The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with non regular boundary conditions

http://jipam.vu.edu.au/

Volume 5, Issue 2, Article 38, 2004

BOUNDARY VALUE PROBLEM FOR SECOND-ORDER DIFFERENTIAL OPERATORS WITH MIXED NONLOCAL BOUNDARY CONDITIONS

M. DENCHE AND A. KOURTA LABORATOIREEQUATIONSDIFFERENTIELLES,

DEPARTEMENT DEMATHEMATIQUES, FACULTE DESSCIENCES, UNIVERSITEMENTOURI,

25000 CONSTANTINE, ALGERIA. denech@wissal.dz

Received 15 October, 2003; accepted 16 April, 2004 Communicated by A.G. Ramm

ABSTRACT. In this paper, we study a second order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called non regular boundary conditions, we prove that the resolvent decreases with respect to the spectral parameter inL^p(0,1), but there is no maximal decreasing at infinity forp≥1. Furthermore, the studied operator generates inL^p(0,1)an analytic semi group with singularities for p ≥ 1. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with non regular boundary conditions.

Key words and phrases: Green’s Function, Non Regular Boundary Conditions, Regular Boundary Conditions, Semi group with Singularities.

2000 Mathematics Subject Classification. 47E05, 35K20.

1. INTRODUCTION

In the spaceL^p(0,1)we consider the boundary value problem (1.1)







L(u) = u⁰⁰ =f(x),

B₁(u) = a₀u(0) +b₀u⁰(0) +c₀u(1) +d₀u⁰(1) = 0, B2(u) = R1

0 R(t)u(t)dt+R1

0 S(t)u⁰(t)dt= 0,

where the functions R, S ∈ C([0,1],C). We associate to problem (1.1) in space L^p(0,1)the operator:

L_p(u) =u⁰⁰,

ISSN (electronic): 1443-5756 c

2004 Victoria University. All rights reserved.

The authors are grateful to the editor and the referees for their valuable comments and helpful suggestions which have much improved the presentation of the paper.

147-03

with domainD(L_p) ={u∈W^2,p(0,1) : B_i(u) = 0, i= 1,2}.

Many papers and books give the full spectral theory of Birkhoff regular differential operators with two point linearly independent boundary conditions, in terms of coefficients of boundary conditions. The reader should refer to [7, 10, 20, 21, 22, 28, 31, 33] and references therein.

Few works have been devoted to the study of a non regular situation. The case of separated non regular boundary conditions was studied by W. Eberhard, J.W. Hopkins, D. Jakson, M.V.

Keldysh, A.P. Khromov, G. Seifert, M.H. Stone, L.E. Ward (see S. Yakubov and Y. Yakubov [33] for exact references). A situation of non regular non-separated boundary conditions was considered by H. E. Benzinger [2], M. Denche [4], W. Eberhard and G. Freiling [8], M.G.

Gasumov and A.M. Magerramov [12, 13], A.P. Khromov [18], Yu. A. Mamedov [19], A.A.

Shkalikov [24], Yu. T. Silchenko [26], C. Tretter [29], A.I. Vagabov [30], S. Yakubov [32] and Y. Yakubov [34].

A mathematical model with integral boundary conditions was derived by [9, 23] in the con- text of optical physics. The importance of this kind of problem has been also pointed out by Samarskii [27].

In this paper, we study a problem for second order ordinary differential equations with mixed nonlocal boundary conditions combined with weighted integral boundary conditions and an- other two point boundary condition. Following the technique in [11, 20, 21, 22], we should bound the resolvent in the spaceL^p(0,1)by means of a suitable formula for Green’s function.

Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called non regular boundary conditions, we prove that the resolvent decreases with respect to the spectral parameter inL^p(0,1), but there is no maximal decreasing at infinity for p≥1. In contrast to the regular case this decreasing is maximal forp= 1as shown in [11]. Fur- thermore, the studied operator generates inL^p(0,1)an analytic semi group with singularities forp≥1. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with non regular non local boundary conditions.

2. GREEN’SFUNCTION

Letλ ∈C,u₁(x) =u₁(x, λ)andu₂(x) =u₂(x, λ)be a fundamental system of solutions to the equation

L(u)−λu= 0.

Following [20], the Green’s function of problem (1.1) is given by

(2.1) G(x, s;λ) = N(x, s;λ)

∆(λ) ,

where∆(λ)it the characteristic determinant of the considered problem defined by

(2.2) ∆(λ) =

B₁(u₁) B₁(u₂) B₂(u₁) B₂(u₂)

and

(2.3) N(x, s;λ) =

u₁(x) u₂(x) g(x, s, λ) B1(u1) B1(u2) B1(g)x

B2(u1) B2(u2) B2(g)x

forx, s∈[0,1].The functiong(x, s, λ)is defined as follows

(2.4) g(x, s;λ) =±1

2

u₁(x)u₂(s)−u₁(s)u₂(x) u⁰₁(s)u2(s)−u1(s)u⁰₂(s), where it takes the plus sign forx > sand the minus sign forx < s.

Given an arbitraryδ ∈ ^π₂, π

, we consider the sector

Σ_δ ={λ ∈C,|argλ| ≤δ, λ6= 0}. Forλ∈P

δ,defineρas the square root ofλwith positive real part. Forλ6= 0, we can consider a fundamental system of solutions of the equationu⁰⁰ = λu = ρ²ugiven byu₁(t) = e^−ρt and u₂(t) = e^ρt.

In the following we are going to deduce an adequate formulae for∆(λ)andG(x, s;λ).First of all, forj = 1,2,we have

B₁(u_j) =a₀+ (−1)^jb₀ρ+c₀e^(−1)jρ+ (−1)^jd₀ρe^(−1)jρ, B₂(u_j) =

Z 1 0

R(t)e^(−1)jρtdt+ (−1)^jρ Z 1

0

S(t)e^(−1)jρtdt.

so we obtain from (2.2)

(2.5) ∆(λ) = a₀−b₀ρ+c₀e^−ρ−d₀ρe^−ρ Z 1

0

(R(t) +ρS(t))e^ρtdt

−(a₀+b₀ρ+c₀e^ρ+d₀ρe^ρ) Z 1

0

(R(t)−ρS(t))e^−ρtdt

, andg(x, s;λ)has the form

g(x, s;λ) =









 1

4ρ(e^ρ(x−s)−e^ρ(s−x)) ifx > s, 1

4ρ(e^ρ(s−x)−e^ρ(x−s)) ifx < s.

Thus we have

B₁(g) = a₀−b₀ρ−c₀e^−ρ+d₀ρe^−ρe^ρs 4ρ + (−a0−b0ρ+c0e^ρ+d0ρe^ρ)e^−ρs

4ρ , B₂(g) = e^ρs

4ρ Z s

0

(R(t)−ρS(t))e^−ρtdt+ Z 1

s

(−R(t) +ρS(t))e^−ρtdt

+e^−ρs 4ρ

− Z s

0

(R(t) +ρS(t))e^ρtdt+ Z 1

s

(R(t) +ρS(t))e^ρtdt

. After a long calculation, formula (2.3) can be written as

(2.6) N(x, s;λ) = ϕ(x, s;λ) +ϕ_i(x, s;λ), where

(2.7) ϕ(x, s;λ) = 1 2ρ

Z 1 0

(a₀ +ρb₀) (R(t) +ρS(t))e^ρtdt

− e^ρ(c₀+ρd₀) Z s

0

(R(t) +ρS(t))e^ρtdt

e^−ρ(x+s) +

Z 1 s

(a₀−ρb₀) (R(t)−ρS(t))e^−ρtdt

− e^ρ(c₀−ρd₀) Z s

0

(R(t)−ρS(t))e^−ρtdt

e^ρ(x+s)

and the functionϕ_i(x, s;λ)is given by

(2.8) ϕ_i(x, s;λ) =







ϕ1(x, s;λ) ifx > s, ϕ₂(x, s;λ) ifx < s, with

(2.9) ϕ₁(x, s;λ) = e^ρ(s−x) 2ρ

Z s 0

(a₀+ρb₀+c₀e^ρ+ρe^ρd₀) (R(t)−ρS(t))e^−ρtdt

− Z 1

s

(a₀−ρb₀) (R(t) +ρS(t))e^ρtdt

+e^ρ(x−s) 2ρ

Z s 0

a₀−ρb₀+c₀e^−ρ−ρe^−ρd₀

(R(t) +ρS(t))e^ρtdt

− Z 1

0

(a₀+ρb₀) (R(t)−ρS(t))e^−ρtdt

, and

(2.10) ϕ₂(x, s;λ) = 1 2ρ

− Z 1

s

(a₀ +ρb₀+c₀e^ρ+ρe^ρd₀) (R(t)−ρS(t))e^−ρtdt +

Z 1 0

(c₀−ρd₀)e^−ρ(R(t) +ρS(t))e^ρtdt

e^ρ(s−x)

− Z 1

s

a0−ρb0+c0e^−ρ−ρe^−ρd0

(R(t) +ρS(t))e^ρtdt

− Z 1

0

(c₀+ρd₀)e^ρ(R(t)−ρS(t))e^−ρtdt

e^ρ(x−s)

.

3. BOUNDS ON THE RESOLVENT

Everyλ ∈ C such that∆(λ) 6= 0 belongs to ρ(L_p), and the associated resolvent operator R(λ, L_p)can be expressed as a Hilbert Schmidt operator

(3.1) (λI−L_p)⁻¹f =R(λ, L_p)f =− Z 1

0

G(·, s;λ)f(s)ds, f ∈L^p(0,1).

Then, for everyf ∈L^p(0,1)we estimate (3.1) kR(λ;Lp)fk_Lp(0,1) ≤

sup

0≤s≤1

Z 1 0

|G(x, s;λ)|^pdx ¹_p

kfk_Lp(0,1), and so we need to bound

sup

0≤s≤1

Z 1 0

|G(x, s;λ)|^pdx ¹_p

= 1

|∆(λ)|

sup

0≤s≤1

Z 1 0

|N(x, s;λ)|^pdx ¹_p

.

3.1. Estimation ofN(x, s;λ). We will denote byk·kthe supremum norm which is defined by kRk= sup

0≤s≤1

|R(s)|. Since

N(x, s;λ) =







ϕ(x, s;λ) +ϕ₁(x, s;λ) ifx > s ϕ(x, s;λ) +ϕ₂(x, s;λ) ifx < s

;

then

(3.2) kN(x, s;λ)k_Lp ≤ kϕ(x, s;λ)k_Lp+kϕ_i(x, s;λ)k_Lp, from (2.8), we have

(3.3) kϕ_i(x, s;λ)k_Lp ≤2^1/p

(Z s 0

|ϕ₂(x, s;λ)|^pdx _p¹

+ Z 1

s

|ϕ₁(x, s;λ)|^pdx ¹_p)

. From (2.7) we have

|ϕ(x, s;λ)| ≤ e^{−(x+s) Reρ} 2|ρ|

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) Z 1

s

e^tReρdt + (kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|)e^Reρ

Z s 0

e^tReρdt

+e^{(x+s) Reρ} 2|ρ|

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) Z 1

s

e^−tReρdt + e^−Reρ(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|)

Z s 0

e^−tReρdt

then

|ϕ(x, s;λ)| ≤ e^{−(x+s) Reρ} 2|ρ|Reρ

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) e^Reρ−e^sReρ + (kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^{(s+1) Reρ}−e^Reρ

+e^{(x+s) Reρ} 2|ρ|Reρ

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) e^−sReρ−e^−Reρ + (kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^−Reρ−e^{−(s+1) Reρ} and

Z 1 0

|ϕ(x, s;λ)|^pdx

≤ 2^pR1

0 e^−pxReρdx (|ρ|Reρ)^p

h(kRk+|ρ| kSk)^p(|a₀|+|ρ| |b₀|)^p e^{(1−s) Reρ}−1^p + (kRk+|ρ| kSk)^p× (|c₀|+|ρ| |d₀|)^p e^Reρ−e^{(1−s) Reρp}i

+2^pR1

0 e^pxReρdx

(|ρ|Reρ)^p [(kRk+|ρ| kSk)^p×(|a₀|+|ρ| |b₀|)^p 1−e^{(s−1) Reρ}p

+ (kRk+|ρ| kSk)^p(|c₀|+|ρ| |d₀|)^p e^{(s−1) Reρ}−e^{−1 Reρ}pi ,

after calculation we obtain Z 1

0

|ϕ(x, s;λ)|^pdx ¹p

≤ 2^1+p2e^Reρ

|ρ|(Reρ)^1+1pp^1/p

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|)

e^−sReρ−e^−Reρ

× 1−e^−pReρ1_p

+ 1−e^−pReρ1_p

1−e^{(s−1) Reρ}i

+ (kRk+|ρ| kSk)

×(|c₀|+|ρ| |d₀|)h

1−e^−pReρ1_p

1−e^−sReρ + 1−e^−pReρ1_p

e^{(s−1) Reρ}−e^−Reρii asReρ >0so

(3.4) sup

0≤s≤1

Z 1 0

|ϕ(x, s;λ)|^pdx _p¹

≤ 2^1+2pe^Reρ(kRk+|ρ| kSk) [(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)]

|ρ|(Reρ)^1+1pp^1/p

.

From (2.9) we have

|ϕ₁(x, s;λ)| ≤ e^{(s−x) Reρ} 2|ρ|

(|a₀|+|ρ| |b₀|) +e^Reρ(|c₀|+|ρ| |d₀|)

×(kRk+|ρ| kSk) Z s

0

e^−tReρdt + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk)

Z 1 0

e^tReρdt

+ e^{(x−s) Reρ} 2|ρ|

(kRk+|ρ| kSk)

[(|a₀|+|ρ| |b₀|) +e^Reρ(|c₀|+|ρ| |d₀|)]

Z s 0

e^tReρdt + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk)

Z 1 0

e^−tReρdt

then

|ϕ1(x, s;λ)| ≤ e^{(s−x) Reρ} 2|ρ|Reρ

(|a0|+|ρ| |b0|) (kRk+|ρ| kSk) e^Reρ−e^−sReρ + (|c0|+|ρ| |d0|) (kRk+|ρ| kSk) e^Reρ−e^{(1−s) Reρ}

+ e^{(x−s) Reρ} 2|ρ|Reρ

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) e^sReρ−e^−Reρ + (|c₀|+|ρ| |d₀|) (kRk+|ρ| kSk) e^{(s−1) Reρ}−e^−Reρ

and Z 1

s

|ϕ₁(x, s;λ)|^pdx

≤ 2^pR1

s e^−pxReρdx

|ρ|^p(Reρ)^p

h(|a₀|+|ρ| |b₀|)^p(kRk+|ρ| kSk)^p e^{(1+s) Reρ}−1^p + (|c₀|+|ρ| |d₀|)^p(kRk+|ρ| kSk)^p e^{(1+s) Reρ}−e^Reρpi

+2^pR1

s e^pxReρdx

|ρ|^p(Reρ)^p

h(|a₀|+|ρ| |b₀|)^p(kRk+|ρ| kSk)^p 1−e^{−(1+s) Reρp} + (|c₀|+|ρ| |d₀|)^p(kRk+|ρ| kSk)^p e^−Reρ−e^{−(1+s) Reρp}

, this yields

Z 1 s

|ϕ₁(x, s;λ)|^pdx≤ 2^p+1e^pReρ

|ρ|^p(Reρ)^p+1 [(|a₀|+|ρ| |b₀|)^p(kRk+|ρ| kSk)^p

× 1−e^{−(1+s) Reρp}

1−e^{p(s−1) Reρ}

+ (|c₀|+|ρ| |d₀|)^p(kRk+|ρ| kSk)^p 1−e^−sReρp

1−e^{p(s−1) Reρ} asReρ >0we obtain

(3.5) sup

0≤s≤1

Z 1 s

|ϕ1(x, s;λ)|^pdx ¹_p

≤ 2^1+2pe^Reρ(kRk+|ρ| kSk) [(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)]

|ρ|(Reρ)^1+1pp^1/p

. From (2.10) we have

|ϕ₂(x, s;λ)| ≤ e^{(x−s) Reρ} 2|ρ|

(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) Z 1

0

e^{(1−t) Reρ}dt + (kRk+|ρ| kSk)

(|a₀|+|ρ| |b₀|) +e^−Reρ(|c₀|+|ρ| |d₀|) Z 1

0

e^tReρdt

+e^{(s−x) Reρ}

2|ρ| [(|c₀|+|ρ| |d₀|) (kRk+|ρ| kSk) Z 1

0

e^{(t−1) Reρ}dt +

(|a0|+|ρ| |b0|) +e^Reρ(|c0|+|ρ| |d0|)

(kRk+|ρ| kSk) Z 1

s

e^−tReρdt

then

|ϕ₂(x, s;λ)| ≤ e^xReρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^{(1−s) Reρ}−e^−Reρ + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk) e^{(1−s) Reρ}−1

+ e^−xReρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^Reρ−e^{(s−1) Reρ} + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk) 1−e^{(s−1) Reρ}

.

So Z s

0

|ϕ₂(x, s;λ)|^pdx≤ 2^p+1e^pReρ p|ρ|^p(Reρ)^p+1

h(kRk+|ρ| kSk)^p(|c₀|+|ρ| |d₀|)^p 1−e^{(s−2) Reρp}

× 1−e^−psReρ

+ (|a₀|+|ρ| |b₀|)^p

×(kRk+|ρ| kSk)^p 1−e^−psReρ

1−e^{(s−1) Reρp}i asReρ >0we obtain

(3.6) sup

0≤s≤1

Z s 0

|ϕ2(x, s;λ)|^pdx ¹_p

≤ 2^1+2pe^Reρ(kRk+|ρ| kSk) [(|a0|+|c0|) +|ρ|(|b0|+|d0|)]

|ρ|(Reρ)^1+1pp^1/p

. From (3.4), (3.5) and (3.6), we obtain

kR(λ, L_p)k ≤ 2^1+2pe^Reρ(kRk+|ρ| kSk) [(|a0|+|c0|) +|ρ|(|b0|+|d0|)]

|4(λ)| |ρ|(Reρ)^1+1pp^1/p

2^1+2p + 1 , forρ∈Σδ

2 =

ρ∈C:|argρ| ≤ ^δ₂, ρ6= 0 , we have(Reρ)⁻¹ < ¹

|ρ|cos(^δ₂), then kR(λ, L_p)k ≤

2^1+2p

2^1+2p + 1

e^Reρ(kRk+|ρ| kSk) [(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)]

|4(λ)| |ρ|

|ρ|^1+1p

cos^δ₂1+¹_p

p^1/p

.

Finally, we obtain forλ=ρ² ∈Σ_δ (3.7) kR(λ, L_p)k_Lp ≤ c e^Reρ

|4(λ)| |ρ|^1+1p

kRk

|ρ| +kSk

[(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)], where

c=

2^1+2p

2^1+p2 + 1 p^1/p cos^δ₂1+_p¹ .

3.2. Estimation of the Characteristic Determinant. The next step is to determine the cases for which|∆(λ)|remains bounded below. It will then be necessary to bound|∆(λ)|appropri- ately. However, formula (2.5) is not useful for this purpose. It will be then necessary to make some additional regularity hypotheses on the functions R and S, and so we assume that the functionsRandSare inC²([0,1],C).

Integrating twice by parts in (2.5), we obtain (3.8) ∆(λ) =e^ρh

ρ(d₀S(0)−b₀S(1)) + (d₀S⁰(0) +b₀S⁰(1) +a₀S(1)

−b₀R(1)−d₀R(0) +c₀S(0)) + 1

ρ(a₀R(1)−c₀R(0) +b₀R⁰(1)

−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0)) + 1

ρ²(−a₀R⁰(1)−c₀R⁰(0)) + Φ(ρ)

,

where

(3.9) Φ(ρ) =

(a0−ρb0)e^−ρ+ (c0−ρd0)e^−2ρ Z 1

0

R⁰⁰(t)

ρ² −S⁰⁰(t) ρ

e^ρtdt +

(a₀+ρb₀)e^−ρ+ (c₀+ρd₀) Z 1

0

R⁰⁰(t)

ρ² − S⁰⁰(t) ρ

e^−ρtdt +e^−ρ

1 ρ

−b₀R⁰(0) +d₀R⁰(1)−c₀S⁰(1) +a₀S⁰(0) +c₀R(1)−a₀R(0) + ρ(b₀S(0)−d₀S(1))] + 2e^−2ρ[(a₀+ρb₀)(R(1)−ρS(1))

−(c₀−ρd₀)(R(0) +ρS(0)) + 1

ρ²(a₀+ρb₀)(R⁰(1)−ρS⁰(1)) +1

ρ²(c₀−ρd₀)(R⁰(0) +ρS⁰(0))

. After a straightforward calculation, we obtain the following inequality valid for ρ ∈ Σ^δ

2,with

|ρ|sufficiently large,

|Φ(ρ)| ≤

(|a₀|+|ρ| |b₀|)e^−Reρ+ (|c₀|+|ρ| |d₀|)e^{−2 Reρ}

kR⁰⁰k

|ρ|² + kS⁰⁰k

|ρ|

e^Reρ−1 Reρ

+

(|a₀|+|ρ| |b₀|)e^−Reρ+ (|c₀|+|ρ| |d₀|)

kR⁰⁰k

|ρ|² +kS⁰⁰k

|ρ|

1−e^−Reρ Reρ

+ 2e^−Reρ 1

|ρ|

−b0R⁰(0) +d0R⁰(1)−c0S⁰(1) +a0S⁰(0) +c0R(1)−a0R(0) + |ρ| |b₀S(0)−d₀S(1)|] +e^{−2 Reρ}

1

|ρ|(|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk) + 1

|ρ|(|c₀|+|ρ| |d₀|) (kRk+|ρ| kSk) + (|a₀|+|ρ| |b₀|)

R⁰

+|ρ|

S⁰

|ρ|²

!

+ (|c₀|+|ρ| |d₀|)

R⁰

+|ρ|

S⁰

|ρ|²

!#

.

Then

|Φ(ρ)| ≤ 2

|ρ|cos^δ₂

(|a₀|+|c₀|)

|ρ| + (|b0|+|d0|)

kR⁰⁰k

|ρ| +kS⁰⁰k

+ 1

|ρ|cos^δ₂ 1

|ρ|²

−b₀R⁰(0) +d₀R⁰(1)−c₀S⁰(1) +a₀S⁰(0) +c₀R(1)−a₀R(0)

+ |b₀S(0)−d₀S(1)|] + 1

|ρ|(cos^δ₂)²

|a₀|

|ρ| +|b₀| kRk

|ρ| +kSk

+ |c₀|

|ρ| +|d₀| kRk

|ρ| +kSk

+ |a₀|

|ρ| +|b₀|

R⁰

|ρ|² + S⁰

|ρ|

!

+ |c₀|

|ρ| +|d₀|

R⁰

|ρ|² + S⁰

|ρ|

!#

(3.10) .

Where we have used thatRe(ρ)≥ |ρ|cos(^δ₂),(1−e^−Reρ)<1, e^−Reρ≤ 2

|ρ|²(cos^δ₂)²ande^{−2 Reρ} ≤ 1

2|ρ|²(cos^δ₂)², e^−Reρ<1.

There are several cases to analyze depending on the functionsRandS.

• Case 1.

Suppose thatS6= 0,d₀ 6= 0, b₀ 6= 0, d₀S(0)−b₀S(1) = 0and

k₁ =d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0)6= 0.

From (3.8), we have for|ρ|sufficiently large

|4(λ)| ≥e^Reρh

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0)

− 1

|ρ|

a₀R(1)−c₀R(0) +b₀R⁰(1)−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0)

− 1

|ρ|²

a₀R⁰(1) +c₀R⁰(0)

− |Φ(ρ)|

. From (3.10) we deduce forρ∈Σδ

2,|ρ| ≥r₀ >0.

|Φ(ρ)| ≤ c(r₀)

|ρ| . Then, we have

|4(λ)| ≥e^Reρ h

d0S⁰(0) +b0S⁰(1) +a0S(1)−b0R(1)−d0R(0) +c0S(0)

− 1

|ρ|

a₀R(1)−c₀R(0) +b₀R⁰(1)−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0)

− 1

|ρ|²

a₀R⁰(1) +c₀R⁰(0)

− c(r₀)

|ρ|

. we can now chooser₀ >0, such that

1 r₀

a₀R(1)−c₀R(0) +b₀R⁰(1)−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0) + 1

r²₀

a₀R⁰(1) + c₀R⁰(0)

+ c(r₀)

|ρ|

≤ 1 2

d0S⁰(0) +b0S⁰(1) +a0S(1)−b0R(1)−d0R(0) +c0S(0) , then, forρ∈Σ^δ

2,|ρ| ≥r₀ >0, we get

|4(λ)| ≥ e^Reρ 2 |k₁|.

From (3.7), we deduce the following bound, valid for every|argρ| ≤ ^δ₂, ρ6= 0 kR(λ, L_p)k ≤ 2c

|ρ|^1p|k₁|

|a₀|+|c₀|

|ρ|

+ (|b₀|+|d₀|) kRk

|ρ| +kSk

, then

kR(λ, L_p)k ≤ c₁

|λ|^2p1 ,

as|λ| −→+∞,where

c₁ = 2c kSk(|b0|+|d0|)

|k₁| .

• Case 2.

Ifb₀ =d₀ = 0, S 6= 0anda₀S(1) +c₀S(0) = 0,with

k₂ =a₀R(1)−c₀R(0)−a₀S⁰(1) +c₀S⁰(0)6= 0, we have the following bound, valid forλ ∈Σ_δ and|λ|big enough,

kR(λ, L_p)k ≤ 2c(|a₀|+|c₀|)

|ρ|^1p|k₂|

kRk

|ρ| +kSk

, then

kR(λ, Lp)k ≤ c₁

|λ|^2p1 , as|λ| −→+∞,where

c1 = 2c (|a₀|+|c₀|)kSk

|k₂| .

• Case 3.

IfS = 0,b₀ 6= 0,d₀ 6= 0andb₀R(1) +d₀R(0) = 0,with

k₃ =a₀R(1)−c₀R(0) +b₀R⁰(1)−d₀R⁰(0) 6= 0.

Similarly, we get

kR(λ, Lp)k ≤ 2ckRk

|ρ|^1+1p |k₃|

(|a₀|+|c₀|)

|ρ| + (|b0|+|d0|)

, then, we have

kR(λ, L_p)k ≤ c₁

|λ|^2p1 , as|λ| −→+∞,where

c₁ = 2ckRk(|b₀|+|d₀|)

|k3| .

• Case 4.

Ifb₀ =d₀ = 0,S = 0anda₀R(1)−c₀R(0) = 0with k₄ =a₀R⁰(1)−c₀R⁰(0) 6= 0.

Again in this case, we have

kR(λ, L_p)k ≤ 2c(|a0|+|c0|)kRk

|ρ|^1p|k₄| , then

kR(λ, L_p)k ≤ c₁

|λ|^2p1 , as|λ| −→+∞,where

c₁ = ckRk(|a₀|+|c₀|)

|k₄| .

Definition 3.1. The boundary value conditions in (1.1) are called non regular if the functions R,S ∈C²([0,1],C)and if and only if one of the following conditions holds

1-d₀S(0)−b₀(1) = 0, b₀ 6= 0, d₀ 6= 0, S 6= 0and

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0) 6= 0 2-b₀ =d₀ = 0,kSk 6= 0, a₀S(1) +c₀S(0) = 0with

a₀R(1)−c₀R(0)−a₀S⁰(1) +c₀S⁰(0) 6= 0 3-S = 0,b0 6= 0, d0 6= 0, b0R(1) +0R(0) 6= 0with

a₀R(1)−c₀R(0) +b₀R⁰(1)−d₀R⁰(0)6= 0 4-b₀ =d₀ = 0,S = 0,a₀R(1)−c₀R(0) = 0with

a0R⁰(1)−c0R⁰(0) 6= 0.

This proves the following theorem

Theorem 3.1. If the boundary value conditions in (1.1) are non regular, thenΣδ ⊂ ρ(Lp)for sufficiently large|λ|and there existsc >0such that

kR(λ, L_p)k ≤ c

|λ|^2p1 .

Remark 3.2. From Theorem 3.1 it follows that the operator L_p, for p 6= ∞, generates an analytic semigroup with singularities [25] of typeA(2p−1, 4p−1).

3.3. Application. In the following, we apply the above obtained results to the study of a class of a mixed problem for a parabolic equation with an weighted integral boundary condition combined with another two point boundary condition of the form

(3.11)



















∂u(t, x)

∂t −a∂²u(t, x)

∂x² =f(t, x)

L₁(u) =a₀u(0, t) +b₀u⁰(0, t) +c₀u(1, t) +d₀u⁰(1, t) = 0, L₂(u) =R1

0 R(ξ)u(t, ξ)dξ+R1

0 S(ξ)u⁰(t, ξ)dξ = 0, u(0, x) =u0(x),

where(t, ξ) ∈ [0, T]×[0,1]. Boundary value problems for parabolic equations with integral boundary conditions are studied by [1, 3, 5, 6, 14, 15, 16, 17, 35] using various methods. For instance, the potential method in [3] and [17], Fourier method in [1, 14, 15, 16] and the energy inequalities method has been used in [5, 6, 35]. In our case, we apply the method of operator differential equation. The study of the problem is then reduced to a cauchy problem for a parabolic abstract differential equation, where the operator coefficients has been previously studied. For this purpose, letE,E₁, andE₂ be Banach spaces. Introduce two Banach spaces

C_µ((0, T], E) = (

f /f ∈C((0, T], E), kfk= sup

t∈(0,T]

kt^µf(t)k<∞ )

, µ≥0,

C_µ^γ((0, T], E) = (

f /f ∈C((0, T], E), kfk= sup

t∈(0,T]

kt^µf(t)k + sup

0<t<t+h≤T

kf(t+h)−f(t)kh^−γt^µ<∞

, µ≥0, γ ∈(0,1],

and a linear space

C¹((0, T], E₁, E₂) =

f /f ∈C((0, T], E₁)∩C¹((0, T], E₂) , E₁ ⊂E₂,

where C((0, T], E) and C¹((0, T], E) are spaces of continuous and continuously differen- tiable, respectively, vector-function from(0, T]intoE. We denote, for a linear operator Ain a Banach spaceE, by

E(A) =n

u/u∈D(A), kuk_E(A) = kAuk²+kuk²¹₂o , and

C¹((0, T], E(A), E) = n

f /f ∈C((0, T], E(A)), f⁰ ∈C((0, T], E)) o

.

Let us derive a theorem which was proved by various methods in [25] and [31, 33]. Consider, in a Banach spaceE, the Cauchy problem

(3.12)

( u⁰(t) =Au(t) +f(t), t∈[0, T], u(0) =u₀,

where Ais, generally speaking, unbounded linear operator in E, u0 is a given element of E, f(t)is a given vector-function andu(t)is an unknown vector-function inE.

Theorem 3.3. Let the following conditions be satisfied:

(1) Ais a closed linear operator in a Banach spaceEand for someβ ∈(0,1], α >0 kR(λ, A)k ≤C|λ|^−β, |argλ| ≤ π

2 +α, |λ| → ∞;

(2) f ∈C_µ^γ((0, T], E)for someγ ∈(1−β,1], µ∈[0, β) ; (3) u0 ∈D(A).

Then the Cauchy problem (3.12) has a unique solution

u∈C([0, T], E)∩C¹((0, T], E(A), E) and for the solutionuthe following estimates hold

ku(t)k ≤C

kAu₀k+ku₀k+kfk_C

µ((0,t],E)

, t∈(0, T],

u⁰(t)

+kAu(t)k ≤C

t^β−1(kAu₀k+ku₀k) +t^β−µ−1kfk_C^γ

µ((0,t],E)

, t∈(0, T]. As a result of this we get the following theorem

Theorem 3.4. Let the following conditions be satisfied (1) a6= 0,|arga|< π

2,

(2) the functionsR(t), S(t)∈C²([0,1],C)and one of the following conditions is satisfied d0S(0)−b0S(1) = 0,b0 6= 0, d0 6= 0,andS6= 0

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0)6= 0, orb₀ = 0, d₀ = 0, S 6= 0, a₀S(1) +c₀S(0) = 0with

a₀R(1)−c₀R(0)−a₀S⁰(1) +c₀S⁰(0)6= 0, orb₀ 6= 0, d₀ 6= 0, S = 0,b₀R(1) +d₀R(0) = 0with

a₀R(1)−c₀R(0) +b₀R⁰(1)−d₀R⁰(0) 6= 0,

orb₀ =d₀ = 0,S = 0,a₀R(1)−c₀R(0) = 0with a₀R⁰(1)−c₀R⁰(0) 6= 0.

(3) f ∈C_µ^γ((0, T], L^q(0,1))for someγ ∈

1− _2q¹,1i

and someµ∈h 0,_2q¹

, (4) u0 ∈W_q² (0,1), Liu= 0, i= 1,2

. Then the problem (3.11) has a unique solution

u∈C((0, T], L^q(0,1))∩C¹ (0, T], W_q²(0,1), L^q(0,1) and for this solution we have the estimates:

(3.13) ku(t,·)k_Lq(0,1) ≤c

ku₀k_W2

q(0,1)+kfk_C

µ((0,t],L^q(0,1))

, t∈(0, T], (3.14) ku⁰⁰(t,·)k_Lq(0,1)+ku⁰(t,·)k_Lq(0,1)

≤c

t^2q1−1ku₀k_W2

q(0,1)+t^2q1−1−µkfk_C^γ

µ((0,t],L^q(0,1))

, t∈(0, T]. Proof. We consider in the spaceL^q(0,1) 1≤q <∞, the operatorAdefined by

A(u) =au⁰⁰(x), D(A) =

u∈W_q²(0,1), Li(u) = 0, i= 1,2 . Then problem (3.11) can be written as

( u⁰(t) = Au(t) +f(t), u(0) =u₀,

whereu(t) = u(t,·), f(t) = f(t,·), and u₀ = u₀(·) are functions with values in the Banach spaceL^q(0,1).From Theorem 3.1 we conclude thatkR(λ, A)k ≤c|λ|^−2q1 ,for|argλ| ≤ π

2+α, as|λ| → ∞.

Then, from Theorem 3.3 the problem (3.11) has a unique solution

u∈C((0, T], L^q(0,1))∩C¹ (0, T], W_q²(0,1), L^q(0,1) and we have the following estimates

(3.15) ku(t, .)k_Lq(0,1) ≤c

kAu₀k_Lq(0,1)+ku₀k_Lq(0,1)+kfk_C

µ((0,t],L^q(0,1))

, (3.16)

u⁰(t)

Lq(0,1)

+kAu(t,·)k

Lq(0,1)

≤c

t^2q1−1

kAu₀k_Lq(0,1)+ku₀k_Lq(0,1)

+t^2q1−1−µkfk_C^γ

µ((0,t],L^q(0,1))

, wheret ∈[0, T], from (3.15) we get

ku(t,·)k_L

q(0,1) ≤c

ku⁰⁰₀k_Lq(0,1)+ku₀k_Lq(0,1)+kfk_C

µ((0,t],L^q(0,1))

≤c

ku₀k_W2

q(0,1)+kfk_C

µ((0,t],L^q(0,1))

, t∈[0, T]. and from (3.16) we get

u⁰(t)

L^q(0,1)

+ku⁰⁰(t,·)k_Lq(0,1)

≤c

t^2q1−1

kAu₀k_Lq(0,1)+ku₀k_Lq(0,1)

+t^2q1−1−µkfk_C^γ

µ((0,t],L^q(0,1))

≤c

t^2q1−1ku₀k_W2

q(0,1)+t^2q1−1−µkfk_C^γ

µ((0,t],L^q(0,1))

, t∈[0, T].

which gives the desired result.

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