξ‚Ά(𝑑))𝑑π‘₯1/2ξƒ°,βˆ€π‘‘β‰₯0(3.4) can follow from the standard techniques in [17]. Then by taking πœ“βˆ’πœ“ as test function in (2.2), we have ξ€œΞ©||βˆ‡(πœ“βˆ’||πœ“)2+ξ€œΞ©βˆ‡ξ€·πœ“β‹…βˆ‡πœ“βˆ’πœ“ξ€Έ=ξ€œΞ©(ξ€·π‘βˆ’π‘›+𝐢(π‘₯))πœ“βˆ’πœ“ξ€Έ.(3.5) The assumptions we have made and PoincarΓ© inequality yield β€–β€–βˆ‡(πœ“βˆ’β€–β€–πœ“)𝐿2(Ξ©)≀𝐢1+β€–π‘βˆ’π‘›β€–πΏ2(Ξ©)≀𝐢1+‖𝑛‖𝐿2(Ξ©)+‖𝑝‖𝐿2(Ξ©)ξ€Έ.(3.6) Using (π‘›π›Ύβˆ’π‘›π›Ύ)πœ“ as test function for (2.2), and noting that 2<𝛾+1 we get ξ€œπ‘‡0ξ€œΞ©π‘›π›Ύ||||βˆ‡πœ“2=ξ€œπ‘‡0ξ€œΞ©π‘›π›Ύ||||βˆ‡πœ“2+ξ€œπ‘‡0ξ€œΞ©π‘›(π‘βˆ’π‘›+𝐢(π‘₯))ξ€Ίξ€·π›Ύβˆ’π‘›π›Ύξ€Έπœ“ξ€»βˆ’ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›πœ“βˆ‡π›Ύβˆ’π‘›π›Ύξ€Έβ‹…βˆ‡πœ“β‰€πΆβ€–βˆ‡πœ“β€–2𝐿2(𝑄𝑇)+ξ€œπ‘‡0β€–πœ“β€–πΏβˆž(Ξ©)ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0β€–πœ“β€–πΏβˆž(Ξ©)β€–βˆ‡πœ“β€–πΏ2(Ξ©)β€–β€–βˆ‡ξ€·π‘›π›Ύβˆ’π‘›π›Ύξ€Έβ€–β€–πΏ2(Ξ©)ξ€œβ‰€πΆπ‘‡0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έ+ξ€œπ‘‡0ξƒ¬ξ‚΅ξ€œΞ©(𝑛2+𝑝2)ξ‚Ά1/2ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑛𝛾+1ξ€Έξƒ­ξ€œ+𝛿𝑇0ξ€œΞ©||βˆ‡(𝑛𝛾)||2ξƒ¬ξ€œ+𝐢(𝛿)𝑇0ξ‚΅ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έξ‚Ά2ξƒ­ξ€œ+1≀𝛿𝑇0ξ€œΞ©||βˆ‡(𝑛𝛾)||2ξƒ¬ξ€œ+𝐢(𝛿)𝑇0ξ‚΅ξ€œΞ©(𝑛𝛾+1+𝑝𝛾+1)ξ‚Ά2ξƒ­.+1(3.7) A similar estimate for βˆ«π‘‡0βˆ«Ξ©π‘π›Ύ|βˆ‡πœ“|2 follows from a same procedure. Then the proof is completed.

Lemma 3.3. If (πœ“,𝑛,𝑝) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6), there holds that β€–π‘›β€–πΏβˆž(0,𝑇0;𝐿𝛾+1(Ξ©))+β€–βˆ‡(𝑛𝛾)‖𝐿2(𝑄𝑇0)≀𝐢,(3.8)β€–π‘β€–πΏβˆž(0,𝑇0;𝐿𝛾+1(Ξ©))+β€–βˆ‡(𝑝𝛾)‖𝐿2(𝑄𝑇0)≀𝐢,(3.9)β€–πœ“β€–πΏ2(0,𝑇0;𝐻1(Ξ©))+β€–πœ“β€–πΏβˆž(𝑄𝑇0)≀𝐢,(3.10) for some sufficiently small 𝑇0; here positive constant 𝐢 is independent of 𝜏.

Proof. Without loss of generality, we assume 𝑁=3 and the 𝑁=1 or 2 case is easier.Case 1. 2≀𝛾≀5. In this case, from HΓΆlder inequality and (3.3), we have ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έ||||βˆ‡πœ“2ξ‚΅ξ€œβ‰€πΆπ‘‡0ξ€œΞ©(𝑛𝛾+𝑝𝛾)||||βˆ‡πœ“2+ξ€œπ‘‡0ξ€œΞ©||||βˆ‡πœ“2ξ‚Άξ€œβ‰€π›Ώπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚ξƒ¬ξ€œ+𝐢(𝛿)𝑇0ξ‚΅ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έξ‚Άdπ‘₯2ξƒ­.+1(3.11) Noting that 2≀𝛾≀5, we can choose a constant π‘ž such that 6𝛾/5≀2π‘žβ‰€π›Ύ+1. Then by taking 𝑛2π‘žβˆ’π›Ύβˆ’π‘›2π‘žβˆ’π›Ύ and 𝑝2π‘žβˆ’π›Ύβˆ’π‘2π‘žβˆ’π›Ύ as test function in (2.3) and (2.4), respectively, and adding them together, we get 1ξ€œ2π‘žβˆ’π›Ύ+1Ω𝑛2π‘žβˆ’π›Ύ+1+𝑝2π‘žβˆ’π›Ύ+1ξ€Έ+𝛾(2π‘žβˆ’π›Ύ)π‘ž2ξ€œπ‘‡0ξ€œΞ©ξ‚ƒ||βˆ‡(π‘›π‘ž)||2+||βˆ‡(π‘π‘ž)||2ξ‚„=1ξ€œ2π‘žβˆ’π›Ύ+1Ω𝑛02π‘žβˆ’π›Ύ+1+𝑝02π‘žβˆ’π›Ύ+1+ξ€œΞ©ξ‚ƒπ‘›2π‘žβˆ’π›Ύξ€·π‘›βˆ’π‘›0ξ€Έ+𝑝2π‘žβˆ’π›Ύξ€·π‘βˆ’π‘0ξ€Έξ‚„+ξ€œπ‘‡0ξ€œΞ©ξ‚ƒξ‚€ξ‚€π‘›βˆ‡πœ“β‹…π‘›βˆ‡2π‘žβˆ’π›Ύβˆ’π‘›2π‘žβˆ’π›Ύξ‚ξ‚€π‘βˆ’π‘βˆ‡2π‘žβˆ’π›Ύβˆ’π‘2π‘žβˆ’π›Ύ+ξ€œξ‚ξ‚ξ‚„π‘‡0ξ€œΞ©ξ‚ƒβˆ‡(𝑛𝛾)β‹…βˆ‡π‘›2π‘žβˆ’π›Ύξ‚+βˆ‡(𝑝𝛾)β‹…βˆ‡π‘2π‘žβˆ’π›Ύ+ξ€œξ‚ξ‚„π‘‡0ξ€œΞ©ξ€·π‘…(𝑛,𝑝)+π‘”πœ€ξ€Έπ‘›ξ‚ƒξ‚€2π‘žβˆ’π›Ύβˆ’π‘›2π‘žβˆ’π›Ύξ‚+𝑝2π‘žβˆ’π›Ύβˆ’π‘2π‘žβˆ’π›Ύξ‚ξ‚„=𝐼1+β‹―+𝐼4.(3.12) We estimate the right-hand side term by term. Due to the equation of πœ“ and (3.6), we obtain 𝐼2=2π‘žβˆ’π›Ύξ€œ2π‘žβˆ’π›Ύ+1𝑇0ξ€œΞ©ξ‚ƒβˆ‡ξ‚€π‘›βˆ‡πœ“β‹…2π‘žβˆ’π›Ύ+1βˆ’π‘›2π‘žβˆ’π›Ύ+1ξ‚ξ‚€π‘βˆ’βˆ‡2π‘žβˆ’π›Ύ+1βˆ’π‘2π‘žβˆ’π›Ύ+1+ξ€œξ‚ξ‚„π‘‡0ξ€œΞ©ξ‚ƒξ€·βˆ‡πœ“β‹…ξ€Έβˆ‡ξ‚€π‘›βˆ’π‘›π‘›2π‘žβˆ’π›Ύξ‚βˆ’ξ€·ξ€Έβˆ‡ξ‚€π‘βˆ’π‘π‘2π‘žβˆ’π›Ύβ‰€ξ‚ξ‚„2π‘žβˆ’π›Ύξ€œ2π‘žβˆ’π›Ύ+1𝑇0ξ€œΞ©ξ‚ƒξ€·π‘›(π‘βˆ’π‘›+𝐢(π‘₯))2π‘žβˆ’π›Ύ+1βˆ’π‘2π‘žβˆ’π›Ύ+1ξ€Έβˆ’ξ‚€π‘›2π‘žβˆ’π›Ύ+1βˆ’π‘2π‘žβˆ’π›Ύ+1ξ€œξ‚ξ‚„+𝐢𝑇0β€–βˆ‡πœ“β€–πΏ2(Ξ©)ξ€·β€–β€–β€–β€–π‘›βˆ’π‘›πΏ2(Ξ©)+β€–β€–β€–β€–π‘βˆ’π‘πΏ2(Ξ©)ξ€Έξ‚Έξ€œβ‰€πΆπ‘‡0ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έξ‚Ή,+1(3.13) where we use that (π‘βˆ’π‘›)(𝑛2π‘žβˆ’π›Ύ+1βˆ’π‘2π‘žβˆ’π›Ύ+1)≀0 and 2π‘žβˆ’π›Ύ+1≀𝛾+1 when 𝛾β‰₯2. 𝐼3 and 𝐼4 can be bounded as 𝐼3β‰€πœ†14ξ€œπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚ξ€·πœ†+𝐢1ξ€Έ,𝐼(3.14)4β‰€πœ†14ξ€œπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚ξ€œ+𝐢𝑇0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έξ€·πœ†+𝐢1ξ€Έξ‚Έξ€œπ‘‡0ξ€œΞ©ξ€·π‘›4π‘žβˆ’2𝛾+𝑝4π‘žβˆ’2𝛾+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έ||||βˆ‡πœ“2ξ‚Ή,(3.15) where we have used the nonnegativity of 𝑛 and 𝑝. Inserting (3.13)–(3.15) into (3.12) we conclude that ξ€œΞ©ξ€·π‘›2π‘žβˆ’π›Ύ+1+𝑝2π‘žβˆ’π›Ύ+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ‚ƒ||βˆ‡(π‘›π‘ž)||2+||βˆ‡(π‘π‘ž)||2ξ‚„β‰€πœ†1ξ‚Έξ€œπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2+ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έξ‚Ήξ€·πœ†+𝐢1ξ€Έξ‚Έξ€œπ‘‡0ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έ||||βˆ‡πœ“2ξ‚Ή.+1(3.16) Similarly, taking π‘›π›Ύβˆ’π‘›π›Ύ and π‘π›Ύβˆ’π‘π›Ύ as test function in (2.3) and (2.4), we conclude that 1ξ€œπ›Ύ+1Ω𝑛𝛾+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ‚ƒ||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚„=1ξ€œπ›Ύ+1Ω𝑛0𝛾+1+𝑝0𝛾+1+ξ€œΞ©ξ€Ίπ‘›π›Ύξ€·π‘›βˆ’π‘›0ξ€Έ+π‘π›Ύξ€·π‘βˆ’π‘0+ξ€œξ€Έξ€»π‘‡0ξ€œΞ©ξ€Ίβˆ‡(𝑛𝛾)β‹…βˆ‡π‘›π›Ύξ€Έ+βˆ‡(𝑝𝛾)β‹…βˆ‡π‘π›Ύ+ξ€œξ€Έξ€»π‘‡0ξ€œΞ©βˆ‡πœ“β‹…ξ€Ίξ€·ξ€Έβˆ‡ξ€·π‘›βˆ’π‘›π‘›π›Ύξ€Έβˆ’ξ€·ξ€Έβˆ‡ξ€·π‘βˆ’π‘π‘π›Ύ+π›Ύξ€Έξ€»ξ€œπ›Ύ+1𝑇0ξ€œΞ©ξ‚ƒβˆ‡ξ‚€π‘›βˆ‡πœ“β‹…π›Ύ+1βˆ’π‘›π›Ύ+1ξ‚ξ‚€π‘βˆ’βˆ‡π›Ύ+1βˆ’π‘π›Ύ+1+ξ€œξ‚ξ‚„π‘‡0ξ€œΞ©ξ€·π‘…(𝑛,𝑝)+π‘”πœ€π‘›ξ€Έξ€Ίξ€·π›Ύβˆ’π‘›π›Ύξ€Έ+ξ€·π‘π›Ύβˆ’π‘π›Ύξ€Έξ€»β‰€πœ†1ξ‚Έξ€œπ‘‡0ξ€œΞ©ξ‚€||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2+ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έξ‚Ήξ€·πœ†+𝐢1ξ€Έξ‚Έξ€œπ‘‡0ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2𝛾+𝑝2𝛾+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έ||||βˆ‡πœ“2ξ‚Ή.+1(3.17) Choosing πœ†1 sufficiently small, and then summing (3.16) and (3.17), we have ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ‚ƒ||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚„+ξ€œΞ©ξ€·π‘›2π‘žβˆ’π›Ύ+1+𝑝2π‘žβˆ’π›Ύ+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ‚ƒ||βˆ‡(π‘›π‘ž)||2+||βˆ‡(π‘π‘ž)||2ξ‚„ξ€·πœ†β‰€πΆ1ξ€Έξ‚Έξ€œπ‘‡0ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2𝛾+𝑝2𝛾+ξ€œπ‘‡0ξ€œΞ©ξ€·π‘›2+𝑝2ξ€Έ||||βˆ‡πœ“2ξ‚Ή.+1(3.18) Now we estimate the term βˆ«π‘‡0βˆ«Ξ©π‘›2𝛾 (or βˆ«π‘‡0βˆ«Ξ©π‘2𝛾 ). Due to Lemma 2.2 we get ξ€œπ‘‡0ξ€œ0π‘₯0200𝑑Ω𝑛2𝛾=ξ€œπ‘‡0ξ€œΞ©(π‘›π‘ž)π‘ β‰€ξ€œπ‘‡0ξ€œΞ©β€–π‘›π‘žβ€–π»(1βˆ’π›Ό)𝑠1(Ξ©)β€–π‘›π‘žβ€–πΏπ›Όπ‘ 2(Ξ©)β‰€πœ†2ξ€œπ‘‡0β€–βˆ‡(π‘›π‘ž)β€–2𝐿2(Ξ©)ξ€·πœ†+𝐢2ξ€Έξƒ¬ξ€œπ‘‡0ξ‚΅ξ€œΞ©π‘›2π‘žξ‚Άdπ‘₯𝛽,+1(3.19) where 𝑠,𝛼,𝛽 satisfy 𝑠=2π›Ύπ‘žξ‚΅1,0<1βˆ’π›Ό=32βˆ’π‘žξ‚Ά2𝛾<1,0<(1βˆ’π›Ό)𝑠<2,𝛽=𝛼𝑠2βˆ’(1βˆ’π›Ό)𝑠(3.20) Consequently we obtain, taking into account (3.11) and the choice of π‘ž, ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έ+ξ€œπ‘‡0ξ€œΞ©ξ‚ƒ||βˆ‡(𝑛𝛾)||2+||βˆ‡(𝑝𝛾)||2ξ‚„ξ€·πœ†β‰€πΆ1,πœ†2ξ€Έξƒ¬ξ€œ,𝛿1+𝑇0ξ‚΅ξ€œΞ©ξ€·π‘›π›Ύ+1+𝑝𝛾+1ξ€Έξ‚Άdπ‘₯max{2,𝛽}ξƒ­(3.21) for sufficiently small πœ†2 and 𝛿 such that (πœ†2+𝛿)𝐢(πœ†1)<1, where 𝛽 only depends on 𝛾.
This proves that
maxπ‘‘βˆˆ[