Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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Lemma 3.1.

i) If (u, v) is a solution of (S

p,q

(λ)) then (u, v) ∈ L

∞

(Ω) × L

∞

(Ω).

ii) For all λ ∈ IR let (u, v) be an eigenvector associated of µ

(λ) such that

u ≥ 0, v ≥ 0 and B(u, v) = 1 then: u > 0 , v > 0 and (u, v) ∈ C

1,η

loc

(Ω)×

1,η

loc

(Ω).

Proof. i) deduced from the results of Th

elin.

ii) Let (u, v) be a solution of (S

p,q

(λ)) associated of (λ, µ(λ)). Thus (|u|, |v|)

is also a solution of (2.1). Indeed, A(|u|, |v|) ≤ A(u, v) and B(|u|, |v|) = 1.

Hence, we can suppose that u ≥ 0 and v ≥ 0, on the other hand by (i) and

Strong Maximum Principle of Vazquez

we conﬁrm that u > 0 , v > 0.

The results of regularity follow from (i) and local regularity of Dibinedetto.

Now, let

(u, φ) =

Ω

|∇u|

+(p−1)

Ω

|∇φ|



|u|



−p

Ω

|∇φ|

p−2

∇φ∇u



|u|

p−2

p−1



for all (u, φ) ∈



1,p

(Ω)



with φ > 0 in Ω.

Lemma 3.2. For all (u, φ) ∈ (W

1,p

(Ω) ∩C(Ω))

with φ > 0 in Ω, we have

(u, φ) ≥ 0 and if Γ

(u, φ) = 0 there is c ∈ IR such that u ≡ cφ.

Proof. By Young’s inequality, we have for  > 0,

∇u|∇φ|

p−2

∇φ

u|u|

p−2

p−1

≤ |∇u||∇φ|

p−1

(

)

p−1

≤



|∇u|

p−1

p

|∇φ|. (3.3)

For  = 1 we have by integration over Ω,

Ω

|∇φ|

p−2

∇φ∇u(

)

p−1

≤

Ω

|∇u|

+ (p −1)

Ω

|∇φ|

thus

(u, φ) ≥ 0.

On the other hand, if Γ

(u, φ) = 0 by (3.3), we obtain

Ω

|∇φ|

p−2

∇φ∇u(

|u|

p−2

p−1

) −

Ω

|∇u|

− (p − 1)

Ω

|∇φ|

= 0 (3.4)

and

Ω



∇u∇φ|∇φ|

p−2

u|u|

p−2

p−1

− |∇u||∇φ|

p−1

(

)

p−1



dx = 0. (3.5)

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By (3.4) we ﬁnd |∇u| = |

∇φ| thus from (3.5), it follows that ∇u = 

∇φ,

where || = 1. Hence Γ

(u, φ) = 0 implies  = 1 and ∇(

) = 0. Therefore,

there is c ∈ IR such that u = cφ.

Theorem 3.2. For all λ ∈ [

−λ

2kak

∞

2kak

∞

] , if (u(λ), v(λ)) is the eigenvec-

tor corresponding to µ

(λ)satisfying B(u, v) = 1, u > 0, v > 0 then (u, v)

is unique.

Proof. Let (u, v) , (φ, ψ) be two positive eigenvectors associated with µ

(λ)

therefore,



−∆

u = λa(x)|u|

α−1

|v|

β+1

u + µ|u|

α−1

|v|

β+1

u (a)

−∆

v = λa(x)|u|

α+1

|v|

β−1

v + µ|u|

α+1

|v|

β−1

v (b)

and



−∆

φ = λa(x)|φ|

α−1

|ψ|

β+1

φ + µ|φ|

α−1

|ψ|

β+1

φ (c)

−∆

ψ = λa(x)|φ|

α+1

|ψ|

β−1

ψ + µ|φ|

α+1

|ψ|

β−1

ψ. (d)

For any  > 0, let



(u, φ) =

(φ + )

p−1



(v, ψ) =

(ψ + )

q−1

Hence, multiplying (a) by u and (c) by φ



(u, φ), integrating by parts over

Ω, and taking the diﬀerence, we obtain

(u, φ) =

Ω

(λa(x) + µ

)|φ|

α+1

|ψ|

β+1

α+1

β+1

− |

], (3.6)

multiplying (b) by v and (d) by ψ



(v, ψ), integrating by parts over Ω and

taking the diﬀerence, we obtain

(v, ψ) =

Ω

(λa(x) + µ

)|φ|

α+1

|ψ|

β+1

α+1

β+1

− |

]. (3.7)

So, multiplying (3.6) by

α+1

and (3.7) by

β+1

, we have

α + 1

(u, φ)+

β + 1

(v, ψ) =

Ω

(λa(x)+µ

)|φ|

α+1

|ψ|

β+1

α+1

β+1

−

α + 1

−

β + 1

]. (3.8)

By Young’s inequality we have

α+1

β+1

−

α + 1

−

β + 1

≤ 0.

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According to Lemma 3.2 and (3.8), we have by using |λ| ≤

2kak

∞

0 ≤

α + 1

(u, φ) +

β + 1

(v, ψ) ≤ 0.

Hence

(u, φ) = Γ

(v, ψ) = 0.

This and Lemma 3.2 imply that φ ≡ tu and ψ ≡ t

v. Then by using the

normalization B(u, v) = 1 we conclude that u ≡ φ, v ≡ ψ.

Acknowledgements. The author gratefully acknowledges the ﬁnan-

cial support provided by Al-Imam Muhammed Ibn Saud Islamic University

during this research.

References

1. A. Anane: Etude des valeurs propres et de la r´esonance pour l’op´erateur

p-Laplacien, th`ese de Doctorat, U.L.B (1987-88).

2. P.A. Binding and Y.X. Huang: The principal eigencurve for p-Laplacian,

Diﬀ. int. Equations, 8, n.2 (1995), 405-415.

3. J. Chabrowski: On multiple solutions for nonhomogeneous system of elliptic

equations, Revista Matimatica de la Universidad Computense de Madrid

volume 9, numero 1, 1996.

4. J.I. Diaz and J.E. Saa: Existence et unicit´e de solutions positives pour cer-

taines ´equations elliptiques quasilin´eaires. C.R. Acad. Sci. Paris, serie I,

Math. 305,(1987), 521-524.

5. E. Dibenedetto: C

1+α

-local regularity of weak solutions of degenerate el-

liptic equations, Nonlinear Analysis T.M.A. 7, (1983), 827-859.

6. F. De Th

elin: Premi`ere valeur propre d’un syst`eme elliptique non lin´eaire,

C.R. Acad. Sci. Paris,t. 311, Serie I, (1990), 603-606.

7. F. De Th

elin and J. V

elin: Existence et non existence de solutions non

triviales pour des syst´emes elliptiques non-lin´eaires, C.R. Acad. Sci. Paris,

313, serie I, (1991), 589-592.

8. A. El Khalil, A. El Manouni and M. Ouanan: Simplicity and stability

of the ﬁrst eigenvalue of a nonlinear elliptic system, International Journal of

Mathematics and Mathematical Sciences, Vol. (2005)10, (2005), 1555-1563.

9. A. El Khalil and A. Touzani: On the ﬁrst eigencurve of the p-Laplacian,

Lecture Note in Pure and Applied Mathematics, Vol. 229 (2002), 195-205.

10. J. Fleckinger, R.F. Manasevich, N.M. Stavrakakis and F. De Th

elin:

Principal eigenvalue for some quasilinear elliptic equations on IR

, Ad-

vances in Diﬀrential Equations, Vol. 2, No.6 (1997), 981-1003.

11. P. Hess and T. Kato: On some liner and nonliner eigenvalue problems with

an indeﬁnite weight function, Comm.P.D.E. 5 (1980), 999-1030.

12. T. Kato: Superconvexity of the spectral radius and convexity of the spectral

bond and the type, Math.Z., 180 (1982), 265-273.

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13. P. Lindqvist: On the equation div(|∇u|

p−2

∇u)+λ|u|

p−2

u = 0, Proc.Amer.

Math. Soc., 109 (1990), 157-164.

14. R.G.D. Richardson: Theorems of oscillation for tow linear diﬀerential equa-

tions of the second order with two parameters, Tran. Amer. Math. SOC.,

13(1992), 22-34.

15. P.H. Rabinowitz: Minimax methods in critical point theory with applications

to diﬀerential equations, AMS Rigional Conference Series in Math. vol. 65

(1986).

16. A. Szulkin: Ljusternik-Schnirelmann theory on C

-manifolds, Ann. Inst.

Henri Poincar´e, Anal. Non. 5 (1988), 119-139.

17. J.L. Vazquez: A strong maximum principle for some quasilinear elliptic

equations, Appl. Math. Optim. 12 (1984), 191-202.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

Parabolic inclusions with nonlocal conditions

Abdelkader Boucherif

King Fahd University of Petroleum and Minerals

Department of Mathematics and Statistics

Box 5046 Dhahran, 31261, Saudi Arabia

E-mail: aboucher@kfupm.edu.sa

In this paper we consider the solvability of nonlinear parabolic diﬀerential equa-

tions with discontinuous nonlinearities, subjected to nonlocal conditions. We

are concerned with the existence of solutions. Our technique is based on the

Green’s function for linear parabolic partial diﬀerential equations, the maxi-

mum principle, ﬁxed point theorems for multivalued maps and the method of

lower and upper solutions.

Keywords: Parabolic problems; Integral representation of solutions; Maxi-

mum principles; Multivalued maps; Nonlocal conditions; Fixed point theorems;

Lower and upper solutions.

1. Introduction

Let Ω be an open bounded domain in R

, N ≥ 2, with a smooth boundary

∂Ω. Let T be a positive real number, D = Ω × (0, T ] and Γ = ∂Ω × [0, T ].

Given continuous functions φ, β

: Ω → R, i = 1, 2, ..., m and numbers

, i = 1, 2, ..., m in (0, T ) with 0 < t

< ... < t

< T, we are concerned

with the existence of solutions of the following parabolic problem with a

multivalued right-hand side and nonlocal conditions

u + Lu ∈ F (x, t, u) (x, t) ∈ D, (1)

u(x, t) = 0 (x, t) ∈ Γ, (2)

u(x, 0) +

i=1

(x)u(x, t

) = φ (x) x ∈ Ω, (3)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

where L is a strongly elliptic operator given by

Lu = −

i,j=1

(x, t)D

u + c(x, t)u.

Parabolic problems with discontinuous nonlinearities arise in the de-

scription of many phenomena in the applied sciences. We can mention,

for instance, chemical reactor theory,

porous medium combustion,

best response dynamics arising in game theory,

. Parabolic problems

with discontinuous nonlinearities have been also investigated in the pa-

pers,

. The importance of nonlocal conditions and their appli-

cations in diﬀerent ﬁeld has been discussed in,

. Several papers have

dealt with parabolic problems with continuous nonlinearities and nonlocal

conditions. See for instance,

In this paper we consider a nonlocal problem for a class of nonlinear

parabolic equations with a multivalued right hand side. We shall convert

problem (1), (2), (3), to an integral inclusion using the properties of the

Green’s function corresponding to the linear problem. We, then, provide

suﬃcient conditions on the data that will enable us to obtain at least one

solution. Our approach is based on ﬁxed point theorems for suitable multi-

valued operators and the method of lower and upper solutions.

The outline of the paper is as follows. In section 2 we introduce notations

and function spaces which will be used in the paper. In section 3, we shall

study the linear nonhomogeneous problem and the properties of the Green’s

function. In section 4 we recall the main properties of multivalued maps.

We state and prove our main results in section 5.

2. Preliminaries

In this section we introduce some notations and function spaces. Let Ω be

an open bounded domain in R

, N ≥ 2, with a smooth boundary ∂Ω. Let

T be a positive real number, D = Ω × (0, T ) and Γ = ∂Ω × [0, T ]. Then

Γ is smooth and any point on Γ satisﬁes the inside (and outside) strong

sphere property, see;

i.e. for any (x

, t

) ∈ Γ there is a closed ball B ⊂ Ω

(and a closed ball

B outside Ω) such that Γ ∩ (B × [0, T ]) = {(x

, t

)},

(and Γ ∩



B × [0, T ]



= {(x

, t

)}). For u : D → R we denote its partial

derivatives (when they exists) by D

u = ∂u/∂t, D

u = ∂u/∂x

, D

u =

∂

u/∂x

∂x

, i, j = 1, ..., N.

D) denotes the Banach space of continuous functions u : D → R,

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

endowed with the norm

|u|

= sup{|u(x, t)|; (x, t) ∈

D}.

We say that u ∈ C

2,1

(D) if u, D

u, D

u and D

u exist and are

continuous on D. In fact, we can write

2,1

(D) = {u ∈ C(D); u(., t) ∈ C

(Ω), t ∈ (0, T ) , u(x, .) ∈ C

(0, T ) , x ∈ Ω}.

u ∈ C(D) is called H¨older continuous of order α ∈ (0, 1] if

(u) = sup{

|u(x, t) − u(ξ, τ)|



kx − ξk

+ |t −τ|



α/2

; (x, t) , (ξ, τ) ∈ D} < +∞.

In this case we write u ∈ C

(D) and we deﬁne its norm by

|u|

= |u|

+ H

(u) .

If α = 1, u is called Lipschitz continuous and we write u ∈ Lip(D). Note

that the natural injection i : C

(D) → C(D) is continuous. We say that

that C

(D) is continuously embedded in C(D), and we write C

(D) ,→

C(D).

Also, u ∈ C

2+α,1+α

(D) if u(., t) ∈ C

2+α

(Ω) for all t ∈ (0, T ) and

u(x, .) ∈ C

1+α

(0, T ) for all x ∈ Ω. For u ∈ C

2+α,1+α

(D) we deﬁne its

norm by

|u|

2+α,1+α

= |u|

i=1

i,j=1

+ |D

We say that ∂Ω is in the class C

`+α

, ` ∈ N, α ∈ [0, 1) if in a neigh-

borhood of each point of ∂Ω there is a local reprentation of ∂Ω having the

form x

= ϑ

, . . . , x

i−1

, x

i+1

, . . . , x

) with ϑ

∈ C

`+α

Also, for 1 ≤ p < +∞, we say that u : D → R is in L

(D) if u is

measurable and

|u(x, t)|

dxdt < +∞, in which case we deﬁne its norm

|u|



|u(x, t)|

dxdt



1/p

3. Linear nonhomogeneous problem

In this section we consider the linear nonhomogeneous problem

u + Lu = f(x, t), (x, t) ∈ D, (4)

u(x, t) = 0, (x, t) ∈ Γ, (5)

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with the following nonlocal initial condition

u(x, 0) +

i=1

(x)u(x, t

) = φ (x) , x ∈ Ω. (6)

We shall assume throughout this paper that the function φ : Ω → R

is continuous and the functions a

, c : D → R are H¨older continuous,

= a

and moreover there exist positive numbers λ

, λ

such that

kξk

≤

i,j=1

(x, t)ξ

≤ λ

kξk

, ∀ξ ∈ R

and ∀(x, t) ∈ D.

The classical problem, i.e.when β

= 0 for all i = 1, 2, ..., m , is well known

and completely solved (see the books,

). Problem (4), (5), (6)

has been investigated by several authors (see for instance,

and the

references therein). The following version of the maximum principle can be

found in,

Lemma 3.1. Let u ∈ C

2,1

(D) ∩ C(

D). Assume that c(x, t) ≥ c

> 0 on D

and β

(x) ≤ 0 on Ω with −1 ≤

i=1

(x) ≤ 0 on Ω. If D

u + Lu ≥ 0 in

D, u(x, t) ≥ 0 on Γ, u(x, 0)+

i=1

(x)u(x, t

) ≥ 0 on Ω. Then u(x, t) ≥ 0

on D. Moreover, either u(x, t) = 0 ∀(x, t) ∈ D, or u(x, t) > 0 on D.

Proof. Suppose there exists (η, τ) ∈ D such that u (η, τ) < 0. It follows

from the continuity of u that u achieves a negative minimum at some point

, t

) ∈ D. By the strong maximum principle (see

) we have either

, t

) ∈ Γ or (x

, t

) = (x

, 0) with x

∈ Ω. From the above assumptions

we see that min

u (x, t) = u (x

, 0) < 0. Hence, if 1 +

i=1

) > 0,

0 ≤ u(x

, 0) +

i=1

)u(x

, t

) ≤ u(x

, 0)

1 +

i=1

)

< 0,

which is a contradiction.

In case 1 +

i=1

) = 0 we let u(x

, t

) = min{u (x

, t

) ; i =

1, 2, ..., m}. Then

u(x

, 0) −u(x

, t

) = u(x

, 0) + u(x

, t

)

i=1

)

≥ u(x

, 0) +

i=1

)u (x

, t

) ≥ 0.

This last inequality shows that u takes on a negative minimum at

, t

) ∈ D, which is not possible. This completes the proof of the Lemma.

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One of the important features of the linear problem is the integral

reprentation of solutions.

The homogeneous problem

u + Lu = 0, (x, t) ∈ D

u(x, t) = 0, (x, t) ∈ Γ

u(x, 0) +

i=1

(x)u(x, t

) = 0, x ∈ Ω

has a only the trivial solution, provided that −1 ≤

i=1

(x) ≤ 0.

There exists a unique function, G(x, t; y, s), called the Green’s function

corresponding to the linear homogeneous problem. This function satisﬁes

the following (,

2228

(i) D

G + LG = δ (t − s) δ (x − y), s < t, x, y ∈ Ω

(ii) G(x, t; y, s) = 0, s > t, x, y ∈ Ω

(iii) G(x, t; y, s) = G(x, t; y, s) = 0, s < t, x, y ∈ ∂Ω,

(iv) G(x, t; y, s) > 0 for (x, t) ∈ D

(v) G, D

G, D

G are continuous functions of (x, t), (y, s) ∈

D, t −s > 0,

(vi) |G(x, t; y, s)| ≤ C (t − s)

−N/2

exp

−a kx −yk

t − s

, for some posi-

tive constants C, a.

Lemma 3.2. Assume that the function f is H¨older continuous and

i=1

|β

(x)| < 1. Then, Problems (4), (5), (6) has a unique strong so-

lution, i.e. a solution u ∈ C

2,1

(D) ∩C(

D).

Proof. Consider the following representation (,

1628

u(x, t) =

Ω

G(x, t; y, 0) u (y, 0) dy +

Ω

G(x, t; y, s) f (y, s) dyds (7)

for (x, t) ∈ D, where u (y, 0) has to be determined. Using condition (6) we

see that u(x, 0) is a solution of the following Fredholm integral equation of

the second kind







u(x, 0) +

i=1

(x)

Ω

G(x, t

; y, 0) u (y, 0) dy =

−

i=1

(x)

Ω

G(x, t

; y, s) f (y, s) dyds + φ (x) .

(8)

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The condition

i=1

|β

(x)| < 1 implies that Eq. (8) with f = 0 and φ = 0

has only the trivial solution. Hence there exists a unique solution u(., 0) of

(8). Consequently (7) gives the unique solution of (4), (5), (6).

From the above discussion we see that for each v ∈ C (Ω) the problem (4),

(5) and

u(x, 0) = v(x) (9)

has a unique solution u ∈ C

2,1

(D) ∩ C(D), which we denote by u(x, t; v)

and has the following representation

u(x, t; v) =

Ω

G(x, t; y, s) f (y, s) dyds +

Ω

G(x, t; y, 0) v (y) dy

for (x, t) ∈ D.

It is clear from the above representation that the functions ϕ

and ψ deﬁned respectively by ϕ (x, t) =

Ω

G(x, t; y, s) dyds

and ψ (x, t) =

Ω

G(x, t; y, 0)dy are continuous on D. Let K

sup

(x,t)

Ω

G(x, t; y, s) dyds.

Following

we deﬁne an operator

Υ : C (Ω) → C (Ω) ,

Υv (x) = −

i=1

(x)u(x, t

; v) + φ (x) . (10)

Lemma 3.3. Assume max

x∈Ω

{

i=1

|β

(x)|

Ω

G(x, t

; y, 0)dy} < 1. Then

Υ is a Lipschitz operator.

Proof. Let v

, v

∈ C (Ω) and let u

(x, t; v

) and u

(x, t; v

) be the

corresponding solutions of (4), (5), (9) respectively.

(Υv

)(x, t) − (Υv

)(x, t) = −[

i=1

(x)u(x, t

; v

) −

i=1

(x)u(x, t

; v

)].

It follows from (7) that

(Υv

)(x, t) − (Υv

)(x, t) =

−

i=1

(x)

Ω

G(x, t

; y, 0) [v

(y) −v

(y)]dy.