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

Periodic solutions of nonlinear parabolic equations with measure

data and polynomial growth in |∇u|

Abderrahmane El Hachimi

∗

and Abdelilah Lamrani Alaoui

†

UFR Math´ematiques Appliqu´ees et Industrielles, Facult´e des Sciences

B.P 20, El Jadida, Morocco

E-mails:

∗

elhachimi@ucd.ac.ma,

†

lamranii@gmail.com

In this paper we are interested in the problem:

(P)











∂u

∂t

− ∆

u = f (x, t, u, ∇u) + λµ in Q

u = 0 on Σ

u(0) = u(T ) in Ω

where f is a Carath´eodory function with polynomial growth in |∇u|, λ is a real

number, µ a bounded Radon measure and p ≥ 2.

Keywords: Periodic solution; Nonlinear parabolic equation.

1. Introduction

Periodic solutions of parabolic problems were studied by many authors and

in many context particulary when the elliptic operator is linear (see

10,12

and the references therein). In the case of measure datum the unique paper

we found is

where (P) with p = 2 was studied. In that work the authors

ﬁrstly give necessary conditions on the datum for existence, they show that

a size and regularity information is consequence of existence in some cases.

Secondly they give existence in the case of special measure, that is an L

function.

In the present work we generalize in diﬀerent directions the results in,

ﬁrstly the elliptic operator is no more linear it is the so called degenerate

p−Laplacian, that is

−∆

u = −div(|∇u|

p−2

∇u), p ≥ 2.

Secondly the existence in the natural growth case is given for suﬃciently

regular measures (in some capacity sense) but not necessarily in L

The notion of capacity considered here is the one adapted to the parabolic

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p−Laplacian operator due to Droniou et al.

Let us mention that the absence of information about the initial value of

the solution causes some technical diﬃculties.

Throughout the paper we adopt the following notations:

• Ω an open bounded subset of IR

, ∂Ω its boundary, T > 0, Q

Ω × (0, T ) and Σ

= ∂Ω × (0, T ),

• V

= L

(0, T ; W

1,p

(Ω)) and V

= L

(0, T ; W

−1,p

(Ω)) its dual where

p −1

• W =



u ∈ V

such that

∂u

∂t

∈ V

+ L

)



• W

1,1

) is the set of all measurable functions u such that



|u|

+ |Du|

+ |

∂u

∂t



dxdt < +∞

• C

(Ω) the space of bounded and continuous function on Ω,

• M

) the space of bounded Radon measures,

• M

) the subspace of M

) which don’t charge sets of parabolic

capacity zero (see section 3),

•  ., .  is the duality between V

and V

• < ., . > is the duality between W

−1,p

(Ω) and W

1,p

(Ω),

• |.| is the Lebesgue measure of a borelian set, the euclidian norm of a

vector or the absolute value of a real number,

• k.k

the norm of an element of a Banach space X. If X = L

with

p ∈ [1, +∞] it will be noted k.k

• C is a generic positive constant where the dependence on parameters

will be indicated only if needed.

The following hypothesis is common to all sections.

f is a Carath´eodory function:

measurable in (t, x)for all(r, ξ) ∈ IR

N+1

(1)

and continuous in (r, ξ) for almost every (t, x) ∈ Q

The paper is organized as follows. Section 2 is devoted to necessary

conditions of existence, regularity of the measure and nonexistence result

are established. Subsequently, we present existence result in the case of

natural growth and some regular measures datum.

The paper is organized as follows. Section 2 is devoted to necessary

conditions of existence, regularity of the measure and nonexistence result

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

are established. Subsequently, we present existence result in the case of

natural growth and some regular measures datum.

2. Necessary conditions for existence

In this section we are interested in the following problem:

for λ ∈ IR ﬁnd u such that:

u ∈ L

(0, T, W

1,p

(Ω)) ∩ C(]0, T [, L

(Ω))

f(x, t, u, ∇u) ∈ L

loc

∂u

∂t

− ∆

u ≥ f (x, t, u, ∇u) + λµ in D

) (2)

u(0) = u(T ) in M

(Ω),

where

µ is nonnegative bounded Radon measure. (3)

The periodicity condition u(0) = u(T ) in M

(Ω) is understood in the fol-

lowing sense

lim

t→0

Ω

(u(x, t) − u(x, T − t))ϕdx = 0

for all ϕ ∈ C

(Ω).

2.1. Nonexistence in supercritical case (q > 2(p − 1))

In this subsection we prove that solutions to (2) does not exists for all values

of λ if the growth of f in |∇u| is suﬃciently large.

We assume that there exists

]ε, σ[ open in ]0, T [, (4)

q > 2(p −1) (5)

and

a constant C

> 0 such thatf (x, t, s, ξ) ≥ C

|ξ|

for almost every (t, x) ∈ ]ε, σ[ ×Ω := Q

ε,σ

and for all (s, ξ) ∈ IR

N+1

(6)

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and ﬁnally

µ(]ε, σ[×Ω) > 0. (7)

The main result of this subsection is the following:

Theorem 2.1. Assume that (1), (3) and (4)-(7) hold, then there exists

ﬁnite λ

∗

such that (2) does not have any solution for λ > λ

∗

Remark 2.1. The condition (5) is reduced to q > 2 in the case p = 2

which is the superquadratic case treated in

and.

Now we prove Theorem 2.1.

Proof. We follow here the proofs of

and

with some changes due to the

nonlinear feature of the problem (2).

If u is a solution of (2), then by (6) we have

∂u

∂t

− ∆

u ≥ C

|∇u|

+ λµ in D

ε,σ

) (8)

let φ ∈ C

∞

(]0, T [×Ω) such that φ ≥ 0 and φ(ε) = φ(σ) = 0.

Multiply (2) by φ and integrate from ε to σ we obtain

ε,σ

φdµ ≤

ε,σ

|∇u|

p−1

|∇φ| − C

ε,σ

|∇u|

φ −

ε,σ

u∂

φ. (9)

This inequality can be extended to φ ∈ C

([0, T ]; L

∞

(Ω)) ∩

∞

(0, T ; W

1,∞

(Ω)), φ ≥ 0 and φ(ε) = φ(σ) = 0.

But φ

= −∆G(φ

) = −∆(Gφ)

where G is the Green Kernel on Ω.

Then from (9) and using twice Young inequality we obtain

ε,σ

φdµ ≤ C

ε,σ

|∇φ|

q−p+1

p−1

q−p+1

ε,σ

|∇(Gφ)

q−1

. (10)

For φ = (σ−t)

(t−ε)

Φ, where Φ is a solution of the eigenvalue problem

−∆φ = λ

φ in Ω, φ > 0 in Ω, φ = 0 on ∂Ω. (11)

Where λ

is the ﬁrst eigenvalue of −∆ in Ω.

We have

(Gφ)

Φ(t − ε)

q−1

(σ − t)

q−1

(σ + ε − 2t)

Using (5) and the fact that φ ∈ W

1,∞

(Ω) and

∈ L

(Ω) for all α < 1.

the two terms of the right hand side of (10) are ﬁnite.

Finally using (3) the existence of a ﬁnite λ

∗

such that: existence for (2)

implies λ ≤ λ

∗

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2.2. Existence implies regularity of the measure

We recall ﬁrst that a compact set K in Q

is of W

1,1

)-capacity zero

where r ≥ 1, if there exists a sequence of C

∞

) functions (φ

)

such

that :

0 ≤ φ

≤ 1 a.e in Q

, φ

≡ 1 on K, φ

→ 0 in W

1,1

) and a.e. in Q

(12)

The main result of this subsection concerns the problem (2) when:

There exists q > p − 1, (13)

and

there exists C

and C

such that f(x, t, s, ξ) ≥ C

|ξ|

−C

in Q

×IR

N+1

(14)

Theorem 2.2. Assume that (1), (3) hold, and that f satisﬁes (14)

and (13). If (2) has a solution then µ does not charge the sets of

1,1

)-capacity zero where r =

q − p + 1

Remark 2.2. The above statement means that:



K compact, W

1,1

)-capacity(K) = 0



⇒ µ(K) = 0.

This result extend to the p−Laplacian the one of

in the Laplacian case

(p = 2) where r was

q − 1

In the case q > 2(p − 1) both size and regularity conditions are necessary

for existence.

Proof. From (2) and (14) we have:

∂u

∂t

− ∆

u ≥ C

|∇u|

− C

+ λµ in D

). (15)

Let K a compact set of W

1,1

)-capacity zero and (φ

)

⊂ C

∞

) such

that (12) is satisﬁed.

Multiplying (15) by ψ

= φ

(where r is given in Theorem 2.2) we obtain

λµ(K)+ C

|∇u|

≤ −

∂ψ

∂t

|∇u|

p−1

|∇ψ

|+ C

(16)

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By Young inequality we have:

|∇u|

p−1

|∇ψ

| ≤ ε

|∇u|

+ C(ε)

|∇φ

. (17)

For ε suﬃciently small and with (17) into (16) we obtain

λµ(K) ≤ −r

u.φ

r−1

∂

+ C

|∇φ

+ C

. (18)

Passing to the limit using Vitali theorem and (12) we have µ(K) = 0

3. Existence in the case of natural growth and regular

measure data

In this section we focus our attention on the problem (Proof.) where f

takes the form

f(x, t, s, ξ) = −g(s)|ξ|

with

g a continuous fonction on IR such that g ∈ L

(IR). (19)

That is :

∂u

∂t

− ∆

u + g(u)|∇u|

= µ in Q

u = 0 on Σ

u(0) = u(T ) in Ω. (20)

Remark 3.1. Because p ≥ 2 we have p ≤ 2(p − 1) then no size condition

on the measure is needed in this section and the parameter λ is dropped.

The lack of initial data by the periodicity condition leads to some diﬃculties

which we overcome by considering the following problem.











u ≥ g

a.e. in Q

∂u

∂t

− ∆

u + g(u)|∇u|

= µ in Q

u = 0 on Σ

u(0) = u(T ) in Ω

(21)

where µ is a regular measure in the sense µ ∈ L(Q

) = V

+ L

) +

∂

∩ L

∞

)) that is

µ = g

+ g

+ ∂

where (g

, g

) ∈ V

× L

) × (V

∩ L

∞

)).

Before giving the meaning of solutions for the problem (21) let us review

some well known results we use here and explain the reason for that choice

of measures.

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3.1. Some known results

We begin with a decomposition for measures which do not charge sets of

parabolic capacity zero.

First, recall that, if U ⊂ Q

is an open set, parabolic capacity of U is

deﬁned as follows:

Cap

(U) = inf {kuk

: u ∈ W, u ≥ χ

a.e in Q

}

Then for any borelian set B ⊂ Q

Cap

(B) = inf {Cap

(U), U open subset of Q

, B ⊂ U}.

For more properties and details on the parabolic capacity see.

Proposition 3.1. (theorem 2.28

) If µ ∈ M

), the subspace of mea-

sures in M

) which don’t charge sets of parabolic capacity zero.

Then there exists (g

, g

) ∈ V

× L

) × V

such that :

φdµ =

< g

, φ > dt+

φdxdt−

< (φ)

, g

> dt ∀φ ∈ D(Q

We now give a trace and integration by part results which make it possible

to deﬁne and treat periodicity in (21).

Proposition 3.2. (theorem 1.1

and Lemma 2

) We have:

W ,→ C([0, T ]; L

(Ω)),

and for v ∈ W, φ Lipschitz bounded real function from IR to IR, with φ(0) =

0 we have for all t ∈ [0, T ]

∂v(σ)

∂t

, φ(v(σ)) > dσ =

Ω

v(x,t)

φ(σ)dσ −

Ω

v(x,0)

φ(σ)dσ

Moreover if v(0) = v(T ) in L

(Ω) we have 

∂

∂vt

, φ(v) = 0.

Now we are ready to give the main result of this section.

3.2. Deﬁnition and main result

and

) Let µ be a measure in L(Q

) with a decomposition (g

, g

a measurable function u ≥ g

is said to be renormalized solution of (21) if

(u) ∈ V

, (22)

(u − g

) ∈ V

, (23)

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for every k > 0, and

lim

n→+∞

{n≤u−g

≤n+1}

|∇u|

dxdt = 0, (24)

and, for every S ∈ W

2,∞

(IR) such that S

has compact support in

, S(0) = 0,

∂

S(u − g

) −div(S

(u −g

)|∇u|

p−2

∇u)

(u −g

)|∇u|

p−2

∇u.∇(u −g

)

+g(u)|∇u|

(u −g

)

= S

(u −g

+ G

(u −g

)∇(u − g

) − div(G

(u − g

)) (25)

in the sense of distributions, with g

= div(G

S(u − g

)(0) = S(u − g

)(T ). (26)

We state our main result

Theorem 3.1. Let µ a measure in L(Q

) and assume that (19) holds,

then (21) has a solution in the sense of deﬁnition above.

The proof of this result is divided into three classical steps:

- Regularization.

- Existence of approximate solutions.

- A priori estimates and passing to the limit.

3.3. Regularization and existence of approximate solutions

Let (g

, g

) be a decomposition of µ, we regularize (21) by the sequence

of problems

∂u

∂t

− ∆

+ g(u

)|∇u

− nT

− g

)

−

= µ

in Q

= 0 on Σ

(0) = u

(T ) in Ω

(27)

where,

= g

+ g

+ (g

)

, with g

∈ C

∞

), i = 1, 2, 3 and kµ

)

≤ C,

(28)

and

→ g

in V

, (29)

→ g

in L

), (30)