Self-similar solutions of fast reaction limit problem with nonlinear diusion Elaine Crooks Yini Du

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Self-similar solutions of fast reaction limit problem with nonlinear

diﬀusion

Elaine Crooks, Yini Du∗

Department of Mathematics, Faculty of Science and Engineering,

Swansea University, Swansea SA1 8EN, UK

Abstract

In this paper, we present an approach to characterising self-similar fast-reaction limits of systems

with nonlinear diﬀusion. For appropriate initial data, in the fast-reaction limit k→ ∞, spatial

segregation results in the two components of the original systems converging to the positive and

negative parts of a self-similar limit proﬁle f(η), where η=x

√t, that satisﬁes one of four ordinary

diﬀerential systems. The existence of these self-similar solutions of the k→ ∞ limit problems

is proved by using shooting methods which focus on a, the position of the free boundary which

separates the regions where the solution is positive and where it is negative, and γ, the derivative

of −φ(f) at η=a. The position of the free boundary gives us intuition about how one substance

penetrates into the other, and for speciﬁc forms of nonlinear diﬀusion, the relationship between

the given form of the nonlinear diﬀusion and the position of the free boundary is also studied.

Keywords: Nonlinear diﬀusion, Reaction diﬀusion system, Fast reaction limit, Self-similar

solution, Free boundary

1. Introduction

In this paper, we will study the self-similar characterisation of two pairs of problems, one on

the half-line and the other one on the whole real line.

The ﬁrst pair of problems, on the half-strip ST={(x,t):0≤x<∞,0<t<T}and with both

ε > 0 and ε=0, are











wt=D(w)xx,(x,t)∈(0,∞)×(0,T),

w(x,0) =w0(x) :=−V0,for x>0,

w(0,t)=U0,for t∈(0,T).

(1.1)

where the function Dis deﬁned as

D(s) :=









φ(s)s≥0,

−εφ(−s)s<0.(1.2)

∗Corresponding author

Email address: yini.du@swansea.ac.uk (Yini Du)

Preprint submitted to Journal of Diﬀerential Equations October 14, 2022

arXiv:2210.06567v1 [math.AP] 12 Oct 2022

The function φ∈C2(R) and φ0are assumed to be strictly increasing with

φ(s)>0 as s>0 and φ0(s)=φ(s)=0 when s=0.(1.3)

We will also require that φsatisﬁes

φ0(f)

fdf<∞and Z∞

φ0(f)

fdf=∞.(1.4)

The function Darises from the k→ ∞ limit of the following reaction diﬀusion systems that

have been studied in [4],











ut=φ(u)xx −kuv,(x,t)∈(0,∞)×(0,T),

vt=εφ(v)xx −kuv,(x,t)∈(0,∞)×(0,T),

u(0,t)=U0, εφ(v)x(0,t)=0,for t∈(0,T),

u(x,0) =uk

0(x),v(x,0) =vk

0(x),for x∈R+.

(1.5)

Here, uand vrepresent concentration of two substances and kis the positive rate constant of the

reaction. As in [5], the initial data for the limiting self-similar solutions are deﬁned as

u∞

0=(U0x=0,

0x>0,v∞

0=(0x=0,

V0x>0,

where U0and V0are positive constants.

From [4], we see that the limits uof ukand vof vksegregate, given by the positive and

negative parts respectively of a function w. It can be shown in [4], by using a strategy inspired

by [11], that wis the unique weak solution of (1.5),

u=w+and v=−w−,

where s+=max{0,s}and s−=min{0,s}. Here we prove that the limit function wsatisﬁes one

of two self-similar problems, depending on whether ε > 0 or ε=0. In each case, a function

f:R+→Rdescribes a self-similar limit solution such that w(x,t)=f(η) where η=x/√tfor

(x,t)∈ST. There is a free boundary at η=awith f(η)>0 when η < aand f(η)<0 when η > a

and the self-similar solution f(η) satisﬁes the boundary conditions f(a)=0 and

γ:=−lim

η%aφ0(f(η)) f0(η).(1.6)

When ε > 0, the existence of self-similar solutions is proved in Section 3 by using a two-

parameter shooting methods focusing on aand γ. When ε=0, γhas a speciﬁc form, namely

γ=aV0

and the existence of self-similar solutions is proved in Section 4 by a one-parameter shooting,

since γdepends on a.

The second pair of problems on the strip QT={(0,t) : x∈R,0<t<T}with Dfrom (1.2)

and both with ε > 0 and ε=0, are











wt=D(w)xx,in QT,

w(x,0) =w0(x) :=









U0,if x<0,

−V0,if x>0.

(1.7)

The function Dand the initial condition come from the k→ ∞ limit problems of











ut=φ(u)xx −kuv,(x,t)∈R×(0,T),

vt=εφ(v)xx −kuv,(x,t)∈R×(0,T),

u(x,0) =uk

0(x),v(x,0) =vk

0(x),for x∈R,

(1.8)

where we deﬁne, as in [5] that

u∞

0=(U0x<0,

0x>0,v∞

0=(0x<0,

V0x>0,

with U0,V0positive constants and kas in (1.5).

We use arguments similar to those in half-line case to prove the existence of a self-similar

solution of (1.7). If ε > 0, we may have a<0, a=0 and a>0 where f(a)=0, since ais not

necessarily positive in the whole-line case.

We take advantage of various ideas from earlier work on self-similar solutions for nonlinear

diﬀusion problems and discuss brieﬂy two previous papers [1, 3] here. Let k(s) be continuous

with k(0) =0 and k(s)>0 as s>0. We introduce the notation k(s) for ease of comparison

with [1, 3], but, kclearly plays the same role as φ0plays here. Then the solutions of the nonlinear

diﬀusion equation ut=(k(u)x)xcan be studied in self-similar form u(x,t)=f(η) where η=x/√t,

and fsatisﬁes the equation

k(f)f00+1

2ηf0=0.(1.9)

In [1], Atkinson and Peletier proved the existence and uniqueness of a self-similar solution

f(η) which satisﬁes (1.9) for 0 < η < a, where a>0, under the boundary conditions f(0) =U,

lim

η→af(η)=0 and lim

η→ak(f(η)) f0(η)=0. They consider two cases in describing the dependence of

aand U,

A.Z∞

k(s)

sds=∞and B.Z∞

k(s)

sds<∞.

They found that as U→ ∞,a=a(U) tends to inﬁnity in Case A whereas a(U) tends to a

ﬁnite limit in Case B. Here, we only consider the case when Z∞

k(s)

sds=∞. The proof in

[1] depends on a discussion of the function b(a), which is deﬁned as the value at η=0 of the

solution f(η)=f(η;a) of (1.9) with boundary conditions lim

η→af(η)=0 and lim

η→ak(f(η)) f0(η)=0.

A similar problem in an unbounded interval 0 <η<∞with boundary conditions f(0) =Uand

lim

η→∞ f(η)=0 is studied in [3] by Craven and Peletier. Note that in [3], f(η)>0 for all η > 0. [3]

proved the existence and uniqueness of a weak solution by a shooting method where the initial

value problem f(0) =U,f0(0) =βis considered. We will adapt ideas of studying the function

b(a) from [1] and shooting methods from [3] to prove existence of self-similar solutions in this

paper.

Note that [1, 3] treated a single equation where the solutions were always non-negative. In

this paper, we have sign-changing solutions since the free boundary separates regions where

the solutions are positive and where the solutions are negative. Here, our self-similar solutions

satisfy a certain equation when they are positive, and a diﬀerent equation where they are negative.

We exploit ideas from [1, 3] to investigate our self-similar limit problems that involve these two

equations.

We will study the existence of a solution fthat satisﬁes (2.10) with given boundary conditions

by splitting it into two parts: η < awhere f(η) is positive, and η > awhere f(η) is negative.

Then we will discuss the existence and properties of lim

η→0f(η) and lim

η→∞ f(η). These results will

be used to study b(a, γ), the value at η→0 of the solution f(η)=f(η;a, γ), and d(a, γ), the

value at η→ ∞ of the solution f(η)=f(η;a, γ), where γas in (1.6), and also to implement a

two-parameter shooting method.

This paper is organised as follows. In Section 2, the limit problem (1.1) is characterised as

a self-similar solution of the problem ﬁrst in Theorem 2.4 when ε > 0, and then in Theorem

2.9 when ε=0. Section 3 focuses on properties of the parameters aand γin the study of the

self-similar solution f, and prove some preliminary results that are useful in deducing existence

of self-similar solutions. The existence of self-similar solutions when ε≥0 is proved in Section

3.5 and Section 4. Section 5 contains the whole-line counterparts of the study of the half-line

problem in Sections 2-4.

In Section 6, we also consider a speciﬁc family of φ0(f)=fm−1where m>2 is a constant,

and investigate how the free boundary position ais aﬀected by m. Note that with ﬁxed U0,V0,

there exists a unique self-similar solution which determines aand γ. We prove some further

results under the additional conditions that U0<1 and m≥2. In particular, if ε=0, we ﬁnd

that if m1>m2, then am1<am2which is proved in Theorem 6.2. This result indicates that when

mgets smaller, one substance penetrates into the other faster, which can also be seen from the

numerical result in [7].

Acknowledgement

The authors gratefully acknowledge funding from the EPSRC EP/W522545/1. This paper is

based on part of corresponding author’s Ph.D thesis at Swansea University

2. Half-line case: preliminaries

First, we give the deﬁnition of the weak solution of problem (1.1). Note that the uniqueness

of the weak solution of (1.1) is proved in [4].

Deﬁnition 2.1. A function w is a weak solution of (1.1) if

(i) w∈L∞(ST),

(ii) D(w)∈ D( ˆw)+L2(0,T;W1,2

0(R+)), where ˆw∈C∞(R+)is a smooth function with ˆw=U0

when x =0and ˆw=−V0when x >1,

(iii) w satisﬁes

ZR+

w0(x)ξ(x,0)dx+ZZST

wξtdxdt=ZZSTD(w)xξxdxdt.(2.1)

for all ξ∈ FT:=nξ∈C1(ST) : ξ(0,t)=ξ(·,T)=0 for t∈(0,T) and supp ξ⊂[0,J]×[0,T]

for some J>0}.

We state a free-boundary problem, including interface conditions, that is satisﬁed by the

solution wof (1.1) under some regularity assumptions and conditions on the form of the free

boundary. The following result follows from a similar approach to that of [12, Theorem 5]. We

sketch the key points here, focusing on the parts where our problem needs a slightly diﬀerent

argument.

Theorem 2.2. Let w be the unique weak solution of problem (1.1). Suppose that there exists a

function β: [0,T]→R+such that for each t ∈[0,T],

w(x,t)>0 if x< β(t) and w(x,t)<0 if x> β(t).

Then if t 7→ β(t)is suﬃciently smooth and the functions u :=w+and v :=−w−are smooth up to

β(t), the functions u,v satisfy











ut=φ(u)xx,in (x,t)∈ST:x< β(t),

vt=εφ(v)xx,in (x,t)∈ST:x> β(t),

hφ(u)i=εhφ(v)i=0,on ΓT:={(x,t)∈ST:x=β(t)},

hviβ0(t)=hφ(u)x−εφ(v)xi,on ΓT:={(x,t)∈ST:x=β(t)},

u=U0,on {0}×[0,T],

u(·,0) =u∞

0(·),v(·,0) =v∞

0(·),in R+,

(2.2)

where h·i denotes the jump across β(t)from {x< β(t)}to {x> β(t)},

hαi:=lim

x&β(t)α(x,t)−lim

x%β(t)α(x,t),

and β0(t)denotes the speed of propagation of the free boundary β(t).

Proof. We recall that (u,v) satisﬁes

ZZST

(u−v)ξtdxdt+ZR+

(u∞

0−v∞

0)ξ(x,0)dx=ZZST

(φ(u)−εφ(v))xξxdxdt,

for all ξ∈ FT, from which we have

h−u+viβ0(t)+h−φ(u)x+εφ(v)xi=0 on ΓT:={(x,t)∈ST:x=β(t)}.(2.3)

Now we know that D(w) is a continuous function of xfor almost every t∈[0,T], since

D(w)∈ D( ˆw)+L2(0,T;W1,2(R+)) by Deﬁnition 2.1 (ii). So hD(w)i=0, which implies

−lim

x&β(t)εφ(−w−)−lim

x%β(t)φ(w+)=−lim

x&β(t)εφ(v)−lim

x%β(t)φ(u)=0.

Therefore we get

hφ(u)i=εhφ(v)i=0.(2.4)

Moreover, since φ∈C2(R) is strictly increasing, u(·,t) is continuous across β(t) and if ε > 0,

v(·,t) is also continuous across β(t), so that

hui=0 if ε≥0,(2.5)

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Self-similarsolutionsoffastreactionlimitproblemwithnonlineardiusionElaineCrooks,YiniDuDepartmentofMathematics,FacultyofScienceandEngineering,SwanseaUniversity,SwanseaSA18EN,UKAbstractInthispaper,wepresentanapproachtocharacterisingself-similarfast-reactionlimitsofsystemswithnonlineardiusion.Forapp...

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Self-similar solutions of fast reaction limit problem with nonlinear diusion Elaine Crooks Yini Du

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