# The Global Existence of  Nonlinear Evolutionary Equation with Small Delay

Authors: Xunwu Yin
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We investigate the global existence of the delayed nonlinear evolutionary equation∂tu+Au=f(u(t),u(t−τ)). Our work space is the fractional powers space Xα. Under the fundamental  theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.


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## Body

## 1. Introduction

On the existence for solutions of evolutionary equations, there are many works and methods [1–7]. For example, the fixed principle [1, 3–5, 7] and Galerkin approximations [2, 6]. They are very classical methods to prove existence and uniqueness. Generally speaking, there are four solution concepts. That is, weak solution, mild solution, strong solution, and classical solution. We can obtain different types for different conditions. For instance [1], consider the following inhomogeneous initial value problem:
(1.1)u′(t)+Au=f(t),t>0,u(0)=x∈X,
where X is Banach space. If the nonlinearity f∈L1(0,T;X), the initial value problem has a unique mild solution. If the nonlinearity f is differentiable a.e. on [0,T] and f′∈L1(0,T;X), then for every x∈D(A) the initial value problem has a unique strong solution. Furthermore, if the nonlinearity f∈L1(0,T;X) is locally Hölder continuous, then the initial value problem has a unique classical solution.In the article [5], the author considered scalar reaction-diffusion equations with small delay
(1.2)u′(t)-Δu=f(u(t),u(t-τ)).
There the nonlinearity is assumed to be locally lipschitz and to satisfy the one-sided growth estimates
(1.3)f(u,v)≤(u+1)γ(v),u≥0,f(u,v)≥-(|u|+1)γ(v),u≤0,
for some continuous γ. To prove existence, he treated the equation stepwise as a nonautonomous undelayed parabolic partial differential equation on the time intervals [(j-1)τ,jτ] by regarding the delayed values as fixed. His strategy was to mimic the results of Henry [3, Theorem 3.3.3 and Corollary 3.3.5], but with his assumption of Hölder continuity in replaced by p-integrability. Many authors had investigated the nondelayed one in [8–10].In this paper, we consider the following nonlinear evolutionary equation with small delay:(1.4)u′(t)+Au=f(u(t),u(t-τ)),t>0,u∣[-τ,0]=φ(t).
Under the hypothesis of (A1), (A2), and (A3) (see Section 2), we firstly make use of the fixed principle to prove the local existence and uniqueness theorem. Then we obtain the global existence and uniqueness by Gronwall inequality. In the whole paper, our work space is fractional powers space Xα. Its definition can be referred to [1, 3, 4].


## 2. Preliminaries

In this section, we will give some basic notions and facts. Firstly, basic assumptions are listed.(A1)
LetA be a positive, sectorial operator on a Banach Space X. e-At is an analytic semigroup generated by -A. Fractional powers operator Aα is well defined. Fractional powers space Xα=D(Aα) with the graph norm ∥u∥α=∥Aαu∥X. For simplicity, we will denote ∥·∥X as ∥·∥.(A2)
For some0<α<1, the nonlinearity f:Xα×Xα→X is locally Lipschitz in (u,v). More precisely, there exists a neighborhood U such that for ui,vi∈U and some constant L(2.1)‖f(u1,v1)-f(u2,v2)‖≤L(‖u1-u2‖α+‖v1-v2‖α).(A3)
The initial valueφ(t) is Hölder continuous from [-τ,0] to Xα.Definition 2.1.
LetI be an interval. A function u is called a (classical) solution of (1.4) in the space Xα provided that u:I→Xα is continuously differentiable on I with ∂tu∈C(I,X) and satisfies (1.4) everywhere in I.Obviously the (classical) solution of (1.4) can be expressed by the variation of constant formula
(2.2)u(t)=e-Atx+∫0te-A(t-s)f(u(s),u(s-τ))ds,fort≥0,
where we let φ(0)=x. Next we come to the main theorem on analytic semigroup which is extremely important in the study of the dynamics of nonlinear evolutionary equations [4].Theorem 2.2 (fundamental theorem on sectorial operators).
LetA be a positive, sectorial operator on a Banach Space X and e-At be the analytic semigroup generated by -A. Then the following statements hold.(i)
For anyα≥0, there is a constant Cα>0 such that for all t>0(2.3)‖Aαe-At‖L(X)≤Cαt-αe-at,(a>0).(ii)
For0<α≤1, there is a constant Cα>0 such that for t≥0 and x∈D(Aα)(2.4)‖e-Atx-x‖≤Cαtα‖Aαx‖.(iii)
For everyα≥0, there is a constant Cα>0 such that for all t>0 and x∈X(2.5)‖(e-A(t+h)-e-At)x‖α≤Cα|h|t-(1+α)‖x‖.Lemma 2.3 (Gronwall’s equality, [2–4]).
Letv(t)≥0 and be continuous on [t0,T]. If there exists positive constants a,b,α(α<1) such that for t∈[t0,T](2.6)v(t)≤a+b∫t0t(t-s)α-1v(s)d,
then there exists positive constant M such that for t∈[t0,T](2.7)v(t)≤Ma.


## 3. Main Results

Theorem 3.1.
Suppose (A1), (A2), and (A3) hold. Then there exists a sufficiently smallT>0 such that (1.4) has a unique solution on [-τ,T].Proof.
For convenience, we still denoteφ(0)=x. Select δ>0 and construct set
(3.1)V={(u,v)∣‖u-x‖α≤δ,‖v-x‖α≤δ}.
Let B=∥f(x,x)∥, choose sufficient small T<τ such that
(3.2)‖(e-At-I)x‖α≤δ2,0≤t<T,(3.3)Cα(B+2Lδ)∫0Tu-αe-audu≤δ2.
Let Y be the Banach space C([-τ,T];X) with the usual supremum norm which we denote by ∥·∥Y. Let S be the nonempty closed and bounded subset of Y defined by
(3.4)S={y:y∈Y,‖y(t)-Aαx‖≤δ}.
On S we define a mapping F by
(3.5)Fy(t)={e-tAAαx+∫0tAαe-(t-s)Af(A-αy(s),A-αy(s-τ))ds,0<t<T,Aαφ(t),-τ≤t≤0.
Next we will utilize the contraction mapping theorem to prove the existence of fixed point. In order to complete this work, we need to verify that F maps S into itself and F is a contraction mapping on S with the contraction constant ≤1/2.
It is easy to see from (3.4) and (3.5) that for -τ≤t≤0, F:S→S. For 0<t<T, considering (2.1), (2.3), (3.2), and (3.3), we obtain
(3.6)‖Fy(t)-Aαx‖≤‖e-tAAαx-Aαx‖+∫0t‖Aαe-(t-s)Af(A-αy(s),A-αy(s-τ))‖ds≤‖e-tAAαx-Aαx‖+∫0t‖Aαe-(t-s)Af(A-αy(s),A-αy(s-τ))-f(x,x)‖ds+∫0t‖Aαe-(t-s)Af(x,x)‖ds≤δ2+Cα(2Lδ+B)∫0t(t-s)-αe-a(t-s)ds≤δ2+Cα(2Lδ+B)∫0Tu-αe-audu≤δ.
Therefore F:S→S. Furthermore if y1,y2∈S then from (3.3) and (3.5)
(3.7)‖Fy1(t)-Fy2(t)‖≤∫0t‖Aαe-(t-s)A[f(A-αy1(s),A-αy1(s-τ))-f(A-αy2(s),A-αy2(s-τ))]‖ds≤∫0t‖Aαe-(t-s)A‖‖[f(A-αy1(s),A-αy1(s-τ))-f(A-αy2(s),A-αy2(s-τ))]‖‍ds≤∫0tCα(t-s)-αe-a(t-s)L‖y1(s)-y2(s)‖ds≤CαL∫0Tu-αe-auds⋅‖(y1-y2)‖Y≤12‖(y1-y2)‖Y,
which implies that
(3.8)‖Fy1(t)-Fy2(t)‖Y≤12‖(y1-y2)‖Y.
By the contraction mapping theorem the mapping F has a unique fixed point y∈S. This fixed point satisfies the following:
(3.9)y(t)=e-tAAαx+∫0tAαe-(t-s)Af(A-αy(s),A-αy(s-τ))ds,0<t<T,(3.10)y(t)=Aαφ(t),-τ≤t≤0.
From (2.1) and the continuity of y it follows that t→f(A-αy(t),A-αy(t-τ)) is continuous on [0,T] and a fortiori bounded on this interval. Let
(3.11)‖f(A-αy(t),A-αy(t-τ))‖≤N.
Next we want to show that t→f(A-αy(t),A-αy(t-τ)) is locally Hölder continuous on (0,T). To this end, we show first that the solution y of (3.9) is locally Hölder continuous on (0,T).
Select[t0,t1]⊂(0,T), t0≤t<t+h≤t1 such that
(3.12)‖y(t+h)-y(t)‖≤‖(e-hA-I)Aαe-tAx‖+∫0t‖(e-hA-I)Aαe-(t-s)Af(A-αy(s),A-αy(s-τ))‖ds+∫tt+h‖Aαe-(t+h-s)Af(A-αy(s),A-αy(s-τ))‖ds=I1+I2+I3.
Considering (2.3) and (2.4), we select β∈(0,1-α) such that
(3.13)I1≤Cβhβ‖Aα+βe-tAx‖≤CβhβCα+βt-(α+β)‖x‖≤M1hβ,I2≤NCβhβ∫0t‖Aα+βe-(t-s)A‖ds≤NChβ∫0t(t-s)-(α+β)ds≤M2hβ,I3≤NCα∫tt+h(t+h-s)-αds=NCα1-αh1-α≤M3hβ.
Synthesizing (3.12) and (3.13), we get
(3.14)‖y(t+h)-y(t)‖≤Chβ,t∈[t0,t1]⊂(0,T).
So we proved the solutiony of (3.9) is locally Hölder continuous on (0,T). Furthermore, in view of (2.1) we have
(3.15)‖f[A-αy(t+h),A-αy(t+h-τ)]-f[A-αy(t),A-αy(t-τ)]‖≤L(‖y(t+h)-y(t)‖+‖y(t+h-τ)-y(t-τ)‖)≤LChβ+‖Aαφ(t+h-τ)-Aαφ(t-τ)‖≤Mhγ.
Lety be the solution of (3.9) and (3.10) and f~(t)=f(A-αy(t),A-αy(t-τ)). In view of locally Hölder continuous on (0,T) of f~(t), consider the inhomogeneous initial value problem
(3.16)u′(t)+Au=f~(t),0<t<T,u(0)=x.
By Corollary 4.3.3 in [1], this problem has a unique solution and the solution is given by
(3.17)u(t)=e-tAx+∫0te-(t-s)Af(A-αy(s),A-αy(s-τ))ds.
Each term of (3.17) is in D(A) and a fortiori in D(Aα). Operating on both sides of (3.17) with Aα we find
(3.18)Aαu(t)=e-tAAαx+∫0tAαe-(t-s)Af(A-αy(s),A-αy(s-τ))ds.
By (3.9) the right-hand side of (3.18) equals y(t) and therefore u(t)=A-αy(t). So for 0<t<T, by (3.17) we have
(3.19)u(t)=e-tAx+∫0te-(t-s)Af(u(s),u(s-τ))ds.
So u is a u∈C1(0,T;X) solution of (1.4). The uniqueness of u follows readily from the uniqueness of the solutions of (3.9) and (3.16), and the proof is complete.Before giving our global existence theorem, we should first prove extended theorem of solution.Theorem 3.2 (extended theorem).
Assume that (A1), (A2), and (A3) hold. And also assume that for every closed bounded setB⊂U, the image f(B) is bounded in X. If u is a solution of (1.1) on [-τ,Tmax), then either Tmax=+∞ or  there exists a sequence tn→Tmax as n→+∞ such that u(tn)→∂U. (If U is unbounded, the point at infinity is included in ∂U.)Proof.
SupposeTmax<+∞, there exists a closed bounded B subset of U and τ0<Tmax such that for τ0≤t<Tmaxu(t)∈B. We prove there exists x*∈B such that
(3.20)lim⁡t→Tmax⁡-u(t)=x*
in Xα, which implies the solution may be extended beyond time Tmax.
Now let(3.21)C=sup⁡{‖f(u,v)‖,(u,v)∈B}.
We show firstly that ∥u(t)∥β remains bounded as t→Tmax- for any β∈[α,1).
Observe that ifα≤β<1, τ0≤t<Tmax, in view of (2.3) and (3.19) we have
(3.22)‖u(t)‖β≤‖Aβ-αe-tA‖‖x‖α+∫0t‖Aβe-(t-s)A‖‖f(u(s),u(s-τ))‖ds≤Cβ-αt-(β-α)‖x‖α+CβC∫0t(t-s)-βds=Cβ-αt-(β-α)‖x‖α+CβC1-βt1-β≤M,0<τ0≤t<Tmax⁡.
Secondly, supposeτ0≤t1<t<Tmax, so
(3.23)u(t)-u(t1)=(e-(t-t1)A-I)u(t1)+∫t1te-(t-s)Af(u(s),u(s-τ))ds.
From (2.3) and (2.4) we get
(3.24)‖u(t)-u(t1)‖α≤‖(e-(t-t1)A-I)Aαu(t1)‖+C∫t1t‖Aαe-(t-s)A‖ds≤Cβ-α(t-t1)β-α‖Aβ-α+αu(t1)‖+CCα∫t1t(t-s)-αds=Cβ-α(t-t1)β-α‖u(t1)‖β+CCα1-α(t-t1)1-αds≤C0(t-t1)β-α.
Thus (3.20) holds, and the proof is completed.Theorem 3.3 (global existence and uniqueness).
Assume that (A1), (A2), and (A3) hold. And for all(u,v)∈Xα×Xα, f satisfies
(3.25)‖f(u,v)‖≤L(‖u‖α+‖v‖α).
Then, the unique solution of (1.4) exists for all t≥-τ.Proof.
We need to verify that∥u(t)∥α is bounded when t→Tmax-. As for 0≤t<Tmax(3.26)u(t)=e-tAx+∫0te-(t-s)Af(u(s),u(s-τ))ds.
Considering (3.25), we can obtain
(3.27)‖u(t)‖α=‖Aαu(t)‖≤‖e-tAAαx‖+∫0t‖Aαe-(t-s)A‖L(‖u(s)‖α+‖u(s-τ)‖α)ds≤C1‖x‖α+LCα∫0t(t-s)-α‖u(s)‖αds+LCα∫0t(t-s)-α‖u(s-τ)‖αds,
For
(3.28)∫0t(t-s)-αu(s-τ)ds=s-τ=w∫-τt-τ(t-τ-w)-αu(w)dw.
Case 1. If Tmax≤τ. Because u(t)=φ(t) for -τ≤t≤0 and φ is Hölder continuous from [-τ,0] to Xα. Let
(3.29)M=max⁡t∈[-τ,0]‖φ(t)‖,(3.30)∫0t(t-s)-α‖u(s-τ)‖αds=∫-τt-τ(t-τ-w)-α‖φ(w)‖αdw≤M∫-τ0(t-τ-w)-αdw≤M1-αt1-α≤M1,t∈[0,Tmax⁡).
From (3.27), we immediately get
(3.31)‖u(t)‖α≤a+b∫0t(t-s)-α‖u(s)‖αds.
From Lemma 2.3, that is, Gronwall’s inequality, we find ∥u(t)∥α≤C.
Case 2. If Tmax>τ, still let
(3.32)M=max⁡t∈[-τ,0]⁡‖φ(t)‖,
because
(3.33)∫0t(t-s)-α‖u(s-τ)‖αds=∫-τt-τ(t-τ-w)-α‖u(w)‖αdw=∫-τ0(t-τ-w)-α‖φ(w)‖αdw+∫0t-τ(t-τ-w)-α‖u(w)‖αdw≤M∫-τ0(t-τ-w)-αdw+∫0t(t-s)-α‖u(s)‖αds≤M0+∫0t(t-s)-α‖u(s)‖αds.
From (3.27) again, we obtain
(3.34)‖u(t)‖α≤a0+b0∫0t(t-s)-α‖u(s)‖αds.
By Gronwall’s inequality again, we get ∥u(t)∥α≤C. This completes the proof of this theorem.



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*Source: 257140-2012-08-10.xml*