A Note on the Parabolic Differential and Difference Equations

Volume 2007, Article ID 61659,16pages doi:10.1155/2007/61659

Research Article

A Note on the Parabolic Differential and Difference Equations

Received 13 August 2006; Revised 12 December 2006; Accepted 6 March 2007 Recommended by Martin J. Bohner

The differential equationu(t) + Au(t)= f(t) (−∞< t <∞) in a general Banach space E with the strongly positive operator A is ill-posed in the Banach spaceC(E)=C(R, E) with normϕ_C(E)=sup_−∞<t<∞ϕ(t)_E. In the present paper, the well-posedness of this equa- tion in the H¨older spaceC^α(E)=C^α(R, E) with normϕ_Cα(E)=sup_−∞<t<∞ϕ(t)_E+ sup_{−∞<t<t+s<∞}(ϕ(t+s)−ϕ(t)_E/s^α), 0< α <1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme inC(Rτ, E) spaces is proved. The well-posedness of this difference scheme inC^α(Rτ, E) spaces is obtained.

Copyright © 2007 Allaberen Ashyralyev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The role played by coercivity inequalities (maximal regularity, well-posedness) in the study of boundary value problems for parabolic and elliptic differential equations is well known (see, e.g., [1–3]).

Coercivity inequalities approach permits to investigate the general boundary value problems for both elliptic and parabolic differential equations.

The coercivity inequalities also hold for various difference analogues of such prob- lems. These inequalities evidently enable us to prove not only the existence of solutions but also the well-posedness of such problems. Main role of the coercivity inequalities for difference problems lies in that they present a special type of stability, which allows the ex- istence of exact, that is, two-sided estimates of rate of convergence approximate solutions with respect to the corresponding coercivity norms.

It is quite possible that there are cases where the difference problem is well-posed, although the differential problem is not.

Well-posedness of local and nonlocal boundary value problems for abstract parabolic differential and difference equations in Banach spaces have been studied extensively by many researchers (see [3–21] and the references therein).

In present paper, the well-posedness of the parabolic equation is investigated. The pa- per is organized as follows. In Section 2, the parabolic differential equation in the Ba- nach space E is considered. The well-posedness of this equation in the H¨older space is presented. In Section 3, the first order of accuracy Rothe difference scheme for para- bolic differential equation is studied. The almost coercivity inequality for solutions of this difference scheme is established.Section 4presents the well-posedness of this difference scheme in difference analogues of H¨older spaces.

2. Well-posedness of the parabolic differential equation

In the arbitrary Banach space E, the parabolic differential equation du(t)

dt + Au(t)=f(t), −∞< t <∞, (2.1) is considered. Hereu(t) and f(t) are unknown and given abstract functions, defined on R=(−∞,∞) with values in E; A is a linear unbounded closed operator acting in E with dense domainD(A)⊂E.

A functionu(t) is called a solution of the problem (2.1) if the following conditions are satisfied:

(i)u(t) is continuously differentiable bounded onR;

(ii) The elementu(t) belongs toD(A) for allt∈Rand the functionAu(t) is contin- uously bounded onR;

(iii)u(t) satisfies (2.1).

A solution of problem (2.1) defined in this manner will from now on be referred to as a solution of problem (2.1) in the space C(E)=C(R, E) of all continuously bounded functionsϕ(t) defined onRwith values in E equipped with the norm

ϕC(E)= sup

−∞<t<∞

ϕ(t)E. (2.2)

We say that the problem (2.1) is well-posed inC(E) if the following conditions are satis- fied.

(1) Problem (2.1) is uniquely solvable for any f(t)∈C(E). This means that an ad- ditive and homogeneous operatoru(t)≡u(t;f(t)) acting fromC(E) toC(E) is defined and gives the solution of problem (2.1) inC(E). Moreover, the operators (d/dt)[u(t;f(t))] and Au(t;f(t)) acting inC(E) have these properties also (see, e.g., [10]).

(2)u(t;f(t)), regarded as an operator fromC(E) toC(E), is continuous. It means that inequality

ut;f(t)_C(E)≤MfC(E) (2.3) holds for some 1≤M <∞, which does not depend on f(t)∈C(E).

In this paper, we will indicate withMpositive constants which can be different from time to time and we are not interested to precise. We will writeM(α,β,...) to stress the fact that the constant depends only onα,β,....

From the well-posedness of problem (2.1) inC(E) it follows that the operatoru(t;f(t)) is continuous inC(E), and the operator Au(t;f(t)) is defined on the entire spaceC(E).

The operator A, which acts in the Banach space E with domainD(A), generates via the formulaᏭu=Au(t) an operatorᏭ, which acts in the Banach spaceC(E) and is defined on the functionsu(t)∈C(E) with the property that Au(t)∈C(E). From the fact that the operator A⁻¹ exists and is bounded, it follows that the operator Ꮽ⁻¹ exists and is bounded, and henceᏭis closed inC(E). As a result, the operator Au(t;f(t))=Ꮽ(·,f) is closed inC(E). By Banach’s theorem, this operator is continuous, that is, for any f(t)∈ C(E) one has the inequality

Aut;f(t)_E≤MfE, (2.4)

whereMdoes not depend f(t).

This leads us to coercivity inequality

uC(E)+Au(t)_C(E)≤M_CfC(E) (2.5) for solution of well-posed inC(E) problem (2.1) with some 1≤M_C<∞, which does not depend on f(t)∈C(E).

It is assumed that the operator−A generates a semigroup exp{−tA}(t≥0) with ex- ponentially decreasing norm whent→+∞, that is, the following estimates hold:

e^−tA_E→E≤Me^−δt. (2.6)

Now let us consider the functionv(t) defined by (2A)⁻¹e^tAv ift <0,

(2A)⁻¹e^−tAv+te^−tAv ift≥0. (2.7) Ifv∈D(A), thenv(t) is the solutionC(E) of (2.1) with f(t)=e^−|t|Av.

Fort >0, using (2.3), (2.4), and (2.6), we get the estimate

tAe^−tAv_E≤2⁻¹e^−tA+tAe^−tAv_E+2⁻¹e^−tAv_E

≤ sup

0_≤t<∞

2⁻¹e^−tA+tAe^−tAv_E+M 2 vE

≤ sup

−∞≤t<∞

Av(t)E+M 2vE

≤Me^−|t|Av_E+M

2 vE≤M1vE.

(2.8)

SinceD(A) is dense in E, this implies that Ae^−tAis bounded and obeys the estimate

Ae^−tA_E→E≤Mt⁻¹. (2.9)

This means the analyticity of the semigroup e^−tA fort >0 [10]. Finally, the foregoing argument shows that the analyticity of the semigroup e^−tAis a necessary condition for the well-posedness of problem (2.1) inC(E) [10].

Letu(t) be a solution of the problem (2.1). Then, for any−∞< s≤t, we have the identity

d ds

e^−(t−s)AA⁻¹u(s)=e^−(t−s)AA⁻¹f(s). (2.10)

Integrating with respect tosover the interval [x,t], we obtain A⁻¹u(t)−e^−(t−x)AA⁻¹u(x)=

_t

xe^−(t−s)AA⁻¹f(s)ds. (2.11) Since A is closed, we have

u(t)=e^−(t−x)Au(x) + _t

xe^−(t−s)Af(s)ds. (2.12) By the fact that the analytic semigroup e^−pAhas norm decaying property asp→ ∞,

u(t)= _t

−∞e^−(t−s)Af(s)ds. (2.13) It is easy to see that formula (2.13) defines solution of problem (2.1) in C(E), if, for example, Af(t)∈C(E) or f(t)∈C(E). It turns out that formula (2.13) defines solution of problem (2.1) inC(E) under essentially less restriction on smoothness of function f(t).

Finally, from (2.6), (2.9), the following estimate follow:

A^βe^−tA−e^−(t+τ)A_E→E≤M τ^α

t^α+β (2.14)

for any 0< t < t+τ, 0≤β≤1, and 0≤α≤1.

The well-posedness of problem (2.1) can be established on the assumption (2.6), (2.9) if one considers this problem in the H¨older spaceC^α(E)=C^α(R, E),α∈(0, 1), of all E- valued abstract functionsϕ(t) defined onRwith the norm

ϕC^α(E)= sup

−∞<t<∞

ϕ(t)_E+ sup

−∞<t<t+τ<∞

ϕ(t+τ)−ϕ(t)_E

τ^α . (2.15)

A functionu(t) is said to be a solution of problem (2.1) inC^α(E) if it is a solution of this problem inC(E) and the functionsu(t), Au(t)∈C^α(E). The well-posedness inC^α(E) of problem (2.1) means that coercivity inequality

uC^α(E)+AuC^α(E)≤M(α)fC^α(E) (2.16) holds for its solutionu(t) inC^α(E) with some 1≤M(α)<∞, which is independent of

f(t)∈C^α(E).

Theorem 2.1. The problem (2.1) is well-posed inC^α(E) and the following coercivity in- equality:

uC^α(E)+AuC^α(E)≤ M

α(1−α)fC^α(E) (2.17) holds for some 1≤M <∞.

Proof. From formula (2.13), it follows that

Au(t)= f(t) + _t

−∞Ae^−A(t−s)f(s)−f(t)ds. (2.18) Let us estimateAuC(E). Using formula (2.18), we have

Au(t)_E≤f(t)_E+ _t

−∞

Ae^−A(t−s)_E→Ef(s)−f(t)_Eds. (2.19)

Using estimates (2.6), (2.9), we get

Ae^−A(t−s)_E→E≤Ae^{−A((t−s)/2)}_E→Ee^{−A(t−s)/2}_E→E≤Me^{−(δ/2)(t−s)} M

(t−s)/2. (2.20) Hence,

Au(t)_E≤ fC^α(E)

1 +M1

_t

−∞

e^{−(δ/2)(t−s)}

(t−s)^1−αds (2.21) for allt∈R.

From the substitutionu=t−s, it becomes _t

−∞

e^{−(δ/2)(t−s)} (t−s)^1−αds=

_∞

0

e^−(δ/2)x x^1−α dx≤

1 0

dx x^1−α+

_∞

1 e^−(δ/2)xdx. (2.22) Hence,

Au(t)_E≤ f_C^α(E)

1 +M1

1 α+e^−δ/2

δ/2

=M(δ)

α f_C^α(E) (2.23) for allt∈R. So, from that it follows

AuC(E)≤M(δ)

α fC^α(E). (2.24)

Now, let us establish similar bound for theα-H¨older normH^α(Au) of Au, whereH^α(ϕ) denotes sup_{−∞<t<t+τ<∞}(ϕ(t+τ)−ϕ(t)E/τ^α). For−∞< t < t+τ <∞, we can write

Au(t+τ)−Au(t)=f(t+τ)−f(t) + _t+τ

t−τAe^{−A(t+τ−s)}f(s)−f(t+τ)ds

− _t

t−τAe^−A(t−s)f(s)−f(t)ds

− _t−τ

−∞Ae^{−A(t+τ−s)}−e^−A(t−s)f(s)−f(t)ds +

_t−τ

−∞Ae^{−A(t+τ−s)}f(t)−f(t+τ)ds

=J1+J2+J3+J4+J5.

(2.25)

Clearly,

f(t+τ)−f(t)E≤τ^αfC^α(E) (2.26) for allt∈R. Then

J1

E≤τ^αfC^α(E). (2.27)

Using estimates (2.6), (2.9), we get J2

E≤ _t+τ

t−τMAe^{−A(t+τ−s)}_E→Ef(s)−f(t+τ)_Eds≤Mf_C^α(E)

_t+τ t−τ

1

(t+τ−s)^1−αds.

(2.28) The use of the substitutionx=t+τ−sgives

_t+τ t−τ

1

(t+τ−s)^1−αds= 2τ

0

1

x^1−αdx=(2τ)^α

α , (2.29)

from that it follows

J2

E≤M(2τ)^α

α fC^α(E) (2.30)

for allt∈R. In a similar manner one establishes the estimate J3

E≤Mτ^α

α f_C^α(E). (2.31)

Using estimate (2.14) forβ=1 andα=1, we get J4

E≤ _t−τ

−∞Me^−δ(t−s)τ

(t−s)² f(s)−f(t)_Eds≤Mf_C^α(E)τ _t−τ

−∞

ds (t−s)^2−α=

Mτ^α

1−αf_C^α(E)

(2.32)

for allt∈R. Then

J4

E≤ Mτ^α

1−αfC^α(E). (2.33)

Finally, using the formula

_t−τ

−∞Ae^{−A(t+τ−s)}ds=e^−2τA, (2.34) and estimates (2.6), (2.9), we get

J5

E≤e^−2τA_E→Ef(t)−f(t+τ)_E≤Mτ^αf_C^α(E) (2.35) for allt∈R. Then

J5

E≤Mτ^αf_C^α(E). (2.36)

Combining all these, and using estimate (2.24), we get Au_C^α(E)≤ M

α(1−α)f_C^α(E). (2.37)

By the triangle inequality, this last estimate and (2.1) yield uC^α(E)≤ M

α(1−α)fC^α(E). (2.38)

Theorem 2.1is proved.

Note that the proof ofTheorem 2.1can also be considered a new proof of a particular case of a well-known result [21]. More precisely, if we assume that

(i)^√−1R⊂ρ(A);

(ii) there isM∈R⁺= {z∈R;z >0}, such that

∀ω∈R, ^√−1ω+A⁻¹_E→E≤

1 +|ω|₋1

, (2.39)

then as a consequence of [21, Theorem 8.2],Theorem 2.1can be obtained.

3. Almost coercivity inequality

The difference analogue of the differential equation (2.1) uk−uk−1

τ + Auk=fk, k∈Z, (3.1)

will be considered. Hereu_k∈D(A) and f_k∈E are unknown and given elements,τis a positive small number.

The Banach spaceᏯ(Rτ, E) of all bounded grid functionsv^τ= {vk}^∞_k=−∞defined on R_τ= {t_k=kτ;k∈Z}with the norm

υ^τ_Ꮿ(Rτ,E)= sup

−∞<k<∞

υ_kE (3.2)

is introduced and the operatorDτ, acting from the spaceᏯ(Rτ, E) into the spaceᏯ(Rτ, E), by the rule

v^τ=D_τu^τ, v_k=uk−uk−1

τ , k∈Z, (3.3)

is defined. Then the difference equation (3.1) will be considered as operator equation

D_τu^τ+ Au^τ= f^τ (3.4)

in the Banach spaceᏯ(Rτ, E). Here Au^τ= {Auk}^∞_k=−∞and f^τ= {fk}^∞_k=−∞.

From the property (2.6), (2.9), it follows that there exists the bounded operator (I+ τA)⁻¹, that is, the resolventR(τA), defined on whole space E. Therefore, for every f^τ, there exists a unique solutionu^τ=u^τ(f^τ) of the problem (3.4) and the following formula holds:

u_k= ^k

i=−∞

R^k−i+1(τA)f_i τ,k∈Z. (3.5)

Let the assumption (2.6), (2.9) be satisfied. Since the semigroup e^−tAobeys the exponen- tial decay estimates (2.6), (2.9), we have that

R^k(τA)_E→E≤M(1 +τδ)^−k, k≥1, (3.6)

kτAR^k(τA)_E→E≤M, k≥1. (3.7)

Actually, from the formula connecting the resolvent of the generator of a semigroup with the semigroup (see [20]) it follows that

(I+τA)^−k= 1 (k−1)!

_∞

0 t^k−1e^−te^−τtAdt. (3.8) Using this formula and (2.6), we get

(I+τA)^−k_E→E≤ M (k−1)!

_∞

0 t^k−1e^−t(1+δτ)dt=M(1 +δτ)^−k. (3.9) Estimate (3.6) is proved. Fork≥2 using (2.6), (2.9), (3.8) and the fact that the operator A is closed, this yields the estimate

A(I+τA)^−k_E→E≤ M τ(k−1)!

_∞

0 t^k−2e^−tdt= M

τ(k−1)(1 +δτ)^{k−1 ≤}

4δM τk(1 +δτ)^k,

(3.10) where the last inequality results from 0≤τ≤1. Therefore, (3.7) is proved fork≥2. For k=1, the estimate is obvious.

From (3.6), (3.7), the following estimates follow:

A^βR^k(τA)−R^k+m(τA)_E→E≤M (mτ)^α

(kτ)^α+β (3.11)

for any 1≤k < k+m, 0≤α≤1, and 0≤β≤1.

The problem (3.4) is said to be stable inᏯ(Rτ, E) if we have the stability inequality u^τ_Ꮿ(Rτ,E)≤Mf^τ_Ꮿ(Rτ,E), (3.12) whereMis independent not only of f^τbut also ofτ.

Theorem 3.1. The problem (3.4) is stable inᏯ(R_τ, E) norm.

The proof ofTheorem 3.1is based on formula (3.5) and estimates (3.6), (3.7).

The problem (3.4) is said to be coercively stable (well-posed) inᏯ(Rτ, E) if we have the coercive stability

Au^τ_Ꮿ(Rτ,E)≤Mf^τ_Ꮿ(Rτ,E), (3.13) whereMis independent not only of f^τbut also ofτ.

Since the problem (2.1) in the spaceC(R, E) is not well-posed for the general posi- tive operator A and space E, then the well-posedness of the difference problem (3.4) in Ꮿ(R_τ, E) norm does not take place uniformly with respect toτ >0. This means that the coercivity norm

u^τ_᏷τ(E)=Au^τ_Ꮿ(Rτ,E)+Dτu^τ_Ꮿ(Rτ,E) (3.14) tends to∞asτ→0⁺. The investigation of the difference problem (3.4) permits to estab- lish the order of growth of this norm to∞.

Theorem 3.2. For the solution of the difference problem (3.4), we have the almost coercivity inequality

u^τ_᏷τ(E)≤Mmin

ln1

τ, 1 +lnAE→E f^τ_Ꮿ(Rτ,E). (3.15) Proof. Using formula (3.5) and the substitutionm=k−i+ 1, we get

Au_k= ^k

i=−∞

AR^k−i+1(τA)f_iτ= ^∞

m=1

AR^m(τA)f_k−m+1τ

=

[1/τ] m=1

AR^m(τA)fk−m+1τ+ ∞ m=[1/τ]+1

AR^m(τA)fk−m+1τ=J1+J2,

(3.16)

where [·] stands for the integer part.

Let us estimateJ2. Using estimates (3.6), (3.7), we obtain J2

E≤ ∞ m=[1/τ]+1

AR^[m/2]R^m−[m/2]E_→Efk−m+1

E

≤Mf^τ_Ꮿ(_Rτ,E)

∞ m=[1/τ]+1

1 (1 +τδ)^m/2m

≤Mf^τ_Ꮿ(Rτ,E) 1 [1/τ] + 1

∞ m=[1/τ]+1

1 (1 +τδ)^m/2

≤M(δ)f^τ_Ꮿ(Rτ,E).

(3.17)

Let us estimateJ1. It is clear that

[1/τ] m=1

τAR^m(τA)f_k−m+1

E≤

[1/τ] m=1

τAR^m(τA)_E→Ef^τ_Ꮿ(Rτ,E). (3.18) By [10, Theorem 1.2, page 87],

[1/τ] m=1

τAR^m(τA)_E→E≤Mmin

ln 1

τ

, 1 +lnAE_→E . (3.19) Thus,

J1

E≤Mmin

ln 1

τ

, 1 +lnAE_→E f^τ_Ꮿ(_Rτ,E). (3.20) Combining the estimates forJ1EandJ2E, we obtain

Auk

E≤Mmin

ln 1

τ

, 1 +lnAE_→E f^τ_Ꮿ(_Rτ,E) (3.21) for allk. It follows from that

Au^τ_Ꮿ(Rτ,E)≤Mmin

ln1

τ, 1 +lnAE_→E f^τ_Ꮿ(Rτ,E). (3.22) By the triangle inequality, this last estimate and (3.4) yield

D_τu^τ_Ꮿ(Rτ,E)≤M1min

ln1

τ, 1 +lnAE_→E f^τ_Ꮿ(Rτ,E). (3.23)

Theorem 3.2is proved.

Finally, in the next section the theorem on the well-posedness of difference scheme (3.1) in the difference analogy ofC^α(R, E), 0< α <1, spaces are established.

4. Well-posedness of difference scheme

Now, the difference equation (3.1) is considered as operator equation (3.4) in the Banach spaceᏯ^α(Rτ, E) (0< α <1) of grid functionsϕ^τ= {ϕk}^∞_k=−∞with norm

ϕ^τ_Ꮿα(_Rτ,E)=ϕ^τ_Ꮿ(_Rτ,E)+ sup

−∞<i<i+r<∞

ϕ_i+r−ϕ_iE

(rτ)^α . (4.1)

The well-posedness of (3.4) in the spaceᏯ^α(R_τ, E) means that for solutionsu^τof (3.4) in Ꮿ^α(Rτ, E) coercive inequality

Dτu^τ_Ꮿα(Rτ,E)+Au^τ_Ꮿα(Rτ,E)≤MC(α)f^τ_Ꮿα(Rτ,E) (4.2)

holds for some 1≤M_C(α)<∞, which is independent of f^τand positive small numberτ.

Theorem 4.1. The difference equation (3.4) is well-possed in Banach spaceᏯ^α(R_τ, E) (0<

α <1) and for its solutions coercivity inequality

D_τu^τ_Ꮿα(Rτ,E)+Au^τ_Ꮿα(Rτ,E)≤ M

α(1−α)f^τ_Ꮿα(Rτ,E) (4.3) holds for some 1≤M <∞, which does not depend on f^τ∈Ꮿ^α(Rτ, E),α∈(0, 1), and posi- tive small numberτ.

Proof. Let us estimateAu^τᏯ(Rτ,E). Using formula (3.5), estimates (3.6), (3.7), and the identity

τAR^k−i+1=R^k−i−R^k−i+1, (4.4) we obtain

Auk= ^k

−1

i=−∞

τAR^k−i+1(τA)fi−fk

+fk. (4.5)

The estimates (3.6), (3.7), (3.11), and the substitutionj=k−i+ 1 will imply Auk

E≤ ^k

−1

i=−∞

τAR^k−i+1(τA)_E→Efi−fk

E+fk

E

= ∞ j=2

τAR^j(τA)E_→Efk−j+1−fk

E+fk

E

≤f^τ_Ꮿα(_Rτ,E)

_∞

j=2

Mτ^α

(1 +τδ)^j/2j^1−α+ 1

(4.6)