# Some Applications of(C0,1)-Semigroups

Authors: Yu-Hsien Chang; Cheng-hong Hong
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This study is originally motivated by the thermal balance equations for the gas and solid interphase heat-transfer for the fast-igniting catalytic converter of automobiles. Instead of solving this problem directly, we proved some results concerning the existence and uniqueness for abstract semilinear initial value problem by means of(C0,1)-semigroup theories on locally convex topological space. The most enjoyable here is that these results not only can be applied to solve the interphase heat-transfer for the fast-igniting catalytic converter of automobile under the situation of preheating at the entry edge of converter, but also can be applied to some other practical problems.


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## Body

## 1. Introduction

Monolithic catalytic reactors are used in a variety of environmental and industrial applications. There has been a great deal of research in catalytic converter technology since in the mid-1970s. The initial few minutes after starting a car when the converter is still cold is very impotent, since during this period the converter is not able to perform its role of converting exhaust carbon monoxide and unburned hydrocarbons to carbon dioxide and water due to low converter temperatures. From an environment point of view how to cope with motor vehicle exhaust emission is an increasing concern in automobile engineering. This concern and many others lead to various mathematical models for the study of interphase heat transfer problem in catalytic converter. There have been suggestions made on how to decrease noxious gas, such as introducing a heater at the inlet. In this study we consider the thermal balance equations for the gas and solid interphase heat-transfer for the fast-igniting catalytic converter of automobiles. This problem can be simplified to the following mathematical model, which was original proposed by Leighton and Chang [1]:(1.1)∂u∂t+a∂u∂x+cu=cv,0<t<T,0<x≤l,∂v∂t+bv=bu+λev,0<t<T,0<x≤l,u(t,0)=η,υ(t,0)=0,0<t<T,u(0,x)=u0(x),υ(0,x)=v0(x),0<x≤l,
where u0 and υ0 are continuous functions on [0,l] with u0(0)=η and υ0(0)=0. The physical meaning of the functions and parameters in this system was given in Chang et al. [2]. The additional initial condition υ(t,0)=0 is understood in the situation of preheating at the entry edge of converter. Instead of solving system (1.1) directly, we proved some results concerning the existence and uniqueness for the classical local solution of the semilinear initial value problem:(1.2)du(t)dt=Au(t)+f(t,u),t≥0,u(0)=u0∈D(A),
where A is a generator of a (C0,1)-semigroup on a locally convex topological space. Before we consider the semilinear initial value problem (1.2), we consider following abstract Cauchy problem on complete locally convex topological linear space firstly:(1.3)du(t)dt=Au(t),t≥0,u(0)=u0∈D(A).It is well known that as long asA is a generator of a C0-semigroup {T(t)}t≥0 on a Banach space X, there exists a nonnegative real number ω such that {e-ωtT(t)x;t≥0} being bounded in X for every given x∈X. But this is not true in general, for example, if X is a complex Hausdorff locally convex topological linear space and {T(t)}t≥0 is a C0-semigroup on X, then {e-ωtT(t)x;t≥0} is not bounded for any nonnegative real number ω. Sometimes, complex Hausdorff locally convex topological linear space (hereafter, we will denote it by lcs) being a more natural domain for some partial differential operators. Partial differential equations are being currently studied on lcs, for example, the spaces of continuous functions, test functions and distributions with nonnormable lcs topology. Moreover, in a norm space endow with locally convex topology, using the locally convex topology sometimes is also better than using the norm in certain cases. Certain example was given in [3].


## 2. Preliminarie

Babalola [4] considers the operator A=x(∂/∂x) for the Cauchy problem (1.3), the author showed that A generates a (C0,1)-semigroup {T(t)}t≥0, when X is a lcs. Roughly speaking, the (C0,1)-semigroup is a C0-semigroup on X which can be characterized as having the property that for each continuous seminorm p on X there exists a positive number σp (which depends on the seminorm p) and a continuous seminorm q on X such that(2.1)p(T(t)x)≤eσptq(x)∀x∈X,t≥0.
Precise definition of (C0,1)-semigroup will be given in Definition 2.1. A semigroup {T(t)}t≥0 is called an equicontinuous semigroup if σp=0 in the above inequality, and it is called a quasiequicontinuous semigroup if eσpt is replaced by eωt for some positive constant ω (independent of seminorm p). Follows from these definitions, (C0,1)-semigroup is a more general than the quasi-equicontinuous semigroup. However, (C0,1)-semigroup lacks some property which the equicontinuous semigroup has. For instance, the resolvent operator R(λ:A) exist for some λ>0 as {T(t)}t≥0 is an equicontinuous C0-semigroup but it is not true, while {T(t)}t≥0 is a (C0,1)-semigroup. In fact, let X=S(R) be the Schwartz set of functions with topology determined by the seminorms {pmn}. The seminorm pmn on X is defined by(2.2)pmn(f)=‖MmDnf‖foreverypairofnonnegativeintegerm,n,
where (Mf)(x)=xf(x), (Df)(x)=(d/dx)f(x), and ∥f∥=(∫R|f(x)|2dx)2 for every f∈X. Let the group G={S(ξ):-∞<ξ<∞} be defined by(2.3)(S(ξ)f)(x)=f(eξx),∀f∈X.Thenpmn(S(ξ)f)=e(n-m-1/2)pmn(f), and hence G is a (C0,1) group with generator A=x(∂/∂x). However, it is impossible to find any positive number ω such that the group {e-ω|ξ|S(ξ):|ξ|≥0} is equicontinuous. This shows that the resolvent operator R(λ:A) does not exist for any λ>0.In view of above evidences, we would like to consider the abstract semilinear initial value problem:(2.4)du(t)dt=Au(t)+f(t,u),t≥0,u(0)=u0∈D(A),
where A is a generator of (C0,1)-semigroup in a lcsX and f:[0,∞)×X→X is a continuous function. For discussing the (C0,1)-semigroup and its underlying space X, we will use the following terminologies and lemmas in this paper. We say a family of continuous seminorms {pα;α∈Γ} is saturated if for any pair α1,α2∈Γ, there exists a α0∈Γ such that(2.5)pαi(x)≤pα0(x)(i=1,2)∀x∈X.
We always assume that X is a lcs endowed with a saturated family of continuous seminorms {pα;α∈Γ} such that the family {ɛVα;α∈Γ,ɛ>0} is a base of neighborhoods at the origin for the topology of X, where Vα is the set {x∈X:pα(x)≤1} for every α∈Γ. We denote LΓ(X) by the family of linear operators T:X→X such that for each α∈Γ there exist positive number λα,T with(2.6)TVα⊂λα,TVα,∀α∈Γ.
It is easy to see that T∈LΓ(X) is continuous and satisfied:(2.7)pα(Tx)≤λα,Tpα(x),∀x∈X,α∈Γ.
Furthermore, with addition of defined pointwise and multiplication by composition, LΓ(X) becomes an algebra, also we can define a topology on LΓ(X). For each pα∈{pα;α∈Γ}, the real-valued function Pα on LΓ(X), defined by(2.8)Pα(T)=inf⁡⁡{λα,T:pα(Tx)≤λα,Tpα(x)∀x∈X}=sup⁡x∈X,pα(x)≤1⁡pα(Tx)
is a seminorm with the additional properties that(2.9)Pα(T1T2)≤Pα(T1)Pα(T2),∀T1,T2∈LΓ(X),α∈Γ,Pα(Tx)≤Pα(T)pα(x),∀T∈LΓ(X),α∈Γ.The topology onLΓ(X) is defined by the family {Pα:α∈Γ} of seminorms on LΓ(X). Under this topology, LΓ(X) becomes a Hausdroff locally multiplicatively convex topological algebra, and LΓ(X) is complete whenever X does.Anlcs can be regarded as a projective limit of Banach spaces (see, e.g., [5, page 231]). Firstly, we express the definition of a projective system of spaces and homomorphism. Let (I,≤) be a directed set, and let (Xα)α∈I be a family of normed spaces. Suppose we have a family of homomorphism for all with following properties:(1)fαα is the identity in Xα;(2)fαγ=fαβ∘fβγ for all α≤β≤γ.Then, the set of pair(Xα,fαβ) is called a projective system of normed spaces and homomorphism over I.Hence, we can define the projective limit (it is also called inverse limit) of the projective system as a particular subspace of the direct product of theXα’s:(2.10)lim⁡←⁡⁡Xα={(Xα)∈∏α∈IXα:xα=fαβ(xβ)∀α≤β}.
For a given lcsX, we consider the normed space Xα and the set of all linear operators L(Xα) as follows. For each α∈Γ, let be the coset of x in the quotient space X/pα-1(0), and let(2.11)‖xα‖α=pα(x)for eachx∈X.
Then, ∥·∥α is a norm on the quotient space X/pα-1(0). Under the topology induced by ∥·∥α, the normed linear space formed by the elements of X/pα-1(0) is denoted by Xα and denoting its completion by Xα¯. For each α∈Γ, the natural homomorphism πα:X→Xα, defined by πα(x)=xα for each x∈X is continuous and onto.Now we can relateLΓ(X) to the uniform norm algebra L(Xα). Let α∈Γ and S∈LΓ(X), then the operator Sα:Xα→Xα defined by Sαxα=(Sx)α. It is easy to see Sα that in L(Xα) (for detail, please see [4, Proposition  1.12]). Moreover, Sα is uniquely extensible to a bounded linear operator Sα¯ on Xα¯ such that(2.12)‖Sα¯‖α=‖Sα‖α=sup⁡‖x‖α≤1‖Sαxα‖α=Pα(S).
To identify LΓ(X) with projective families of operators, we order the index set Γ by α≥α′ if and only if Vα⊂Vα′. Define for α≥α′, an operator πα′α:Xα→Xα′ by πα′α(xα)=xα′ is a continuous homomorphism also it can be extended to Xα¯. The set (Xα¯,πα′α) consists of a projective system of spaces and homomorphism over Γ. Denote the project limit by  lim←Xα¯=X. We call a projective family of Banach spaces is saturated if every finite product of members is still a member. Throughout this paper we always assume that our projective family of Banach spaces {Xα¯:α∈Γ} is saturated, X is a complete lcs, and Xα is a Banach space for each α∈Γ. Let Sα be a linear operator from D(Sα)⊂Xα into Xα(α∈Γ). We call {Sα:α∈Γ} a (saturated) projective family of operators if and only if Sα(πβαxβ)=πβα(Sβxβ) for xβ∈D(Sβ) and β≥α. For such a family, we can define a linear operator S on the project limit D(S) of {Sα:α∈Γ} by πα(Sx)=Sα(παx) for x∈D(S) and α∈Γ, and we call the operator S be the project limit of the family of operators {Sα:α∈Γ}. If Sα∈L(Xα) for each α∈Γ, then S∈LΓ(X). Moreover, the family {Sα¯:α∈Γ} associated with S∈LΓ(X) is projective, and its limit is S.Now, we are ready to define the(C0,1)-semigroup on X.Definition 2.1.
The family of continuous linear operators{T(t)}t≥0⊂L(X) is called a C0-semigroup if and only if:(1)
T(t+s)x=T(t)(T(s)x) for all s,t≥0 and x∈X;(2)
T(0)x=x for all x∈X;(3)
T(t)x→x as t↓0 for every x∈X.It is called an LΓ(X)-operator semigroup of class (C0,1) ((C0,1)-semigroup for shot) if and only if, in addition, for each α and each positive δ, there exists a positive number λ=λ(α,{T(t):0≤t≤δ}) such that T(t)Vα⊂λVα, for all 0≤t≤δ, where Vα={x∈X:pα(x)≤1} for every α∈Γ.Definition 2.2.
We call an operatorS:D(S)⊂X→X s a compartmentalized operator (with respective to {pα:α∈Γ} if for each α∈Γ, the operator Sα:πα(D(S))→Xα, given by Sαxα=(Sx)α for xα∈πα(D(S))), is well defined. Follows from the definitions of LΓ(X) and compartmentalized operator. It is easy to see that every operator S in LΓ(X) is a compartmentalized operator.The following results linked the(C0,1)-semigroup in lcs with the well-know C0-semigroup in Banach space. For details, please see the reference [4].Lemma 2.3 (see [4, Theorem  2.5]).
There is a 1-1 correspondence between(C0,1)-semigroup {S(ξ):ξ≥0} on complete lcs and projective families of C0-semigroup{Sα(ξ)=πα(S(ξ)):ξ≥0} on Banach space Xα such that if A is the generator of a (C0,1) semigroup, and {Aα} is the family of generators associated with the corresponding C0-semigroup on {Xα:α∈Γ}, then {Aα} is the projective family, and its limit is A.Babalola (see [4, Theorem  2.6]) show that the C0-semigroup {S(ξ):ξ≥0} is a (C0,1)-semigroup on a locally convex space X if and only if there exist sets {Mα;α∈Γ} and {σα;α∈Γ} of real numbers such that
(2.13)pα(S(ξ)x)≤Mαeσαξpα(x)∀ξ≥0,x∈X.
The author also showed that, by choosing a suitable seminorm qα on X, the last inequality is equivalent to
(2.14)p(S(ξ)x)≤eσpξq(x)∀x∈X,ξ≥0.
For details, please see [4, page 171].From this property, Definition2.1 is equivalent to the following definition.Definition 2.4.
Let{S(ξ):ξ≥0}⊂L(X) be a family of continuous linear operators on X, it is a (C0,1)-semigroup if and only if it satisfies following conditions (1)
{S(ξ):ξ≥0} is a semigroup of class (C0) in X;(2)
for each continuous seminormp on X, there exist a nonnegative number σp and a continuous seminorm q on X such that(2.15)p(S(ξ)x)≤eσpξq(x)∀ξ≥0,x∈X.
Now we can start to consider the abstract semilinear initial value problem (2.4). Suppose X is a complete lcs which is the saturated projective limit of Banach spaces {Xα;α∈Γ}. We are searching suitable conditions for the function f such that (2.4) has a mild solution. At first, we consider the function f in (2.4) only depends on the variable t. That is f:[0,t]→X. For any t0∈[0,t], in as much as X is the projective limit of Xα, for each α∈Γ, there is a function fα:[0,t]→Xα such that fα(t0)=πα(f(t0))∈Xα, and it satisfies
(2.16)f(t0)=lim⁡←⁡fα(t0)=lim⁡←⁡πα(f(t0)).
We assume that for everyα∈Γ, fα:[0,t]→Xα, is Bochner integrable (integrable for short), and the integration is denoted by ∫0tfα(s)ds. Since X is the saturated projective limit space of {Xα;α∈Γ}, if f:[0,t]→X satisfies that for every α∈Γ, fα=πα(f) is integrable and ∫0tfα(s)ds=zα∈Xα, then
(2.17)zα′=∫0tfα′(s)ds=∫0tπα′α(fα(s))ds=πα′α(∫0tfα(s)ds)=πα′α(zα),(α≥α′).
The same reason also implies that {zα∈Xα;α∈Γ} has a limit z=lim←zα in X. In this case, we say f is integrable and denotes the integration of f on [0,t] by ∫0tf(s)ds=z. Since πα is the projection from X onto Xα and f(s)=(fα(s))α∈Γ, we have
(2.18)∫0tf(s)ds=z=lim⁡←zα=lim⁡←⁡∫0tfα(s)ds=lim⁡←⁡∫0tπα(f(s))ds=lim⁡←⁡πα(∫0tf(s)ds),(2.19)∫0tf(s)ds=∫0tlim⁡←⁡fα(s)ds=∫0tlim⁡←⁡πα(f(s))ds.
Combine (2.18) and (2.19), we have
(2.20)∫0tlim⁡⟵⁡πα(f(s))ds=∫0tf(s)ds=lim⁡←⁡πα(∫0tf(s)ds).
We sayf:[0,T]×X→X is uniformly Lipschitz continuous corresponding to α∈Γ with positive Lipschitz constant Lα (independent of s,u,v) if fα=πα(f) satisfies Lipschitz continuous condition and 0<supα∈ΓLα<∞. Moreover, if there exists a constantc (independent of α∈Γ) such that the Lipschitz constant Lα(t,c) of fα depends on (t,c), satisfies locally Lipschitz condition and 0<supα∈ΓLα(t,c)<∞. Then, we say f is locally Lipschitz continuous.Definition 2.5.
For each fixedα∈Γ and positive real number T, we define a norm ∥.∥α,∞ on the function space {u:[0,T]→X:uiscontinuous} by∥u∥α,∞=∥πα(u)∥α,∞=sups∈[0,T]∥uα(s)∥α.


## 3. Main Result

Firstly, we consider the initial value problem:(3.1)ddtu(t)=Au(t)+f(t),t≥0,u(0)=u0∈D(A),
where A is a generator of a (C0,1)-semigroup on a lcs and f:[0,T]×X→X. We found that (3.1) has a unique mild solution provide the function f is integrable. Furthermore, if f is continuous, then this mild solution is also the solution of the differential equation.Namely, we have following theorem.Theorem 3.1.
IfA is the generator of a (C0,1)-semigroup {T(t)}t≥0, and f is integrable, then (3.1) has a unique mild solution:
(3.2)u(t)=T(t)u0+∫0tT(t-s)f(s)ds.Moreover, iff is continuous then u(t) is a classical solution of (3.1).Proof.
For fixedα∈Γ, we first consider the initial value problem (in Xα):
(3.3)ddtuα(t)=Aαuα(t)+fα(t),t≥0,uα(0)=u0,α∈D(Aα),
where Aα is defined as in Lemma 2.3, u0,α=πα(u0), and fα(s)=πα(f(s)) is integrable. Since Xα is a Banach space, from C0-semigroup theorems, it is well known that (3.3) has a unique mild solution which is given by
(3.4)uα(t)=Tα(t)u0,α+∫0tTα(t-s)fα(s)dsfor everyα∈Γ.
Since X is the projective limit of {Xα;α∈Γ}, the projective limit of {uα;α∈Γ} exists for each fixed t. We denote it by u(t). Furthermore, since
(3.5)u(t)=lim⁡←⁡uα(t)=lim⁡←⁡(Tα(t)u0,α+∫0tTα(t-s)fα(s)ds)=lim⁡←⁡πα(T(t)u0+∫0tT(t-s)f(s)ds)=T(t)u0+∫0tT(t-s)f(s)ds.
This shows the projective limit u(t) satisfies (3.2), and hence u(t) is a unique mild solution of (3.1). To see u(t) is a classical solution of (3.1) for f is a continuous function, we need to check u′(t) exists for all t>0 and satisfies (3.1). In fact, we  have
(3.6)u′(t)=lim⁡h→01h(u(t+h)-u(t))=lim⁡h→0⁡1h(T(t+h)u0+∫0t+hT(t+h-s)f(s)ds-T(t)u0-∫0tT(t-s)f(s)ds)=lim⁡h→01h(T(t+h)u0-T(t)u0)+lim⁡h→01h(∫0t+hT(t+h-s)f(s)ds-∫0tT(t-s)f(s)ds)=AT(t)u0+lim⁡h→01h(T(h)-I)∫0tT(t-s)f(s)ds+lim⁡h→01h∫tt+hT(t+h-s)f(s)ds=AT(t)u0+A∫0tT(t-s)f(s)ds+f(t)=Au(t)+f(t).
Thus, the derivative of u(t) exists and it satisfies (3.1) as we claimed.
Since global (or local) equicontinuous semigroup is a special case of(C0,1)-semigroup, we have the following corollary immediately.Corollary 3.2.
IfA is the generator of a global (locally) equicontinuous semigroup {T(t)}t≥0, and f is integrable, then (3.1) has a unique mild solution u(t). Furthermore, if f is continuous, then the mild solution u(t) is a classical solution of (3.1). Next we consider the semilinear initial value problem
(3.7)ddtu(t)=Au(t)+f(t,u(t)),t≥0,u(0)=u0∈D(A),
where A is a generator of a (C0,1)-semigroup on a lcs and f:[0,T]×X→X. We found that (3.7), having a unique solution, providing the function f; is uniformly Lipschitz continuous. Namely, we have the following theorem.Theorem 3.3.
SupposeA is the generator of a (C0,1) semigroup {T(t)}t≥0, f:[t0,T]×X→X is continuous in t in the interval [t0,T], and it satisfies uniformly Lipschitz continuous condition, then (3.7) has a uniqe mild solution
(3.8)u(t)=T(t)u0+∫0tT(t-s)f(s,u(s))ds.Proof.
For a givenu0∈X, we defined a mapping
(3.9)F:C([t0,T]:X)⟶C([t0,T]:X),
by
(3.10)Fu(t)=T(t-t0)u0+∫t0tT(t-s)f(s,u(s))ds,t0≤t≤T.
The projective family of Banach space {Xα:α∈Γ} is defined in Section 2, then for each α∈Γ the mapping
(3.11)Fα=πα∘F:C([t0,T]:Xα)⟶C([t0,T]:Xα)
is well defined and satisfies
(3.12)Fαuα(t)=Tα(t-t0)u0,α+∫t0tTα(t-s)fα(s,uα(s))ds,t0≤t≤T.
From (3.12)
(3.13)‖Fαuα(t)-Fαvα(t)‖α=‖πα(Fu(t)-Fv(t))‖α=‖∫t0tTα(t-s)(fα(s,uα(s))-fα(s,vα(s)))ds‖α≤∫t0t‖Tα(t-s)(fα(s,uα(s))-fα(s,vα(s)))‖αds≤MαeσαtLα‖uα-vα‖α,∞(t-t0)≤KαLα‖uα-vα‖α,∞(t-t0),
where Kα=MαeσαT. Using (3.12) and (3.13), by induction on n, it follows easily that
(3.14)‖Fαnuα(t)-Fαnvα(t)‖α≤1n!(KαLα(t-t0))n‖uα-vα‖α,∞≤1n!(KαLαT)n‖uα-vα‖α,∞foreveryα∈Γ,
for n large enough (1/n!)(KαLαT)n<1. By a well-known extension of the contraction principle, for every α∈Γ, Fα has a unique fixed point uα in C([t0,T]:Xα) which satisfies
(3.15)uα(t)=Tα(t-t0)u0,α+∫t0tTα(t-s)fα(s,uα(s))ds.
Since X is the projective limit space of {Xα:α∈Γ}, lim←uα(t) exists for each fixed t∈[t0,T]. Denote the projective limit by u(t), then it satisfies
(3.16)u(t)=lim⁡←⁡uα(t)=lim⁡←⁡(Tα(t-t0)u0,α+∫t0tTα(t-s)fα(s,uα(s))ds)=T(t-t0)uα+∫t0tT(t-s)f(s,u(s))ds.
This shows that u(t) is the mild solution of (3.1). The uniqueness of u(t) followed from uα(t) is unique in Xα for each α∈Γ and u(t) is the projective limit of uα(t). Appling the same method as we used in the proof of Theorem 3.1, one may show that u′(t) exists and satisfies the differential equation in (3.7) for all t>0, and hence (3.8) is the solution of (3.7).Remark 3.4.
Letf:[t0,T]×X→X be a continuous function for t in [t0,T] and satisfies locally Lipschitz condition uniformly for t on bounded intervals. If A is the generator of a (C0,1)-semigroup {T(t)}t≥0 on X, then for every u0∈X there is a tmax≤∞ such that (3.7) has a unique solution u(t) on [0,tmax). Furthermore, if tmax is a finite number, then limt→tmax-∥u(t)∥α=∞ for some α∈Γ. This implies that the solution of (3.7) blows up in finite time.Corollary 3.5.
SupposeA is the generator of a global (locally) equicontinuous semigroup {T(t)}t≥0,f:[t0,T]×X→X is continuous in t on the interval [t0,T], and it satisfies uniformly Lipschitz continuous condition, then (3.7) has a unique solution u(t).


## 4. Applications

Example 4.1.
We consider the thermal balance equations for the gas and solid interphase heat-transfer for the fast-igniting catalytic converter of automobiles:(4.1)∂u∂t+a∂u∂x+cu=cv,0<t<T,0<x≤l,∂v∂t+bv=bu+λev,0<t<T,0<x≤l,u(t,0)=η,v(t,0)=0,0<t<T,u(0,x)=u0(x),v(0,x)=v0(x),0<x≤l,
where a,b,c,λ,η,T>0, and l≥0 are arbitrary given constants; u0(x) and v0(x) are known continuous functions on [0,l]. Let Q=[0,T]×(0,l], and let C1(Q) be the set of functions that are continuously differentiable in Q. To solve the boundary-initial value problem (4.1), we are looking for a pair of functions u, v in C1(Q), which satisfy the boundary and initial conditions. Denote Qα=[0,T]×[1/α,l] for every α∈N, then Q=∪α=1∞Qα. Let Xα be the space C1(Qα)×C1(Qα) endow with the norm
(4.2)‖[u1αu2α]‖α=max⁡⁡{sup⁡(t,x)∈Qα(|uiα(t,x)|),sup⁡(t,x)∈Qα(|∂∂tuiα(t,x)|),sup⁡(t,x)∈Qα(|∂∂xuiα(t,x)|)fori=1,2}.
Then Xα is a Banach space for every α∈N. We consider a topological space X=C1(Q)×C1(Q) with the seminorms {pα}α∈N which is defined as
(4.3)pα([u1u2])=‖[u1αu2α]‖α∀[u1u2]∈X,(whereuiα=ui|Xα,i=1,2).
Then X is a complete topological locally convex space. Let a vector value function U⃗(t,x)=[u(t,x)v(t,x)] for all (t,x)∈Q, then (4.1) can be rewritten as
(4.4)ddtU⃗=AU⃗+BU⃗+F(t,U⃗),(t,x)∈Q,U⃗(0,x)=[u0(x)v0(x)]0<x<l,
where A=[-a(∂/∂x)000], B=[-ccb-b], and F(t,U⃗)=[0λev] with domain of A as
(4.5)D(A)={[uv]∈X:[u(t,0)v(t,0)]=[η0]}.
Let uα=ui|Qα,vα=vi|Qα, Uα⃗=U⃗(t,x)|Qα=[uαvα]∈Xα, Bα=B|Xα, Fα(t,U⃗)=[0λevα] and Aα=A|Xα with domain
(4.6)D(Aα)={[uαvα]∈X:[u(t,1α)v(t,1α)]=[η0]}.
It is well known that the Cauchy problem
(4.7)ddtg=-addxg,g(0,x)=g0(x),
on Banach space C1(Qα) has a unique solution g(t,x)=T(t)g0(x)=g0(x-at), where {T(t)}t≥0 is a C0-semigroups on C1(Qα) generated by -a(∂/∂x). This implies that Aα is a generator of a C0-semigroups on Xα for every α∈N. It is obviously that Bα is a bounded operator on Xα, and hence Aα+Bα generates a C0-semigroup on Xα for every α∈N. Follows [4, Theorem  2.6], A+B is a generator of a (C0,1)-semigroup on X. According to Theorem 3.3, (4.1) has a unique solution on Q as long as F(t,U⃗) satisfies the local Lipschitz condition. To see this, we apply the identity
(4.8)ev1-ev2=∑n=0∞1n!((v1)n-(v2)n)=(v1-v2)∑n=0∞1n!(∑k=0n-1(v1)(n-1)-k(v2)k).
Suppose ∥v1α∥α, ∥v2α∥α≤c for some constant c and for all α∈N, where ∥·∥α is the norm on C1(Qα) defined as
(4.9)‖v‖α=max⁡⁡{sup⁡(t,x)∈Qα(|v(t,x)|),sup⁡(t,x)∈Qα(|∂∂tv(t,x)|),sup⁡(t,x)∈Qα(|∂∂xv(t,x)|)},
for every v in C1(Qα), then
(4.10)‖ev1α-ev2α‖α≤‖(v1α-v2α)∑n=0∞1n!(∑k=0n-1(v1α)(n-1)-k(v2α)k)‖α≤‖(v1α-v2α)‖α∑n=0∞1n!‖(∑k=0n-1(v1α)(n-1)-k(v2α)k)‖α≤‖(v1α-v2α)‖α∑n=0∞1n!(c)n≤ec‖(v1α-v2α)‖α.
This shows that Fα(t,U⃗) satisfies local Lipschitz condition on C1(Qα)×C1(Qα) for every α∈N. It is easy to check that F(t,U⃗) satisfies local Lipschitz condition on X, and hence, by Theorem 3.3, (4.1) has a unique local solution in X.Example 4.2.
The Lasota equation(4.11)∂∂tu+x∂∂xu=λu,
describes the process of reproduction and differentiation of a population of red blood cells. Lasota’s equation can be solved by ergodic method (please see Rudnicki [6]). However, we like to apply our Theorem 3.3 to solve this problem. For this purpose, let A=-x(∂/∂x) and f(t,u)=λu, then the Lasota equation is a special case of (2.4). Let S(R) be the set of all rapidly decreasing test functions whose topology is determined by the seminorms {pmn} which is defined as in Section 2. Consider the Lasota equation with the initial value u(0,x)=u0(x)∈X=S(R). It is an example of Theorem 3.3. Since the function f(t,u(t))=λu(t) obviously satisfies uniformly Lipschitz condition corresponding to pmn with Lipschitz constant λ (independent of all nonnegative integers m and n), the Lasota equation has a unique solution:
(4.12)u(t,x)=T(t)u0(x)+∫0tT(t-s)f(s,u(s,x))ds.
In fact, the corresponding (C0,1)-group {T(t)} generated by -x(∂/∂x) is given by for any (T(t)u0)x=u0(e-tx) for f∈X. Moreover, we may consider more general initial value problem
(4.13)ddtu=xddxu+φ(x)u+f(t,u(t,x)),t>0,x∈R,u(0,x)=u0(x),x∈R,u0∈X,
where φ is any given C∞ function on R possessing bounded derivatives of all orders, and f(t,u(t,.))=e-tu2(t,.) for all u∈X. Let Bu=φ(x)u for every u∈X. Babalola [4] showed that (x(d/dx)+B) generates a (C0,1)-semigroup. For any given constant c and nonnegative integers m and n, if pmn(u)≤c and pmn(v)≤c, then
(4.14)pmn(f(t,u2(t,x))-f(t,v2(t,x)))=pmn(e-tu2(t,x)-e-tv2(t,x))=e-tpmn(u(t,x)-v(t,x))(u(t,x)+v(t,x))≤pmn(u(t,x)+v(t,x))pmn(u(t,x)-v(t,x))≤2cpmn(u(t,x)-v(t,x)).
This shows that f is a locally Lipschitz continuous function with Lipschitz constant Lmn(t,c)=2c. Notice that the Lipschitz constant 2c is independent of t, m, and n. According to Theorem 3.3, (4.13) has a unique local solution.



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*Source: 968760-2012-05-07.xml*