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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

Issues

The BV solution of the parabolic equation with degeneracy on the boundary

Huashui Zhan / Shuping Chen
Published Online: 2016-05-10 | DOI: https://doi.org/10.1515/math-2016-0025

Abstract

Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

Keywords: Local BV Solution; Boundary degeneracy; Partial boundary condition; Stability

MSC 2010: 35L65; 35L85; 35R35

1 Introduction and the main results

Yin-Wang [1] first studied the equation

ut=div(ρα|u|p2u),(x,t)QT=Ω×(0,T).(1)

where Ω is a bounded domain in RN with appropriately smooth boundary, ρ(x) = dist(x, ∂Ω), p > 1, α > 0. An obvious character of the equation is that, the diffusion coefficient depends on the distance to the boundary. Since the diffusion coefficient vanishes on the boundary, it seems that there is no heat flux across the boundary. However, Yin-Wang [1] showed that the fact might not coincide with what we image. In fact, the exponent α, which characterizes the vanishing ratio of the diffusion coefficient near the boundary, does determine the behavior of the heat transfer near the boundary. One may refer to [1] for the details.

In our paper, we will consider the following equation

ut=div(ρα|u|p2u)+i=1Nbi(u)xi,(x,t)QT.(2)

The convection term i=1Nbi(u)xi not only brings the difference on operational skills, but more essentially, it makes the nature of boundary condition change. The equation (2) had been originally studied by the authors in [2, 3]. Instead of the whole boundary condition

u(x,t)=0,(x,t)Ω×(0,T),(3)

only a partial boundary condition

u(x,t)=0,(x,t)Σp×(0,T),(4)

matching equation (2) is considered. Here, denoting {ni (x)} as the unit inner normal vector of ∂Ω, when bi(0)ni(x)<0,xΩ, then Σp = ∂Ω. But generally, it is just a portion of ∂Ω. However, we don’t need to pay too much attention to its explicit formula, we only need to remember it is just a subset of ∂Ω. Certainly, the initial value is always necessary,

u(x,0)=u0(x).(5)

In [2, 3], we said a bounded domain Ω has the integral non-singularity, if the constants α, p, satisfy

Ωρ2αp2dxc.

We assumed that there are constants β, c such that

|bi(s)|c|s|1+β,|bi(s)|c|s|β.(6)

If p > 2, we had obtained the existence of the solution of equation (2) with the initial boundary values (4)(5), and if Σp = ∂Ω, we also had obtained the stability of the weak solutions. In our paper, we will promote the existence of the solution without the condition (6) but limiting that α ≥ 1. The most innovation of our paper is that the stability of the weak solutions can be obtained only based on the partial boundary condition (4). Comparing with the case of that Σp = ∂Ω in [2, 3] or Σp = ∅ in [1] (when αp – 1), how to obtain the stability of the weak solutions only based on the partial boundary condition (4) seems very difficult.

Let us give the basic definitions and the main results as following.

A function u(x, t) is said to be a local BV solution of equation (2) with the initial value (5), if uBV(QTλ)L(QT),ρα|u|pL1(QT),

and for any function φC0(QT), the following integral equivalence holds QT(uφt+ρα|u|p2uφ+i=1Nbi(u)φxi)dxdt=0.(7)

The initial value is satisfied in the sense of limt0Ω|u(x,t)u0(x)|dx=0.(8)

Here, Q = {(x, t) ∈ QT : ρ(x) = dist(x, ∂Ω) > λ} for small enough λ > 0.

A function u(x, t) is said to be a local BV solution of equation (2) with the initial boundary values (4)(5), if u satisfies Definition 1.1, and it satisfies the partial boundary condition (4) in the sense of the trace.

If u is a local BV solution of equation (2), satisfies that |u(x,t)|cρ(x),|u|cρα(x),(9)

when x is near ∂Ω, then we say u is a regular solution.

If bi(s) ≡ 0, we had proved that the solution of equation (1) is regular in [4].

Let 1 < p, 1 ≤ α < p – 1, bi(s) ∈ C2(R1). If u0(x)C0(Ω),(10)

then equation (2) with initial boundary values (4)(5) has a local BV solution u, and utL(QT).

Let α < p – 1, u and v be two local BV solutions of equation (1) with the same partial homogeneous boundary value u|Σp×(0,T)=0=υ|Σp×(0,T),(11)

and with the different initial values u(x, 0) = v(x, 0) respectively. If bi(s) is a Lipschitz function, and moreover |u|cρα(x),|υ|cρα(x)(12)

then Ω|u(x,t)v(x,t)|dxΩ|u0v0|dx+cΣp|uv|dΣ+limsupnΣpgn(uv)|uv|dΣ,t[0,T).(13)

Here, n > 0 is a nature number, the details of the definition and the properties of the function gn(s) is in Section 3, in particular, |gn(s)s| ≤ c.

Let αp – 1, and u, v be two local BV solutions of (1) with the initial values u0(x), v0(x) respectively. If u and v are regular, and |bi(u)bi(v)|c|uv|α+2,(14)

then Ω|u(x,t)v(x,t)|2dxcΩ|u0v0|2dx,a.e.t(0,T).(15)

The most important character of Theorem 1.7 is in that we obtain the stability (15) without any boundary value condition. However, since the solutions considered in the theorem are regular, we can easily obtain the conclusion (15) in a similar way as in [1]. So we omit the details of the proof of the theorem in our paper.

Recently, the author has been interested in the initial-boundary value problem of the following strongly degenerate parabolic equation

ut=xi(aij(u,x,t)uxj)+bi(u,x,t)xi,(x,t)QT.(16)

The stability of the solutions based on a partial boundary condition (4) has been established in [57] et. al. Actually, many mathematicians have been interested in the problem, and have obtained many important results of the the stability of the solutions based on a partial boundary condition, one may see the Refs. [811]. Unlike the equation (16), to the best knowledge of the authors, considering the parabolic equation related to the p-Laplacian, our paper is the first one to study the stability of the solutions based on a partial boundary condition (4). Of course, whether the condition (12) in Theorem 1.6 and the assumption that u, v are regular in Theorem 1.7 are necessary or not? This is a very interesting problem to be studied in the future. Some other related references, one can refer to Refs. [1216]. The paper is arranged as following. In Section 1, we have introduced the problem and given the main results of the paper. In Section 2, we prove the existence of the local BV solution. In Section 3, only based on a partial boundary condition, we prove the Theorem 1.6.

2 The localBV solution

To study equation (2), we consider the following regularized problem

uεtdiv(ρεα(|uε|2+ε)p22uε)i=1Nbi(uε)xi=0,(x,t)QT,(17)uε(x,t)=0,(x,t)Ω×(0,T),(18)uε(x,0)=u0(x),xΩ.(19)

where ρ = ρ * δ + , > 0, δ is the mollifier as usual. It is well-known that the above problem has a unique classical solution [17, 18]. Hence, for any φC0(QT), u satisfies

QT(uεφt+ρεα|uε|p2uεφ+i=1Nbi(uε)φxi)dxdt=0.(20)

If u0C0(Ω),α1, then the solution u of the initial boundary value problem (17)(19) converges locally in BV(QT), and its limit function u is the local BV solution of equation (2) with the initial value (5).

Proof. By the maximum principle, there is a constant c only dependent on ‖u0L(Ω) but independent on ε, such that uεL(QT)c.(21)

Multiplying (17) by uε and integrating it over QT, we have 12Ωuε2dx+QTρεα(|uε|2+ε)p22|uε|2dxdt+QTuεi=1Nbi(uε)xidxdt=12Ωu02dx.

By the fact QTuεi=1Nbi(uε)xidxdt=i=1Nuεxibi(uε)dxdt=i=1NΩxi0uεbi(s)dsdt=0,

then 12Ωuε2dx+QTρεα(|uε|2+ε)p22|uε|2dxdtc.(22)

For small enough λ > 0, let Ωλ = {x ∈ Ω : dist(x, ∂Ω) > λ}. Since p > 1, by (22), 0TΩλ|uε|dxdtc(0TΩλ|uε|pdxdt)1pc(λ).(23)

Differentiating (17) with t, and denoting w = ut, then wt=(p2)ρεα(|uε|2+ε)p42uxkuxiwxkxi+ρεα(|uε|2+ε)p22wxixi+(p2)ρεαuε(|uε|2+ε)p42uxkwxk+ρεα(|uε|2+ε)p22w+(p2)(p4)ρεα(|uε|2+ε)p62uxjuxixjuxiwxkuxk+(p2)ρεα(|uε|2+ε)p42(uxiuxixkwxk+uxkuxkxiwxi+uxkuxixiwxk)+bi(u)uxiw+bi(u)wxi,

rewriting it as wt=aij2wxixj+i=1Nfi(x,t,w)wxi+bi(u)uxiw,

where aij=ρεα(|uε|2+ε)p22(δij+(p2)(|uε|2+ε)1uxiuxj).fi(x,t,w)=(p2)ρεαuε(|uε|2+ε)p42uxi+(|uε|2+ε)p22(ρεα)xi+(p2)(p4)ρεα(|uε|2+ε)p62uxjuxkxjuxkuxi+(p2)ρεα(|uε|2+ε)p42(uxkuxkxi+uxiuxixk+uxiuxkxk)+bi(u),

Clearly, w satisfies that w(x,t)=0,(x,t)Ω×[0,T),w(x,0)=div(ρεα(|u0|2+ε)p22u0)+bi(u0)xi,xΩ.

Denoting that a=(|uε|2+ε)p22,

then min{p1,1}a|ξ|2aijξiξjmax{p1,1}a|ξ|2.

By the maximum principle, due to α ≥ 1, we have supΩ×(0,T)|uεt|supΩ|div(ρεα(|u0|2+ε)p22+bi(u0)xi|c.(24)

By (23)(24), we know that uεBV(QTλ), and 0TΩλ|uε|dxdtc,0TΩλ|uεt|dxdtc.(25)

Then by Kolmogoroff’s theorem, there exists a subsequence (still denoted as uε) of uε, which is strongly convergent to uBV(QTλ). In particular, by the arbitrary of λ, uεu a.e. in QT.

Hence, by (22), (25), there exists a function u and n dimensional vector function ζ=(ζ1,,ζn) satisfying that uBV(QTλ)L(QT),|ζ|Lpp1(QT),

and uε*u,inL(QT),uεuinLlocp(QT),ρεα|uε|p2uεζinLpp1(QT).

In order to prove u satisfies equation (2), we notice that for any function φC0(QT), QT(uεφt+ρεα(|uε|2+ε)p22uεφ+i=1Nbi(uε)φxi)dxdt=0.(26)

and uεu is almost everywhere convergent, so bi(uε) → bi(u) is true. Then QT(utφ+ςφ+i=1Nbi(u)φxi)dxdt=0.(27)

Now, if we can prove that QTρα|u|p2uφdxdt=QTζφdxdt.(28)

for any function φC0(QT), then u satisfies equation (2).

Let 0ψC0(QT) and ψ = 1in suppφ. Let vBV(QTλ) ∩ L(QT), ρα|∇v|pL1(QT). It is well-known that QTψρεα(|uε|p2uε|v|p2v)(uεv)dxdt0.(29)

By choosing φ = ψu in (26), we can obtain QTψρεα(|uε|2+ε)p22|uε|2dxdt=12QTψtuε2dxdtQTρεαuε(|uε|2+ε)p22uεvψdxdti=1NQTbi(uε)(uεxiψ+uεψxi)dxdt.(30)

Noticing that when p ≥ 2, (|uε|2+ε)p22|uε|2|uε|p,(|uε|2+ε)p22|uε|(|uε|p1+1),

and when 1 < p < 2, (|uε|2+ε)p22|uε|2(|uε|2+ε)p2εp2,(|uε|2+ε)p22|uε|(|uε|2+ε)p12,(31)

then in both cases, by (29), we have 12QTψtuε2dxdtQTρεαuε(|uε|2+ε)p22uεψdxdti=1NQTbi(uε)(uεxiψ+uεψxi)dxdt+εp2c(Ω)QTρεαψ|uε|p2uεvdxdtQTρεαψ|v|p2(uεv)dxdt0.(32)

Thus 12QTψtuε2dxdtQTρεαuε(|uε|2+ε)p22uεψdxdti=1NQTbi(uε)(uεxiψ+uεψxi)dxdt+εp2c(Ω)QTρεαψ|uε|p2uεvdxdtQTψρα|v|p2v(uεv)dxdt+QTψ(ραρεα)|v|p2v(uεv)dxdt0.(33)

Noticing |QTψ(ραρεα)|v|p2v(uεv)dxdt|sup(x,t)QT|ψ(ραρεα)|ραQTρα|v|p1|uεv|dxdtsup(x,t)QT|ψ(ραρεα)|ρα(QTρα|v|pdxdt+QTρα|v|p1|uε|dxdt)(34)

and (|uε|2+ε)p22uε=|uε|p2uε+p22ε01(|uε|2+εs)p42dsuε

then limε0QTp22ε01(|uε|2+εs)p42dsuεψuεdxdt=0.(35)

By Hölder inequality, there holds QTρα|v|p1|uε|dxdt(QT(ρm|v|p1)sdxdt)1/s(QT(ρn|uε|)pdxdt)1/p,

where m=α(p1)p,n=αp,s=pp1. Due to ρα|∇|p, ρα|∇|pL1(QT), then QTρα|v|pdxdt+QTρα|v|p1|uε|dxdtc.

Let ε → 0 in (33). It converges to 0.

Thus, we have 12QTψtu2dxdtQTuζψdxdti=1NQTbi(u)(uxiψ+uψxi)dxdtQTψζvdxdtQTψρα|v|p2v(uεv)dxdt0.

Let v = ψu in (2), we get QTψζudxdt12QTu2ψtdxdt+QTuζψdxdt+i=1NQTbi(u)(uxiψ+uψxi)dxdt=0.

Thus QTψ(ζρα|v|p2v)(uv)dxdt0.(36)

Let v = u – λφ, λ > 0, φC0(QT), then QTψ(ζρα|(uλφ)|p2(uλφ))φdxdt0,

If λ → 0, then QTψ(ζρα|u|p2u)φdxdt0.

Moreover, if λ < 0, similarly we can get QTψ(ζρα|u|p2u)φdxdt0.

Thus QTψ(ζρα|u|p2u)φdxdt=0,

Noticing that ψ = 1 on suppφ, (28) holds. At the same time, we can prove (5) as in [19], we omit the details here. The lemma is proved. □

If u0C0(Ω), α ≥ 1 there exists a local BV solution u of equation (2) with the initial value (5), such that utL(QT).

If α < p – 1, u is a weak solution of equation (1) (also equation (2)) with the initial value (5). Then u has trace on the boundary ∂Ω.

Lemma 2.3 had been proved in [1, 2]. Clearly, Theorem 1.5 is the directly corollary of Lemma 2.12.3.

The condition α ≥ 1 is only used to prove that |u∊t| ≤ c, which implies that QTλ|uεt|dxdtc. Maybe one can prove the later conclusion directly. u0C0(Ω) is the simplest condition, but it is not the most general condition. However, we mainly concern with how the degeneracy of diffusion coefficient ρα affects the boundary value condition.

3 The stability when α < p – 1

Proof of Theorem 1.6. For a small positive constant λ > 0, let Ωλ={xΩ:ρ(x)=dist(x,Ω)>λ},

and let ϕ(x)={1,ifxΩ2λ,1λ(ρ(x)λ),xΩλ\Ω2λ0,ifxΩ\Ωλ.(37)

For any given positive integer n, let gn(s) be an odd function, and gn(s)={1,s>1n,n2s2e1n2s2,0s1n.(38)

Clearly, limn0gn(s)=sgn(s),s(,+),(39)

and 0gn(s)cs,0<s<1n,(40)

where c is independent of n.

By a process of limit, we can choose gn(ϕ(uv) as the test function, then Ωgn(ϕ(uv))(uv)tdx+Ωρα(|u|p2u|v|p2vϕ(uv)gndx+Ωρα(u)|p2u|v|p2v)ϕ(uv)gndx+i=1NΩ(bi(u)bi(v))(uv)xignϕdx+i=1NΩ(bi(u)bi(v))(uv)gndx=0.(41)

Thus limnlimλ0Ωgn(ϕ(uv))(uv)tdx=ddtuv1,(42) Ωρα(|u|p2u|v|p2v)(uv)gnϕ(x)dx0.(43)

By L’Hospital rule, limλ0Ωλ\Ω2λgn(ϕ(uv))(uv)dxλ=limλ0λ2λρ=ζgn(ϕ(uv))(uv)dΣdζλ=limλ0ρ=2λgn(uv)(uv)dΣ=Ωgn(2(uv))(uv)dΣ=Σpgn(uv)(uv)dΣ.(44)

Since we assume that ρ|∇u|p < ∞, ρ|∇v|p < ∞, we have limλ0|Ωρα(|u|p2u|v|p2v)ϕ(uv)gn(ϕ(uv))dx|=limλ0|Ωλ\Ω2λρα(|u|p2u|v|p2v)ϕ(uv)gn(ϕ(uv))dx|climλ0|Ωλ\Ω2λgn(ϕ(uv))(uv)dxλ=pgn(uv)(uv)dΣ.(45)

While, |Ω(bi(u)bi(v))gn(ϕ(uv))(uv)ϕxi(x)dx|ΩλΩ2λ|bi(u)bi(v)||uv|gn(ϕ(uv))cλdx.(46)

By |bi(u) – bi(v)| ≤ c|uv|, and by (40), according to the definition of the trace, we have limλ0|Ω(bi(u)bi(v))gn(ϕ(uv))(uv)ϕxi(x)dx|limλ0ΩλΩ2λ|uv|2gn(ϕ(uv))cλdx=Ω|uv|2gn((uv))dcΩ|uv|dΣ=cp|uv|dΣ.(47)

Moreover, limλ0|Ω(bi(u)bi(v))gn(ϕ(uv))(uv)xiϕ(x)dx|=|{xΩ:|uv|<1n}[bi(u)bi(v)]gn(uv)(uv)xidxc{xΩ:|uv|<1n}|bi(u)bi(v)uv||(uv)xi|dx=c{xΩ:|uv|<1n}|ραpbi(u)bi(v)uv||ραp(uv)xi|dxc[{xΩ:|uv|<1n}(|ραpbi(u)bi(v)uv|)pp1dx]pp1[{xΩ:|uv|<1n}|ρα(uv)|pdx]1p.(48)

Since α < p – 1, {xΩ:|uv|<1n}(|ραpbi(u)bi(v)uv|)pp1dxcΩραp1dxc.(49)

In (47), let n → ∞. If {x ∈ Ω : |uv| = 0} is a set with 0measure, then limn{xΩ:|uv|<1n}ραp1dx={xΩ:|uv|=0}ραp1dx=0.(50)

If the set {x ∈ Ω : |uv| = 0} has a positive measure, then, limn{xΩ:|uv|<1n}ρα|(uv)|pdx={xΩ:|uv|=0}ρα|(uv)|pdx=0.(51)

Therefore, in both cases, we have limnlimλ0Ω(bi(u)bi(v))gn(ϕ(uv))(uv)xiϕ(x)dx=0.(52)

Now, after letting λ → 0, let n → ∞ in (41). Then ddtuv1cΣp|uv|dΣ+limsupnΣpgn(uv)|uv|dΣ.

It implies that Ω|u(x,t)v(x,t)|dxΩ|u0v0|dx+cΣp|uv|dΣ+limsupnΣpgn(uv)|uv|dΣ,t[0,T).

Theorem 1.6 is proved.

Acknowledgement

The paper is supported by NSF of China (no.11371297, 11302184), Natural Science Foundation of Fujian province (no: 2015J01592), supported by Science Foundation of Xiamen University of Technology, China (no:XYK201448).

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About the article

Received: 2016-01-15

Accepted: 2016-04-09

Published Online: 2016-05-10

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0025.

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© 2016 Zhan and Chen, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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