JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 257140 10.1155/2012/257140 257140 Research Article The Global Existence of Nonlinear Evolutionary Equation with Small Delay Yin Xunwu Yao Yonghong School of Science Tianjin Polytechnic University Tianjin 300387 China tjpu.edu.cn 2012 10 8 2012 2012 11 04 2012 09 05 2012 2012 Copyright © 2012 Xunwu Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the global existence of the delayed nonlinear evolutionary equation tu+Au=f(u(t),u(tτ)). 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.

1. Introduction

On the existence for solutions of evolutionary equations, there are many works and methods . For example, the fixed principle [1, 35, 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 , consider the following inhomogeneous initial value problem: (1.1)u(t)+Au=f(t),t>0,u(0)=xX, where X is Banach space. If the nonlinearity fL1(0,T;X), the initial value problem has a unique mild solution. If the nonlinearity f is differentiable a.e. on [0,T] and fL1(0,T;X), then for every xD(A) the initial value problem has a unique strong solution. Furthermore, if the nonlinearity fL1(0,T;X) is locally Hölder continuous, then the initial value problem has a unique classical solution.

In the article , 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),u0,f(u,v)-(|u|+1)γ(v),u0, 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 .

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.

Let A 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 uα=AαuX. For simplicity, we will denote ·X as ·.

For some 0<α<1, the nonlinearity f:Xα×XαX is locally Lipschitz in (u,v). More precisely, there exists a neighborhood U such that for ui,viU and some constant L(2.1)f(u1,v1)-f(u2,v2)L(u1-u2α+v1-v2α).

The initial value φ(t) is Hölder continuous from [-τ,0] to Xα.

Definition 2.1.

Let I be an interval. A function u is called a (classical) solution of (1.4) in the space Xα provided that u:IXα is continuously differentiable on I with tuC(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,for  t0, 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 .

Theorem 2.2 (fundamental theorem on sectorial operators).

Let A 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.

For any α0, there is a constant Cα>0 such that for all t>0(2.3)Aαe-AtL(X)Cαt-αe-at,(a>0).

For 0<α1, there is a constant Cα>0 such that for t0 and xD(Aα)(2.4)e-Atx-xCαtαAαx.

For every α0, there is a constant Cα>0 such that for all t>0 and xX(2.5)(e-A(t+h)-e-At)xαCα|h|t-(1+α)x.

Lemma 2.3 (Gronwall’s equality, [<xref ref-type="bibr" rid="B2">2</xref>–<xref ref-type="bibr" rid="B4">4</xref>]).

Let v(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+bt0t(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 small T>0 such that (1.4) has a unique solution on [-τ,T].

Proof.

For convenience, we still denote φ(0)=x. Select δ>0 and construct set (3.1)V={(u,v)u-xαδ,v-xαδ}. Let B=f(x,x), choose sufficient small T<τ such that (3.2)(e-At-I)xαδ2,0t<T,(3.3)Cα(B+2Lδ)0Tu-αe-auduδ2. 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:yY,y(t)-Aαxδ}. 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),-τt0. 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 -τt0, F:SS. For 0<t<T, considering (2.1), (2.3), (3.2), and (3.3), we obtain (3.6)Fy(t)-Aαxe-tAAαx-Aαx+0tAαe-(t-s)Af(A-αy(s),A-αy(s-τ))dse-tAAαx-Aαx+0tAαe-(t-s)Af(A-αy(s),A-αy(s-τ))-f(x,x)ds+0tAαe-(t-s)Af(x,x)dsδ2+Cα(2Lδ+B)0t(t-s)-αe-a(t-s)dsδ2+Cα(2Lδ+B)0Tu-αe-auduδ. Therefore F:SS. Furthermore if y1,y2S then from (3.3) and (3.5) (3.7)Fy1(t)-Fy2(t)0tAαe-(t-s)A[f(A-αy1(s),A-αy1(s-τ))-f(A-αy2(s),A-αy2(s-τ))]ds0tAαe-(t-s)A[f(A-αy1(s),A-αy1(s-τ))-f(A-αy2(s),A-αy2(s-τ))]ds0tCα(t-s)-αe-a(t-s)Ly1(s)-y2(s)dsCαL0Tu-αe-auds(y1-y2)Y12(y1-y2)Y, which implies that (3.8)Fy1(t)-Fy2(t)Y12(y1-y2)Y. By the contraction mapping theorem the mapping F has a unique fixed point yS. 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),-τt0.

From (2.1) and the continuity of y it follows that tf(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 tf(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), t0t<t+ht1 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+hAαe-(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)I1CβhβAα+βe-tAxCβhβCα+βt-(α+β)xM1hβ,I2NCβhβ0tAα+βe-(t-s)AdsNChβ0t(t-s)-(α+β)dsM2hβ,I3NCαtt+h(t+h-s)-αds=NCα1-αh1-αM3hβ. 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 solution y 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.

Let y 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 , 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 uC1(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 set BU, 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 tnTmax as n+ such that u(tn)U. (If U is unbounded, the point at infinity is included in U.)

Proof.

Suppose Tmax<+, there exists a closed bounded B subset of U and τ0<Tmax such that for τ0t<Tmaxu(t)B. We prove there exists x*B such that (3.20)limtTmax-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 tTmax- for any β[α,1).

Observe that if αβ<1, τ0t<Tmax, in view of (2.3) and (3.19) we have (3.22)u(t)βAβ-αe-tAxα+0tAβe-(t-s)Af(u(s),u(s-τ))dsCβ-αt-(β-α)xα+CβC0t(t-s)-βds=Cβ-αt-(β-α)xα+CβC1-βt1-βM,0<τ0t<Tmax.

Secondly, suppose τ0t1<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)+Ct1tAαe-(t-s)AdsCβ-α(t-t1)β-αAβ-α+αu(t1)+CCαt1t(t-s)-αds=Cβ-α(t-t1)β-αu(t1)β+CCα1-α(t-t1)1-αdsC0(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(uα+vα). Then, the unique solution of (1.4) exists for all t-τ.

Proof.

We need to verify that u(t)α is bounded when tTmax-. As for 0t<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+0tAαe-(t-s)AL(u(s)α+u(s-τ)α)dsC1xα+LCα0t(t-s)-αu(s)αds+LCα0t(t-s)-αu(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 Tmaxτ. Because u(t)=φ(t) for -τt0 and φ is Hölder continuous from [-τ,0] to Xα. Let (3.29)M=maxt[-τ,0]φ(t),(3.30)0t(t-s)-αu(s-τ)αds=-τt-τ(t-τ-w)-αφ(w)αdwM-τ0(t-τ-w)-αdwM1-αt1-αM1,t[0,Tmax). From (3.27), we immediately get (3.31)u(t)αa+b0t(t-s)-αu(s)αds. From Lemma 2.3, that is, Gronwall’s inequality, we find u(t)αC.

Case 2. If Tmax>τ, still let (3.32)M=maxt[-τ,0]φ(t), because (3.33)0t(t-s)-αu(s-τ)αds=-τt-τ(t-τ-w)-αu(w)αdw=-τ0(t-τ-w)-αφ(w)αdw+0t-τ(t-τ-w)-αu(w)αdwM-τ0(t-τ-w)-αdw+0t(t-s)-αu(s)αdsM0+0t(t-s)-αu(s)αds. From (3.27) again, we obtain (3.34)u(t)αa0+b00t(t-s)-αu(s)αds. By Gronwall’s inequality again, we get u(t)αC. This completes the proof of this theorem.

Acknowledgments

This work is supported by NNSF of China (11071185) and NSF of Tianjin (09JCYBJC01800).

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