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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions

Xuping Zhang / Qiyu Chen / Yongxiang Li
Published Online: 2018-02-23 | DOI: https://doi.org/10.1515/math-2018-0012

Abstract

This paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.

Keywords: Functional partial differential equation; Nonlocal condition; Strong solution; Analytic compact semigroup; Fractional power operator; α-norm

MSC 2010: 34G20; 34K30; 35D35

1 Introduction

Let X be a Banach space with norm |·| and α be a constant such that 0 < α< 1. We denote by C(J, D(Aα)) the Banach space of all the continuous functions from J to D(Aα) provided with the uniform norm topology and D(Aα) the domain of the linear operator Aα to be defined later.

In this paper, by using fractional power of operators and Schauder’s fixed point theorem, we study the existence and regularity of mild solutions in some interpolation to the following functional partial differential equations with nonlocal initial conditions

u(t)+Au(t)=f(t,u(t),u(a1(t)),u(a2(t)),,u(am(t))),t(0,a],(1)

u(0)=i=1pciu(ti),(2)

where u(·) takes values in a subspace spaces of Banach space X, A : D(A) ⊂ XX is a sectorial operator, m is a positive integer number, J = [0, a], a > 0 is a constant, aj : JJ are continuous functions such that 0 ≤ aj(t) ≤ t for j = 1, 2, …, m, f : J × (D(Aα))m+1X is Carathéodory continuous nonlinear function, 0 < t1 < t2 < … < tp < a, ci are real numbers, ci ≠ 0, i = 1, 2, …, p.

Nonlocal initial conditions can be applied in physics with better effect than the classical initial condition u(0) = u0. For example, in [1] Deng used the nonlocal condition (2) to describe the diffusion phenomenon of a small amount of gas in a transparent tube. In this case, condition (2) allows additional measurements at ti, i = 1, 2, …, p, which is more precise than the measurement just at t = 0. In [2], Byszewski pointed out that if ci ≠ 0, i = 1, 2, …, p, then the results can be applied to kinematics to determine the location evolution tu(t) of a physical object for which we do not know the positions u(0),u(t1), …, u(tp), but we know that the nonlocal condition (2) holds. Consequently, to describe some physical phenomena, the nonlocal condition can be more useful than the standard initial condition u(0) = u0. The importance of nonlocal conditions have also been discussed in [3,4,5,6].

In [7], Fu and Ezzinbi studied the following neutral functional evolution equation with nonlocal conditions

ddt[x(t)+F(t,x(t)),x(b1(t)),,x(bm(t)))]+Ax(t)=G(t,x(t),x(a1(t)),,x(an(t))),0ta,x(0)+g(x)=x0X,(3)

where the operator –A : D(A) ⊂ XX generates an analytic compact semigroup T(t) (t ≥ 0) of uniformly bounded linear operators on a Banach space X, f : [0, a] × Xm+1X, G : [0, a] × Xn+1X, ai, bj, i = 1, 2, …, n, j = 1, 2, …, m and g are given functions satisfying some assumptions. The authors have proved the existence and regularity of mild solutions. In the subsequent years, various similar results have been established by many authors, see for example [8,9].

Recently, Chang and Liu [10] studied the existence of mild and strong solutions in some interpolation spaces between X and the domain of the linear part for the following semilinear evolution problem with nonlocal initial conditions:

u(t)=Au(t)+f(t,u(t)),t[0,T],u(0)+g(u)=u0X,(4)

where T > 0, the linear part A is a sectorial operator in X, f and g are given X-valued functions.

The aim of this paper is to establish some existence results of (1) and (2) without assuming Lipschitz condition on the nonlinear term and complete continuity on the nonlocal condition. The result obtained is a partial continuation of some results reported in [2,7,10,11,12]. It is worth mentioning that the theory of fractional power and α-norm are used to discuss the problems so that the results obtained in this chapter can be applied to the systems in which the nonlinear terms involve derivatives of spatial variables, and therefore, they have broader applicability.

The rest of this paper is organized as follows: We introduce some basic definitions and preliminary facts which will be used throughout this paper in section 2. The existence results of mild solutions are discussed in Section 3 by applying fixed point theorem. In Section 4, we provide some sufficient conditions to guarantee the regularity of solutions, that is, we obtain the existence of strong solutions. Finally, an example is presented in Section 5 to show the applications of the abstract results obtained.

2 Preliminaries

Assume A : D(A) ⊂ XX be a sectorial operator and –A generates an analytic compact semigroup T(t) (t ≥ 0) on X. It is easy to see that T(t) (t ≥ 0) is exponentially stable, i.e. there exist constants M ≥ 1 and δ < 0 such that

|T(t)|Meδt,for eacht0,(5)

the infimum of δ

v0=inf{δ<0:M1,|T(t)|Meδt,t0}

is called a growth index of semigroup T(t) (t ≥ 0), and at this point v0 < 0. For each v ∈ (0,|v0|), by the definition of v0, there exists a constant M ≥ 1, such that

|T(t)|Mevt,for eacht0.(6)

Define an equivalent norm in X by

x=supt0|evtT(t)x|,(7)

then |x| ≤ ∥x∥ ≤ M|x|. We denoted by ∥T(t)∥ the norm of the operator T(t) in space (X,∥·∥). By (6), we have

T(t)evt1,for eacht0.(8)

and

T(ti)evtievt1<1,i=1,2,,p.(9)

It is well known [13, Chapter 4, Theorem 2.9] that for any u0D(A) and hC1(J, X), the initial value problem of linear evolution equation (LIVP)

u(t)+Au(t)=h(t),tJ,u(0)=u0,(10)

has a unique classical solution uC1((0, a], X) ∩ C(J, D(A)) expressed by

u(t)=T(t)u0+0tT(ts)h(s)ds.(11)

If u0X and hL1(J, X), the function u given by (11) belongs to C(J, X), which is known as a mild solution of the LIVP (10). If a mild solution u of the LIVP (10) belongs to W1, 1(J, X) ∩ L1(J, D(A)) and satisfies the equation for a.e. tJ, we call it a strong solution.

Throughout this paper, we assume that: (P0)i=1p|ci|<1.

Applying (9) and assumption (P0), we get i=1ciT(ti)i=1p|ci|evt1<1. Combining this with the operator spectrum theorem, we know that

B:=(Ii=1pciT(ti))1(12)

exists and it is bounded. Furthermore, by Neumann expression, B can be expressed by

B=n=0(i=1pciT(ti))n.(13)

Therefore,

Bn=0i=1pciT(ti)n=11i=1pciT(ti)11i=1p|ci|.(14)

To prove our main results, for any hC(J, X), we consider the linear evolution equation nonlocal problem (LNP)

u(t)+Au(t)=h(t),tJ,(15)

u(0)=i=1pciu(ti).(16)

Lemma 2.1

If the condition (P0) holds, then the LNP (15)-(16) has a unique mild solution uC(J, X) given by

u(t)=i=1pciT(t)B0tiT(tis)h(s)ds+0tT(ts)h(s)ds,tJ.(17)

Proof

By (10) and (11), we know that Eq. (15) has a unique mild solution uC(J, X) which can be expressed by

u(t)=T(t)u(0)+0tT(ts)h(s)ds.(18)

It follows from (18) that

u(ti)=T(ti)u(0)+0tiT(tis)h(s)ds,i=1,2,,p.(19)

Combining (16) with (19), we have

u(0)=i=1pciT(ti)u(0)+i=1pci0tiT(tis)h(s)ds.(20)

Since Ii=1pciT(ti) has a bounded inverse operator B,

u(0)=i=1pciB0tiT(tis)h(s)ds.(21)

By (18) and (21), we know that u satisfies (17).

Inversely, we can verify directly that the function uC(J, X) given by (17) is a mild solution of the LNP (15)-(16). This completes the proof. □

We recall some concepts and conclusions on the fractional powers of A. Because A : D(A) ⊂ XX is a sectorial operator, it is possible to define the fractional powers Aα for 0 < α ≤ 1. Now we define (see [13]) the operator Aα by

Aαx=1Γ(α)0+tα1T(t)xdt,xX,

where Γ denotes the gamma function. The operator Aα is bijective and the operator Aα is defined by

Aα=(Aα)1.

We denote by D(Aα) the domain of the operator Aα.

Furthermore, we have the following properties which appeared in [13].

Lemma 2.2

Let 0 < α< 1. Then:

  1. T(t) : XD(Aα) for each T > 0.

  2. AαT(t)x = T(t)Aαx, for each xD(Aα) and t ≥ 0.

  3. For every T > 0, the linear operator AαT(t) is bounded on X and there exist Mα such that

    AαT(t)Mαtα.

  4. For every T > 0, there exists a constant Cα such that

    (T(t)I)AαCαtα.

  5. For 0 < α < β ≤ 1, we get D(Aβ) ↪ D(Aα).

D(Aα) endowed with the norm ∥xα = ∥Aαx∥ for all xD(Aα) is a Banach space. We denote it by Xα.

From now on, for the sake of brevity, we rewrite that

(t,u(t),u(a1(t)),,u(am(t))):=(t,x(t)).(22)

Definition 2.3

A function f:J×Xαm+1X is said to be Carathéodory continuous provided that

  1. for all xXαm+1,f(,x):JX is measurable,

  2. for a.e. t[0,a],f(t,):×Xαm+1X is continuous.

3 Existence of mild solutions

This section is devoted to the study of the existence of mild solutions for a class of functional partial differential equations with nonlocal initial conditions (1)-(2). In what follows, we will make the following hypotheses on the data of our problem (1)-(2).

  • (P1)

    The function f:J×Xαm+1X is Carathéodory continuous and for some positive constant r, there exist constants q ∈ [0, 1 – α), γ > 0 and function φrL1q(J,R+) such that for any tJ and ujXα satisfying ∥ujαr for j = 0, 1, 2, …, m,

    f(t,u0,u1,,um)φr(t),lim infr+φrL1q([0,a])r:=γ<+.

Theorem 3.1

Assume that the hypotheses (P0) and (P1) are satisfied. Then the problem (1)-(2) has at least one mild solution on C(J, Xα) provided that

a1αqMαγ1i=1p|ci|(1q1αq)1q<1.(23)

Proof

We consider the operator Q on C(J, Xα) defined by

(Qu)(t)=i=1pciT(t)B0tiT(tis)f(s,x(s))ds+0tT(ts)f(s,x(s))ds,tJ.(24)

With the help of Lemma 2.2, we know that the mild solution of the problem (1)-(2) is equivalent to the fixed point of the operator Q defined by (24). In what follows, we shall prove that the operator Q has at least one fixed point by applying the famous Schauder’s fixed point theorem.

For this purpose, we first prove that there exists a positive constant R such that the operator Q defined by (24) maps the bounded closed convex set

DR={uC(J,Xα):u(t)αR,tJ}

to DR. If this is not true, there would exist urDr and trJ such that ∥(Qur)(tr)∥α > r for each r > 0. However, by (9), the condition (P1), Lemma 2.1 and Hölder inequality, we get that

r<(Qur)(tr)αi=1pciT(tr)B0tiT(tis)f(s,x(s))dsα+0trT(trs)f(s,x(s))dsαBi=1p|ci|0tiAαT(tis)f(s,x(s))ds+0trAαT(trs)f(s,x(s))ds(25)

i=1p|ci|1i=1p|ci|0tiMα(tis)αφr(s)ds+0trMα(trs)αφr(s)dsMαi=1p|ci|1i=1p|ci|(0ti(tis)α1qds)1q(0tiφr1q(s)ds)q+Mα(0tr(trs)α1qds)1q(0trφr1q(s)ds)qMαi=1p|ci|1i=1p|ci|(1q1αq)1qa1αqφrL1q([0,a])+Mα(1q1αq)1qa1αqφrL1q([0,a])a1αqMα1i=1p|ci|(1q1αq)1qφrL1q([0,a]).

Divided by r on both sides of (25) and then take the lower limits as r → +∞ we get

a1αqMαγ1i=1p|ci|(1q1αq)1q1,

which contradicts with the inequality (23). Therefore, there exists a positive constant R such that the operator Q maps DR to DR.

Below we will verify that Q : DRDR is a completely continuous operator. From the definition of operator Q and the assumption (P1) we note that Q is obviously continuous on DR. Next, we shall prove that {Qu : uDR} is a family of equi-continuous functions. Let uDR and t′, t″ ∈ J, t′ < t″. By (24) one get that

(Qu)(t)(Qu)(t)(T(t)T(t))i=1pciB0tiT(tis)f(s,x(s))ds+ttT(ts)f(s,x(s))ds+0t[T(ts)T(ts)]f(s,x(s))ds:=B1+B2+B3.

It is obvious that

(Qu)(t)(Qu)(t)αk=13Bkα.

Therefore, we only need to check ∥Bkα tend to 0 independently of uDR when t″ – t′ → 0 for k = 1, 2, 3.

For B1, by the condition (P1), Lemma 2.1 and Hölder inequality, we have

i=1pciB0tiT(tis)f(s,x(s))dsαi=1p|ci|1i=1p|ci|0tiAαT(tis)f(s,x(s))dsi=1p|ci|1i=1p|ci|0tiMα(tis)αφR(s)dsMαi=1p|ci|1i=1p|ci|(1q1αq)1qa1αqφRL1q([0,a]).(26)

Combining (26) and the strong continuity of the semigroup T(t) (t ≥ 0), one can easily get that ∥B1α → 0 as t″ – t′ → 0.

For B2, taking assumption (P1), Lemma 2.1 and Hölder inequality into account, we obtain

B2αttAαT(ts)f(s,x(s))dsttMα(ts)αφR(s)dsMα(1q1αq)1q(tt)1αqφRL1q([0,a])0astt0.

For t′ = 0, 0 < t″ ≤ a, it is easy to see that ∥B3∥ = 0. For t′ > 0 and 0 < ε < t′ small enough, by the condition (P1), Lemma 2.1, Hölder inequality and the equi-continuity of T(t) (t > 0), we know that

B3α0ϵ[T(tt+s2)T(s2)]AαT(s2)f(ts,x(ts))ds+ϵt[T(tt+s2)T(s2)]AαT(s2)f(ts,x(ts))ds20ϵAαT(s2)f(ts,x(ts))ds+sups[ϵ,t]T(tt+s2)T(s2)ϵtAαT(s2)f(ts,x(ts))ds2Mα(1q1αq)1q(ϵ2)1αqφRL1q[0,a]+sups[ϵ,t]T(tt+s2)T(s2)Mα(1q1αq)1q((t2)1αq1q(ϵ2)1αq1q)1qφRL1q([0,a])2Mα(1q1αq)1q(ϵ2)1αqφRL1q([0,a])+sups[ϵ,t]T(tt+s2)T(s2)Mα(1q1αq)1q(tϵ2)1αqφRL1q([0,a])0astt0andϵ0.

As a result, ∥(Qu)(t″) – (Qu)(t′)∥α → 0 independently of uDR as t″ – t′ → 0, which means that Q maps DR into a family of equi-continuous functions.

It remains to prove that V(t) = {(Qu)(t): uDR} is relatively compact in Xα. Obviously it is true in the case t = 0. Fix t ∈ (0, a], for each ε ∈ (0, t) and uDR, define

(Qϵu)(t)=i=1pciT(t)B0tiT(tis)f(s,x(s))ds+0tϵT(ts)f(s,x(s))ds=T(t)i=1pciB0tiT(tis)f(s,x(s))ds+T(ϵ)0tϵT(tsϵ)f(s,x(s))ds.

The compactness of T(t) (t > 0) ensures that the sets Vε(t) = {(Qεu)(t): uDR} are relatively compact in Xα. Since

(Qu)(t)(Qϵu)(t)αtϵtT(ts)f(s,x(s))αdstϵtAαT(ts)f(s,x(s))dstϵtMα(ts)αφR(s)dsMα(1q1αq)1qϵ1αqφRL1q([0,a])

for every uDR. Therefore, there are relatively compact sets Vε(t) arbitrarily close to V(t) for t > 0. Hence, V(t) is also relatively compact in Xα for t ≥ 0.

Thus, the Ascoli-Arzela theorem guarantees that Q : DRDR is a completely continuous operator. According to the famous Schauder’s fixed point theorem we know that the operator Q has at least one fixed point uDR, and this fixed point is the desired mild solution of the problem (1)-(2) on C(J, Xα). This completes the proof.  □

If we replace the condition (P1) by the following condition:

  • (P2)

    The function f:J×Xαm+1X is Carathéodory continuous and there exist a function ϕL1q(J,R+) (q ∈ [0, 1 – α)) and a nondecreasing continuous function ψ : ℝ+ → ℝ+ such that

    f(t,u0,u1,,um)ϕ(t)ψ(j=0mujα),

    for all ujC(J, Xα), j = 0, 1, 2, …, m, and tJ,

then we have the following existence result.

Theorem 3.2

Assume that the hypotheses (P0) and (P2) are satisfied. Then the problem (1)-(2) has at least one mild solution on C(J, Xα) provided that there exists a positive constant R such that

a1αqψ((m+1)R)Mα1i=1p|ci|(1q1αq)1qϕL1q([0,a])R.(27)

Proof

From the proof of Theorem 3.1, we know that the mild solution of the problem (1)-(2) is equivalent to the fixed point of the operator Q defined by (24). In what follows, we prove that there exists a positive constant R such that the operator Q maps the set DR to itself. For any ujDR, j = 0, 1, 2, …, m, and tJ, by (8), (14), (24), (27), the hypothesis (P2) and Hölder inequality, we have

(Qu)(t)αi=1pciT(t)B0tiT(tis)f(s,x(s))dsα+0tAαT(ts)f(s,x(s))dsi=1p|ci|1i=1p|ci|0tiψ((m+1)R)Mα(tis)αϕ(s)ds+0tψ((m+1)R)Mα(ts)αϕ(s)dsψ((m+1)R)Mαi=1p|ci|1i=1p|ci|(0ti(tis)α1qds)1q(0tiϕ1q(s)ds)q+ψ((m+1)R)Mα(0t(ts)α1qds)1q(0tϕ1q(s)ds)qa1αqψ((m+1)R)Mα1i=1p|ci|(1q1αq)1qϕL1q([0,a])R,

which implies Q(DR) ⊂ DR. By adopting a completely similar method which used in the proof of Theorem 3.1, we can prove that the problem (1)-(2) has at least one mild solution on C(J, Xα). This completes the proof.  □

Similarly to Theorem 3.2, we have the following result.

Corollary 3.3

Assume that the hypotheses (P0) and (P2) are satisfied. Then the problem (1)-(2) has at least one mild solution on C(J, Xα) provided that

lim infr+ψ((m+1)r)r<1i=1p|ci|a1αqMαϕL1q[0,a](1q1αq)q1.(28)

4 The regularity of solutions

In this section, we discuss the existence of strong solutions for the problem (1)-(2), that is, we shall provide conditions to allow the differential for mild solutions of the problem (1)-(2). To do this, we need the following lemma:

Lemma 4.1

([12]). If X is a reflexive Banach space, then Xα is also a reflexive Banach space.

Theorem 4.2

Let X be a reflexive Banach space. If there exists α′ ∈ (α, 1), such that the hypothesis (P0),

  • (P3)

    there exists a positive constant L, such that for any t″, t′ ∈ J and xj, yjXα, j = 0, 1, 2, …, m,

    f(t,x0,x1,,xm)f(t,y0,y1,,ym)L¯(|tt|αα+j=0mxjyjα)

  • (P4)

    There exist constants lj > 0, j = 1, 2, …, m, such that for any t″, t′ ∈ J,

    |aj(t)aj(t)|lj|tt|,j=1,2,,m,

  • (P5)

    a1αL¯Mα1α(1+j=1mljαα)<1,

hold, then the problem (1)-(2) has a strong solution.

Proof

Let Q be the operator defined in the proof of Theorem 3.1. By the conditions (P0), (P3) and (P5), one can use the same argument as in the proof of Theorem 3.1 to deduce that there exists a constant R > 0, such that Q(DR) ⊂ DR. For this R, consider the set

D={uC(J,Xα):uαR,u(t)u(t)αL|tt|αα,t,tJ,|tt|<1},(29)

for some L* large enough. It is clear that D is a convex, closed and nonempty set. We shall prove that Q has a fixed point on D. For any uD and t′, t″ ∈ J, 0 < t″ – t′ < 1, we have

(Qu)(t)(Qu)(t)α(T(t)T(t))i=1pciB0tiT(tis)f(s,x(s))dsα+0tT(s)(f(ts,x(ts))f(ts,x(ts)))dsα+0ttT(ts)f(s,x(s))dsα:=I1+I2+I3.(30)

By (8), (30), Lemma 2.1 and the condition (P3), we know that

I1=(T(t)T(t))i=1pciB0tiT(tis)f(s,x(s))dsα=T(t)(T(tt)I)A(αα)i=1pciB0tiAαT(tis)f(s,x(s))dsCαα(tt)ααi=1pciB0tiAαT(tis)f(s,x(s))dsCααi=1p|ci|1i=1p|ci|0tiMα(tis)αf(s,x(s))ds(tt)ααa1αMαCααi=1p|ci|{L¯(a+(1+m)R)+f(0,0)}(1α)(1i=1p|ci|)(tt)αα.(31)

According to the assumptions (P3) and (P4), Lemma 2.1, (29) and (30), we have

I2=0tT(s)[f(ts,x(ts))f(ts,x(ts))]dsα=0tAαT(s)(f(ts,x(ts))f(ts,x(ts)))dsMα0tsαf(ts,x(ts))f(ts,x(ts))dsL¯Mα0tsα(|tt|αα+u(ts)u(ts)α+j=1mu(aj(ts))u(aj(ts))α)dsL¯Mα0tsα(|tt|αα+L|tt|αα+j=1mL|aj(ts)aj(ts)|αα)dsa1αL¯Mα1α[|tt|αα+L(1+j=1mljαα)|tt|αα].(32)

Using the condition (P3), Lemma 2.1, (30) and (29), we get that

I3=0ttAαT(ts)f(s,x(s))dsMα{L¯(a+(1+m)R)+f(0,0)}1α|tt|αα.(33)

Thus, from (30)-(33) we get that

(Qu)(t)(Qu)(t)α{K0+L[a1αL¯Mα1α(1+j=1mljαα)]}|tt|αα,(34)

where K0 is a constant independent of L. Since the condition (P5) implies that

K:=a1αL¯Mα1α(1+j=1mljαα)<1,

then

(Qu)(t)(Qu)(t)αL|tt|αα,

whenever

LK01K.

Therefore, Q has a fixed point u which is a mild solution of the problem (1)-(2). By the above calculation, we see that for this u(·) and the following function

F(t):=0tT(ts)f(s,x(s))ds

are Hölder continuous. Since the space Xα is reflexive by assumption and Lemma 4.1, u(·) is almost everywhere differentiable on (0, a] and u′(·) ∈ L1(J, X). A similar argument shows that F also have this property. Furthermore, we can obtain that

F(t)=f(t,x(t))A0tT(ts)f(s,x(s))ds.(35)

Therefore, by (35) we get for almost all tJ that

ddtu(t)=ddt(i=1pciT(t)B0tiT(tis)f(s,x(s))ds+0tT(ts)f(s,x(s))ds)=A(i=1pciT(t)B0tiT(tis)f(s,x(s))ds)+f(t,x(t))A0tT(ts)f(s,x(s))ds=Au(t)+f(t,x(t)).

This shows that u is a strong solution of the problem (1)-(2). This completes the proof. □

5 Example

In this section we apply some of the results established in this paper to the following first order parabolic partial differential equation with homogeneous Dirichlet boundary condition and nonlocal initial condition

tw(x,t)2x2w(x,t)=f(x,t,w(x,t),xw(x,t),w(x,a1(t)),xw(x,a1(t))),(x,t)[0,π]×J,w(0,t)=w(π,t)=0,tJ,w(x,0)=i=1parctan12i2w(x,i),x[0,π],(36)

where the functions f and a1 will be described below.

Set X = L2([0, π],ℝ) with the norm ∥ · ∥L2. Then X is reflexive Banach space. Define an operator A in reflexive Banach space X by

D(A)=W2,2(0,π)W01,2(0,π),Au=2x2u.

From [13] we know that –A generates a strong continuous semigroup T(t) (t ≥ 0), which is compact, analytic and exponentially stable in X. Furthermore, A has discrete spectrum with eigenvalues xn = n2, n ∈ ℕ, associated normalized eigenvectors en(x)=2πsin(nx). Then the following properties hold:

  1. If uD(A), then

    Au=n=0n2u,enen.

  2. For each uX,

    A12u=n=11nu,enen.

    In particular, A12=1. Hence, it follows that ∥A–1∥ ≤ 1.

  3. The operator A12 is given by

    A12=n=1nu,enen

    on the space D(A12)={u()X,n=1nu,enenX}.

To prove the main result of this section, we need the following lemma.

Lemma 5.1

([14]). If uD(A12), then u is absolutely continuous with u′ ∈ X andu′∥L2 = A12uL2.

According to Lemma 5.1, we can define the Banach space X12=(D(A12),12). Then for uX12, we have

u12=A12uL2=uL2.

We assume that the nonlinear function g satisfies the following assumption:

  • (P6)

    The function g : [0, π] × J × ℝ4 → ℝ is continuous and there is a function hL (J, ℝ) such that |g(x, t, ζ, ξ, η, ρ)| ≤ h(t), for tJ and ζ, ξ, η, ρ ∈ ℝ.

For each tJ and uX12, we define

α=12,m=1,u(t)=w(,t),f(t,u(t),u(a1(t)))(x)=g(x,t,w(x,t),xw(x,t),w(x,a1(t)),xw(x,a1(t))),ci=i=1parctan12i2,ti=i,i=1,2,,p.

Then f:[0,1]×X12×X12X, and the parabolic partial differential equation with homogeneous Dirichlet boundary condition and nonlocal initial conditions (36) can be rewritten into the abstract form of problem (1)-(2) for m = 1. Since i=1p|ci|i=1parctan12i2=π4<1, the condition (P0) is satisfied. Below we will verify that f satisfies the condition (P1). In fact, it follows from assumption (P6) that

supu12rf(t,u(t),u(a1(t)))h(t)andlim infr+hL1q([0,a])r=0<+.

Therefore, Theorem 3.1 ensures the following existence result.

Theorem 5.2

If the nonlinear function g satisfies the assumption (P6), then the parabolic partial differential equation with homogeneous Dirichlet boundary condition and nonlocal initial conditions (36) has at least one mild solution.

In order to obtain the existence of strong solutions to the parabolic partial differential equation with homogeneous Dirichlet boundary condition and nonlocal initial conditions (36), the following assumptions are also needed.

  • (P7)

    The function g : [0, π] × J × ℝ4 → ℝ is continuous and there is constant L > 0 and α′ ∈ (α, 1) such that

    |g(x,t,ζ2,ξ2,η2,ρ2)g(x,t,ζ1,ξ1,η1,ρ1)|L(|tt|αα+|ζ2ζ1|+|ξ2ξ1|+|η2η1|+|ρ2ρ1|),

  • (P8)

    There exist constants l1 > 0 such that

    |a1(t)a1(t)|l1|tt|,fort,tJ.

For each ϕj, ψj, ∈ X12, j = 1, 2 and t′, t″ ∈ J, we have

f(t,ϕ2,ψ2)f(t,ϕ1,ψ1)L2=[0π(g(x,t,ϕ2(x,t),xϕ2(x,t),ψ2(x,t),xψ2(x,t))g(x,t,ϕ1(x,t),xϕ1(x,t),ψ1(x,t),xψ2(x,t)))2dx]12[0πL2(|tt|αα+|ϕ2(x,t)ϕ1(x,t)|+|xϕ2(x,t)xϕ2(x,t)|+|ψ2(x,t)ψ1(x,t)|+|xψ2(x,t)xψ2(x,t)|)2dx]12L(π|tt|αα+ϕ2ϕ1L2+(ϕ2ϕ1)L2+ψ2ψ1L2+(ψ2ψ1)L2)L(π|tt|αα+(ϕ2ϕ1)L2+(ϕ2ϕ1)L2+(ψ2ψ1)L2+(ψ2ψ1)L2)2L(|tt|αα+ϕ2ϕ112+ψ2ψ112).

Hence (P3) holds with L = 2L. Therefore, it from Theorem 4.2, we have the following result.

Theorem 5.3

If the assumptions (P7) and (P8) are satisfied, then the parabolic partial differential equation with homogeneous Dirichlet boundary condition and nonlocal initial conditions (36) has a strong solution solution provided that 4a1/2LMα(1+l1α1/2)<1.

Acknowledgement

This work is supported by NNSF of China (11661071), NNSF of China (11501455) and Key project of Gansu Provincial National Science Foundation (1606RJZA015).

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About the article

Received: 2017-02-21

Accepted: 2018-01-05

Published Online: 2018-02-23


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 113–126, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0012.

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© 2018 Zhang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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