# Positive Solutions of a Nonlinear Parabolic Partial Differential Equation

Authors: Chengbo Zhai; Shunyong Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We deal with the existence and uniqueness of positive solutions to a class of nonlinear parabolic partial differential equations, by using some fixed point theorems for mixed monotone operators with perturbation.


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## Body

## 1. Introduction

In this paper, we consider a class of nonlinear parabolic partial differential equations of the form(1)ut-2Δu=f(t,u,u)+g(t,u),(x,t)∈Ω×(0,T),u(x,t)=0,(x,t)∈Ω×{0}∪∂Ω×(0,T),
where Ω is a bounded smooth domain in Rn and 0<T<+∞; we denote QT=Ω×(0,T).Problems related to nonlinear parabolic equations arise in many mathematical models of applied science, such as nuclear science, chemical reactions, heat transfer, population dynamics, and biological sciences, and have attracted a great deal of attention in the literature; see [1–6] and the references therein. In recent years, there are many results about existence, uniqueness, blowing-up, global existence, critical exponent, and other properties of the solution; see [4, 5, 7–12], among others. Some of the authors who investigated parabolic equations were using the method of upper and lower solutions; see [11], for example. Different from the works mentioned above, in the present paper, we will utilize some fixed point theorems for mixed monotone operators with perturbation to study the existence and uniqueness of positive solutions to the nonlinear parabolic partial differential equation (1).With this context in mind, the outline of this paper is as follows. In Section2 we will recall certain results from the theory of notations and results of monotone operators. In Section 3, we will provide some conditions under which problem (1) will have a unique positive solution. Finally, in Section 4, we will provide an example, which explicates the applicability of our result.


## 2. Preliminaries

In this sequel, we present some basic concepts in ordered Banach spaces and two fixed point theorems which we will use later. For convenience, we suggest that one refers to [13–16] for details.Suppose that(E,∥·∥) is a real Banach space which is partially ordered by a cone P⊂E; that is, x≤y if and only if y-x∈P. If x≤y and x≠y, then we denote x<y or y>x. By θ we denote the zero element of E. Recall that a nonempty closed convex set P⊂E is a cone if it satisfies (i)x∈P,λ≥0⇒λx∈P;(ii)x∈P,-x∈P⇒x=θ. P is called normal if there exists a constant N>0 such that, for all x,y∈E,θ≤x≤y implies that ∥x∥≤N∥y∥; in this case N is called the normality constant of P. We say that an operator A:E→E is increasing (decreasing) if x≤y implies that Ax≤Ay(Ax≥Ay). For all x,y∈E, the notation x~y means that there exist λ>0 and μ>0 such that λx≤y≤μx. Clearly, ~ is an equivalence relation. Given h>θ (i.e., h≥θ and h≠θ), we denote by Ph the set Ph={x∈E∣x~h}. It is easy to see that  Ph⊂P.Definition 1 (see [13, 14]).
A
:
P
×
P
→
P is said to be a mixed monotone operator if A(x,y) is increasing in x and decreasing in y; that is, ui,vi(i=1,2)∈P, u1≤u2, v1≥v2 imply that A(u1,v1)≤A(u2,v2). Element x∈P is called a fixed point of A if A(x,x)=x.Definition 2.
An operatorA:P→P is said to be subhomogeneous if it satisfies A(tx)≥tAx, ∀t∈(0,1), x∈P.Definition 3.
LetD=P and let β be a real number with 0≤β<1. An operator A:D→D is said to be β-concave if it satisfies A(tx)≥tβA(x), ∀t∈(0,1), x∈D.Lemma 4 (see Theorem  2.1 in [16]).
Letβ∈(0,1). A:P×P→P is a mixed monotone operator and satisfies A(tx,t-1y)≥tβA(x,y), ∀t∈(0,1), x,y∈P.B:P→P is an increasing subhomogeneous operator. Assume that (i) there is h0∈Ph such that A(h0,h0)∈Ph and Bh0∈Ph and (ii) there exists a constant δ0>0 such that A(x,y)≥δ0Bx, ∀x,y∈P. Then we have the following:(1)
A:Ph×Ph→Ph and B:Ph→Ph;(2)
there existu0,v0∈Ph and r∈(0,1) such that
(2)rv0≤u0<v0,u0≤A(u0,v0)+Bu0≤A(v0,u0)+Bv0≤v0;(3)
the operator equationA(x,x)+Bx=x has a unique solution x* in Ph;(4)
for any initial valuesx0,y0∈Ph, constructing successively the sequences
(3)xn=A(xn-1,yn-1)+Bxn-1,yn=A(yn-1,xn-1)+Byn-1,kkkkkkkkkkkkn=1,2,…,one has xn→x* and yn→x* as n→∞.Lemma 5 (see Theorem  2.4 in [16]).
Letβ∈(0,1). A:P×P→P is a mixed monotone operator and satisfies A(tx,t-1y)≥tA(x,y), ∀t∈(0,1), x,y∈P.B:P→P is an increasing β-concave operator. Assume that (i) there is h0∈Ph such that A(h0,h0)∈Ph and Bh0∈Ph and (ii) there exists a constant δ0>0 such that A(x,y)≤δ0Bx, ∀x,y∈P. Then we have the following:(1)
A:Ph×Ph→Ph and B:Ph→Ph;(2)
there existu0,v0∈Ph and r∈(0,1) such that
(4)rv0≤u0<v0,u0≤A(u0,v0)+Bu0≤A(v0,u0)+Bv0≤v0;(3)
the operator equationA(x,x)+Bx=x has a unique solution x* in Ph;(4)
for any initial valuesx0,y0∈Ph, constructing successively the sequences
(5)xn=A(xn-1,yn-1)+Bxn-1,yn=A(yn-1,xn-1)+Byn-1,kkkkkkkkkkkkn=1,2,…,one has xn→x* and yn→x* as n→∞.


## 3. Existence and Uniqueness of Positive Solutions

In this section, we will apply Lemmas4 and 5 to study the problem (1), and we obtain some new results on the existence and uniqueness of positive solutions. The method used here is relatively new to the literature and so are the existence and uniqueness results of the nonlinear parabolic partial differential equations.In our considerations, we work in the Banach spaceE=C(QT¯)={u:QT¯→Riscontinuous} with the standard norm ∥u∥=max⁡⁡{|u(x,t)|:(x,t)∈QT¯}. Notice that this space can be equipped with a partial order given by
(6)u,v∈E,u≤v⟺u(x,t)≤v(x,t),(x,t)∈QT¯.
Set P={u∈C(QT¯)∣u(x,t)≥0,(x,t)∈QT¯}, as the standard cone. It is clear that P is a normal cone in C(QT¯) and the normality constant is 1.The following assumptions on the nonlinear functionsf(x,u,v) and g(x,u) are as follows:(H1)
f:[0,T]×[0,+∞)×[0,+∞)→[0,+∞) is continuous and g:[0,T]×[0,+∞)→[0,+∞) is continuous;(H2)
f(t,u,v) is increasing in u∈[0,+∞), for fixed t∈[0,T] and v∈[0,+∞), and decreasing in v∈[0,+∞), for fixed t∈[0,T] and u∈[0,+∞), and g(t,u) is increasing in u∈[0,+∞), for fixed t∈[0,T];(H3)
g(t,λu)≥λg(t,u), for λ∈(0,1), t∈[0,T], and u∈[0,+∞), and there exists a constant β∈(0,1) such that f(t,λu,λ-1v)≥λβf(t,u,v), ∀t∈[0,T], λ∈(0,1), u,v∈[0,+∞);(H4)
there exists a constantδ0>0 such that f(t,u,v)≥δ0g(t,u), t∈[0,T], u,v≥0;(H5)
min⁡(x,t)∈QT¯⁡f(t,φ,φ)>0, min⁡(x,t)∈QT¯⁡g(t,φ)>0, where φ is the positive solution of the equation
(7)ut-Δu=1,(x,t)∈Ω×(0,T),u(x,t)=0,(x,t)∈Ω×{0}∪∂Ω×(0,T);that is, φ(x,t)≥0 for (x,t)∈QT¯. The existence of φ follows from Theorem  7 in [17, Chapter 7].Theorem 6.
Assume that (H1)-(H5) are satisfied. Then the nonlinear parabolic partial differential equation (1) has a unique positive solution u* in Pφ.Proof.
We divide the proof into three steps.
Step
1. By the existence and uniqueness of linear parabolic partial differential equations, we consider the operator A:E×E→E as the solution of (8) for given u,v∈Pφ as follows:
(8)∂A(u,v)∂t-ΔA(u,v)=f(t,u,v)inQT,A(u,v)=0on⁡Ω×{0}∪∂Ω×(0,T).
Indeed, for t∈[0,T],u,v∈E, we have f(t,u,v)∈Ln(QT¯). From the regularity theory of the heat equation, we conclude that A(u,v)∈W2,1(QT¯), and there exists some α>0 such that W2,1(QT¯)⊂C2α,α(QT¯), so the operator A(u,v) is well defined as the solution of (8) for given u,v∈Pφ. For u,v∈Pφ, ∃ru,rv∈(0,1) such that
(9)ruφ(x,t)≤u(x,t)≤1ruφ(x,t),rvφ(x,t)≤v(x,t)≤1rvφ(x,t),
for (x,t)∈QT¯. Let r0=min⁡{ru,rv}. Then from (H3),
(10)f(t,u(x,t),v(x,t))≥f(t,r0φ(x,t),1r0φ(x,t))≥r0βf(t,φ(x,t),φ(x,t)),f(t,u(x,t),v(x,t))≤f(t,1r0φ(x,t),r0φ(x,t))≤1r0βf(t,φ(x,t),φ(x,t)),
for (x,t)∈QT¯, u,v∈Pφ. By (H3) and the above inequalities, we have
(11)∂A(u,v)∂t-ΔA(u,v)=f(t,u,v)≥r0βf(t,φ,φ)≥r0βmin⁡(x,t)∈QT¯f(t,φ,φ)=r0βmin⁡(x,t)∈QT¯f(t,φ,φ)(∂φ∂t-Δφ),∂A(u,v)∂t-ΔA(u,v)=f(t,u,v)≤1r0βf(t,φ,φ)≤1r0βmax⁡(x,t)∈QT¯f(t,φ,φ)=1r0βmax⁡(x,t)∈QT¯f(t,φ,φ)(∂φ∂t-Δφ),
for (x,t)∈QT, u,v∈Pφ. By the comparison principle for parabolic partial differential equations [18, Lemma 3.1.8, page 108], we know that
(12)1r0βmax⁡(x,t)∈QT¯f(t,φ,φ)φ≥A(u,v)≥r0βmin⁡(x,t)∈QT¯f(t,φ,φ)φ≥0,
for u,v∈Pφ, which is satisfied by (H5). So A:Pφ×Pφ→P.
Letr1=r0βmin⁡(x,t)∈QT¯⁡f(t,φ,φ) and r2=(1/r0β)max⁡(x,t)∈QT¯⁡f(t,φ,φ); we get that r2φ≥A(u,v)≥r1φ, for u,v∈Pφ; that is, A(φ,φ)∈Pφ. By (H2), we have
(13)∂Au(u,v)∂t-ΔAu(u,v)=fu(t,u,v)≥0,∂Av(u,v)∂t-ΔAv(u,v)=fv(t,u,v)≤0,
for (x,t)∈QT, u,v∈Pφ. Also, by the comparison principle for parabolic partial differential equations, we observe that Au≥0,Av≤0; that is, A is a mixed monotone operator. Moreover, by (H3), we have
(14)∂A(λu,λ-1v)∂t-ΔA(λu,λ-1v)=f(t,λu,λ-1v)≥λβf(t,u,v)=∂(λβA(u,v))∂t-Δ(λβA(u,v)),
for (x,t)∈QT, u,v∈Pφ. Also, by the comparison principle for parabolic partial differential equations, we conclude that A(λu,λ-1v)≥λβA(u,v), for u,v∈Pφ.
Step
2. Similar to Step 1, we can also construct the operator B:E→E as the solution of (15) for given u∈Pφ:
(15)∂Bu∂t-ΔBu=g(t,u)inQT,Bu=0on⁡Ω×{0}∪∂Ω×(0,T),
and we get the fact that the operator Bu is well defined as the solution of (15) for given u∈Pφ. From (H3), we get
(16)g(t,u(x,t))≥g(t,r0φ(x,t))≥r0g(t,φ(x,t)),g(t,u(x,t))≤g(t,1r0φ(x,t))≤1r0g(t,φ(x,t)),
for (x,t)∈QT¯, u∈Pφ. By (H3) and the above inequalities, we have
(17)∂Bu∂t-ΔBu=g(t,u)≥r0g(t,φ)≥r0min⁡(x,t)∈QT¯g(t,φ)=r0min⁡(x,t)∈QT¯g(t,φ)(∂φ∂t-Δφ),∂Bu∂t-ΔBu=g(t,u)≤1r0g(t,φ)≤1r0max⁡(x,t)∈QT¯g(t,φ)=1r0max⁡(x,t)∈QT¯g(t,φ)(∂φ∂t-Δφ),
for (x,t)∈QT, u∈Pφ. Also, by the comparison principle for parabolic partial differential equations, we know that
(18)1r0max⁡(x,t)∈QT¯g(t,φ)φ≥Bu≥r0min⁡(x,t)∈QT¯g(t,φ)φ≥0,
for u∈Pφ, which is satisfied by (H5). So B:Pφ→P.
Letr3=r0min⁡(x,t)∈QT¯⁡g(t,φ) and r4=(1/r0)max⁡(x,t)∈QT¯⁡g(t,φ); we get that r4φ≥Bu≥r3φ, where u∈Pφ; that is, Bφ∈Pφ. By (H2), we have
(19)∂Buu∂t-ΔBuu=gu(t,u)≥0,
for (x,t)∈QT, u∈Pφ. By the comparison principle for parabolic partial differential equations, we observe that Bu≥0; that is, B is an increasing operator. Moreover, by (H3), we have
(20)∂B(λu)∂t-ΔB(λu)=g(t,λu)≥λg(t,u)=∂(λBu)∂t-Δ(λBu),
for (x,t)∈QT,u∈Pφ. By the comparison principle for parabolic partial differential equations, we conclude that B(λu)≥λBu, for u∈Pφ.
Step
3. By (H4), we have
(21)∂A(u,v)∂t-ΔA(u,v)=f(t,u,v)≥δ0g(t,u)=∂(δ0Bu)∂t-Δ(δ0Bu),
for (x,t)∈QT, u,v∈Pφ. By the comparison principle for parabolic partial differential equations, we know that A(u,v)≥δ0Bu, where u,v∈Pφ.
Therefore, the operatorsA and B satisfy all the conditions in Lemma 4. Thus, the operator equation A(u,u)+Bu=u has a unique solution u* in Pφ. By
(22)∂u*∂t=∂A(u*,u*)∂t+∂Bu*∂t=(ΔA(u*,u*)+f(t,u*,u*))+(ΔBu*+g(t,u*))=2Δu*+f(t,u*,u*)+g(t,u*),kkkkkkkkkk(x,t)∈Ω×(0,T),kkiiu*=A(u*,u*)+Bu*=0,(x,t)∈Ω×{0}∪∂Ω×(0,T),
we obtain that the nonlinear parabolic partial differential equation (1) has a unique positive solution u* in Pφ. The proof is complete.Further, we make some other assumptions on the nonlinear functionsf(t,u,v) and g(t,u):(H6)
there exists a constantβ∈(0,1) such that g(t,λu)≥λβg(t,u), ∀t∈[0,T], λ∈(0,1), u∈[0,+∞), and f(t,λu,λ-1v)≥λf(t,u,v), for λ∈(0,1), t∈[0,T], u,v∈[0,+∞);(H7)
there exists a constantδ0>0 such that f(t,u,v)≤δ0g(t,u), t∈[0,T], u,v≥0.By using Lemma5, we can also easily prove the following conclusion.Theorem 7.
Assume that (H1), (H2), (H5), (H6), and (H7) are satisfied. Then the nonlinear parabolic partial differential equation (1) has a unique positive solution u* in Pφ.


## 4. An Example

We now present one example to illustrate Theorem6.Example 1.
Consider the following parabolic partial differential equation:(23)ut-2Δu=u1/2+(u+1)-1/3+uu+1a(t)+b(t)+c,(x,t)∈Ω×(0,1),u(x,t)=0,(x,t)∈Ω×{0}∪∂Ω×(0,1),
where c>0 is a constant, a,b:[0,1]→(0,+∞) are continuous, and Ω is a bounded smooth domain in Rn; we denote Q1=Ω×(0,1). Set φ to be the positive solution of the following:
(24)ut-Δu=1,(x,t)∈Ω×(0,1),u(x,t)=0,(x,t)∈Ω×{0}∪∂Ω×(0,1);
that is, φ(x,t)≥0, for (x,t)∈Q1¯. We can conclude that the nonlinear parabolic partial differential equation (23) has a unique positive solution u* in Pφ.Proof .
In this example, we take0<d<c and let
(25)f(t,u,v)=u1/2+(v+1)-1/3+b(t)+d,g(t,u)=uu+1a(t)+c-d,β=12,a*=max⁡{a(t):t∈[0,1]}.
Obviously, a*>0, f:[0,1]×[0,+∞)×[0,+∞)→[0,+∞) is continuous, and g:[0,1]×[0,+∞)→[0,+∞) is continuous. And f(t,u,v) is increasing in u∈[0,+∞), for fixed t∈[0,1] and v∈[0,+∞), and decreasing in v∈[0,+∞), for fixed t∈[0,1] and u∈[0,+∞), and g(t,u) is increasing in u∈[0,+∞), for fixed t∈[0,1]. Besides, for λ∈(0,1), t∈[0,1], and u,v∈[0,+∞), we have
(26)g(t,λu)=λuλu+1a(t)+c-d≥λuu+1a(t)+λ(c-d)=λg(t,u),f(t,λu,λ-1v)=(λu)1/2+(λ-1v+1)-1/3+b(t)+d≥λ1/2[u1/2+(v+1)-1/3+b(t)+d]=λ1/2f(t,u,v).
Moreover, if we take δ0∈(0,d/(a*+c-d)], then we obtain
(27)f(t,u,v)=u1/2+(v+1)-1/3+b(t)+d≥d=da*+c-d·(a*+c-d)≥δ0[uu+1a(t)+c-d]=δ0g(t,u).
Further, min⁡(x,t)∈Q1¯⁡f(t,φ,φ)≥d>0 and min⁡(x,t)∈Q1¯⁡g(t,φ)≥c-d>0.
Hence, all the conditions of Theorem6 are satisfied. An application of Theorem 6 implies that the nonlinear parabolic partial differential equation (23) has a unique positive solution u* in Pφ, where Pφ={u∈C(Q1¯)∣thereexistλ=λ(u,φ), μ=μ(u,φ)>0suchthatλφ≤u≤μφ}.



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