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298 D. Guidetti

Let

v ∈ C

([0,δ]; N



((Ω)) ∩ B([0,δ]; N

2m+β

(Ω)),

with

v ∈ B([0,δ]; N

(Ω)) and v(0) = 0,

and let η ∈ R

Then, by Theorem 3.8 and (J3), we have, ∀τ ∈ [0, 1], if δ is suﬃciently small,

τ{[A(s + ., ., ∂

) − A(s

,.,∂

)]v + P (s + .)v}

B([0,δ];N

(Ω))

≤ ηv

B([0,δ];N

2m+β

(Ω))

+ C

v

B([0,δ];N

(Ω))

(3.38)

for some α ∈ [β, 2m + β). Analogously, employing (J4), we have

τ{[B

(s + ., ., ∂

) −B

,.,∂

)]v}

B([0,δ];N

2m+β−μ

(Ω))

≤ ηv

B([0,δ];N

2m+β

(Ω))

+ C

v

B([0,δ];N

(Ω))

(3.39)

Now we estimate

τ{[B

(s + ., ., ∂

) − B

,.,∂

)]v}

2m+β−μ

−θ

([0,δ];N

(Ω))

≤{[B

(s + ., ., ∂

) − B

,.,∂

)]v}

2m+β−μ

−θ

([0,δ];N

(Ω))

If t, t



∈ [0,δ],

|[B

(s + t, ., ∂

) − B

,.,∂

)]v(t, .)

− B

(s + t



,.,∂

) − B

,.,∂

)]v(t



,.)

θ,p,Ω

≤[B

(s + t, ., ∂

) − B

(s + t



,.,∂

)]v(t, .)

θ,p,Ω

+ [B

(s + t



,.,∂

) −B

,.,∂

)][v(t, .) − v(t



,.)]

θ,p,Ω

:= I

+ I

By Theorem 3.8, we have, if |r|≤μ

,using(J4),

[b

(s + t, .) − b

(s + t



,.)]∂

v(t, .))

θ,p,Ω

≤ Cb

(s + t, .) − b

(s + t



,.)

θ,∞,Ω

v(t, .))

+θ,p,Ω

≤ C|t −t



2m+β−μ

−θ

v

C([0,δ];N

+θ

(Ω))

[b

(s + t



,.) − b

,.)]∂

[v(t, .) − v(t



,.)]

θ,p,Ω

≤ C[b

(s + t



,.) − b

,.)

C(Ω)

v(t, .) − v(t



,.)

+θ,p,Ω

+ b

(s + t



,.) − b

,.)

θ,∞,Ω

v(t, .) − v(t



,.)

,p,Ω

]

≤ C

η|t − t



2m+β−μ

−θ

[v

2m+β−μ

−θ

([0,δ];N

+θ

(Ω))

+ v

2m+β−μ

−θ

([0,δ];N

(Ω))

On Linear Elliptic and Parabolic etc. 299

for ﬁxed α

∈ (0,μ

+ θ). So, applying Proposition 3.7, we obtain the following a

priori estimate, in case δ is suﬃciently small:

v

B([0,δ];N

2m+β

(Ω))

+ D

v

B([0,δ];N

(Ω))

+ δ

α−β

−1

v

B([0,δ];N

(Ω))

+ δ



−β

v

B([0,δ];N

2m+β



(Ω))

k=1

−μ

−θ

v

2m+β−μ

−θ

([0,δ];N

(Ω)(Ω))

≤ C



r

B([0,δ];N

(Ω))

k=1

r



B([0,δ];N

2m+β−μ

(Ω))

k=1

r



2m+β−μ

−θ

([0,δ];N

(Ω))

+ ηv

B([0,δ];N

2m+β

(Ω))

+ v

B([0,δ];N

2m+β



(Ω))

+ v

B([0,δ];N

(Ω))

+ η

k=1

v

2m+β−μ

−θ

([0,δ];N

+θ

(Ω))

k=1

v

2m+β−μ

−θ

([0,δ];N

(Ω)(Ω))



(3.40)

and the conclusion follows from the inequalities

v

2m+β−μ

−θ

([0,δ];N

+θ

(Ω))

≤ C(v

B([0,δ];N

2m+β

(Ω)

+ D

v

B([0,δ];N

(Ω)

(3.41)

with C independent of δ,choosingη and δ suitably small. 

References

[1] R. Adams, J.F. Fournier, Sobolev spaces, 2nd Edition, Academic Press, 2003.

[2] R. Denk, M. Hieber, J. Pr¨uss, R-Boundedness, Fourier Multipliers, and Problems of

elliptic and parabolic Type, Mem. Am. Math. Soc., 2003.

[3] R. Denk, M. Hieber, J. Pr¨uss, Optimal L

− L

Estimates for parabolic Boundary

Value Problems with inhomogeneous Data, Math. Z. 257 (2007), 193–224.

[4] G. Dore, A. Venni, On the Closedness of the Sum of two closed Operators,Math.Z.

196 (1987), 189–201.

[5] D. Guidetti,A Maximal Regularity Result with Applications to Parabolic Problems

with Nonhomogeneous Boundary Conditions, Rend. Sem. Mat. Univ. Padova 64

(1990), 1–37.

[6] D. Guidetti, On Elliptic Problems in Besov Spaces,Math.Nachr.152 (1991), 247–

275.

[7] D. Guidetti, On Interpolation with Boundary Conditions,Math.Z.207 (1991), 439–

460.

[8] J.L. Lions, J. Peetre, Sur une classe d’espaces d’interpolation, Publ. Math. I.H.E.S.,

1963.

[9] A. Lunardi, Analytic Semigroups and optimal Regularity in parabolic Problems,

Birkh¨auser, 1995.

300 D. Guidetti

[10] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and

Nonlinear Partial Diﬀerential Equations, De Gruyter Series in Nonlinear Analysis

and Applications, 1996.

[11] V.A. Solonnikov, On Boundary Value Problems for linear parabolic Systems of dif-

ferential Equations of general Form, Proc. Steklov Inst. Math. 83 (1965), 1–184.

[12] M. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space.

I: Principal properties, Journal Math. Mech. 13 (1964), 407–479.

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systems, Studia Math. 92 (1989), 387–401.

[14] H. Triebel, Theory of Function Spaces,Birkh¨auser, 1983.

Davide Guidetti

Dipartimento di matematica

Piazza di Porta S. Donato 5

I-40126 Bologna, Italy

e-mail: guidetti@dm.unibo.it

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 301–311

 2011 Springer Basel AG

Parabolic Equations in Anisotropic

Orlicz Spaces with General N -functions

Piotr Gwiazda and Agnieszka

Swierczewska-Gwiazda

Abstract. In the present paper we study the existence of weak solutions to an

abstract parabolic initial-boundary value problem. On the operator appearing

in the equation we assume the coercivity conditions given by an N -function

(i.e., convex function satisfying conditions speciﬁed in the paper). The main

novelty of the paper consists in the lack of any growth restrictions on the N -

function combined with an anisotropic character of the N -function, namely

we allow the dependence on all the directions of the gradient, not only on its

absolute value.

Mathematics Subject Classiﬁcation (2000). Primary 35K55; Secondary 35K20.

Keywords. Orlicz spaces, modular convergence, nonlinear parabolic equations.

1. Introduction

We concentrate on an abstract parabolic equation

=divA(t, x, ∇u)in(0,T) ×Ω, (1.1)

u(0,x)=u

in Ω, (1.2)

u(t, x)=0 on(0,T) × ∂Ω, (1.3)

where Ω ⊂ R

is an open, bounded set with a Lipschitz boundary ∂Ω, (0,T)is

the time interval with T<∞, u :(0,T) ×Ω → R and the operator A satisﬁes the

following conditions:

(A1) A is a Carath´eodory function (i.e., measurable w.r.t. t and x and continuous

w.r.t. the last variable).

P.G. is the coordinator and A.

S-G is the mentor in the International Ph.D. Projects Programme of

Foundation for Polish Science, operated within the Innovative Economy Operational Programme

2007–2013 (Ph.D. Programme: Mathematical Methods in Natural Sciences). The authors ac-

knowledge the Grant of Ministry of Science and Higher Education 2007–2010, Nr N201 033

32/2269.

302 P. Gwiazda and A.

Swierczewska-Gwiazda

(A2) There exists a function M : R

→ R

and a constant c>0 such that

A(t, x, ξ) · ξ ≥ c(M(ξ)+M

∗

(A(t, x, ξ)))

where M is an N-function, i.e., it is convex, has superlinear growth, M (ξ)=0

iﬀ ξ =0andM (ξ)=M (−ξ), M

∗

(η)=sup

ξ∈R

(η · ξ − M (ξ)).

(A3) For all ξ, η ∈ R

(A(t, x, ξ) − A(t, x, η)) · (ξ − η) ≥ 0.

We consider the problem of existence of weak solutions to the initial boundary

value problem (1.1)–(1.3). In the present paper we assume a full anisotropy of

the N-function, namely it depends on a vector-valued argument and we do not

put any assumptions on the growth of an N-function. Indeed we do not assume

that neither M nor M

∗

satisﬁes the so-called Δ

condition

. The following pair of

functions can serve as an example in d =2:

M(ξ)=M(ξ

,ξ

)=e

|ξ

−|ξ

|−1+(1+|ξ

|)ln(1+|ξ

|) −|ξ

∗

(η)=M

∗

(η

,η

)=(1+|η

|)ln(1+|η

|) −|η

|+ e

|ξ

−|η

|−1.

The result in the anisotropic case is not a straightforward extension of the isotropic

one. The new diﬃculty arising here concerns the density of compactly supported

smooth functions w.r.t. the modular topology of the gradients. The detailed ana-

lysis of this issue appears in Section 3.

We can cite some nontrivial examples of the operator A:

• A(t, x, ξ)=a(t, x)ξ exp(|ξ|

)where0<c

≤ a(t, x) ≤ c

< ∞,

• A(t, x, ξ)=a(t, x)ξ ln(1 + |ξ|)where0<c

≤ a(t, x) ≤ c

< ∞,

• A(t, x, ξ

,ξ

)=a

(t, x)ξ

exp(|ξ

)+a

(t, x)ξ

ln(1 + |ξ

|)where0<c

≤

(t, x) ≤ c

< ∞ for i =1, 2.

In [2] the operator A was assumed to be an elliptic second-order operator in diver-

gence form and monotone. The growth and coercivity conditions were more general

than the standard growth conditions in L

,namelytheN-function formulation

was stated

. Under the assumptions on the N-function M : ξ

<< M (|ξ|) (i.e., ξ

grows essentially less rapidly than M(|ξ|)) and M

∗

satisﬁes a Δ

-condition, the

existence results to (1.1)–(1.3) was established. The restrictions on the growth of

M were abandoned in [3], but still M had an isotropic character.

The review paper [7] summarizes the monotone-like mappings techniques in

Orlicz and Orlicz–Sobolev spaces

We say that an N -function M satisﬁes the Δ

-condition if for some constant C>0itholds

that M (2ξ) ≤ CM(ξ) for all ξ ∈ R

Assumption (A2) is in fact a generalization of the growth and coercivity conditions assumed in

[2] for the case of M dependent on a vector-valued argument.

is the Orlicz–Sobolev space of functions in L

with all distributional derivatives up

to order m in L

Parabolic Equations in Anisotropic Orlicz Spaces 303

Before stating the main result let us recall the standard notation. For brevity

we write Q := Ω ×(0,T). By the generalized Orlicz class L

(Q)wemeantheset

of all measurable functions ξ : Q → R

for which the modular

(ξ)=



M(ξ(t, x))dxdt

is ﬁnite. By L

(Q) we denote the generalized Orlicz space which is the set of all

measurable functions ξ : Q → R

for which ρ

(αξ) → 0asα → 0. This is a

Banach space with respect to the norm

ξ

=sup





η ·ξdxdt : η ∈ L

∗

(Q),



∗

(η)dxdt ≤ 1



By E

(Q) we denote the closure of all bounded functions in L

(Q). The space

∗

(Q) is the dual space of E

(Q). We will say that a sequence z

converges

modularly to z in L

(Q)ifthereexistsλ>0 such that



− z



→ 0.

We will use the notation z

−→ z for the modular convergence in L

(Q). Contrary

to [6] we consider the N-function M not dependent only on |ξ|, but on the whole

vector ξ. Below, we formulate the main result of the present paper. The space Z

appearinginthetheoremisdeﬁnedinSection3.

Theorem 1.1. Let M be an N-function and let A satisfy conditions (A1)–(A3).

Given u

∈ L

(Ω) there exists u ∈ Z

such that



(−uϕ

+ A(t, x, ∇u) ·∇ϕ) dxdt =



(x)ϕ(0,x)dx (1.4)

holds for all ϕ ∈D((−∞,T) × Ω).

2. Useful facts about Orlicz spaces

In this short section we collect some facts about N-functions and Orlicz spaces,

which are used in the proof of the main theorem. All the proofs to the collected

lemmas and propositions can be found, e.g., in [5].

Lemma 2.1. Let z

: Q → R

be a measurable sequence. Then z

−→ z in L

(Q)

modularly if and only if z

→ z inmeasureandthereexistsomeλ>0 such that

the sequence {M(λz

)} is uniformly integrable, i.e.,

lim

R→∞

sup

j∈N



{(t,x):|M(λz

)|≥R}

M(λz

)dxdt

=0.

Lemma 2.2. Let M be an N-function and for all j ∈ N let



M(z

) ≤ c.Then

the sequence {z

} is uniformly integrable.

304 P. Gwiazda and A.

Swierczewska-Gwiazda

Proposition 2.3. Let M be an N-function and M

∗

its complementary function.

Suppose that the sequences ψ

: Q → R

and φ

: Q → R

are uniformly bounded

in L

(Q) and L

∗

(Q) respectively. Moreover ψ

−→ ψ modularly in L

(Q) and

∗

−→ φ modularly in L

∗

(Q).Thenψ

· φ

→ ψ ·φ strongly in L

(Q).

Proposition 2.4. Let 

be a standard molliﬁer, i.e.,  ∈ C

∞

(R),has a compact

support and



(τ)dτ =1,(t)=(−t). We deﬁne 

(t)=j(jt). Moreover let

∗ denote a convolution in the variable t. Then for any function ψ : Q → R

such

that ψ ∈ L

(Q) it holds that

(

∗ ψ)(t, x) → ψ(t, x) in measure.

Proposition 2.5. Let 

be deﬁned as in Proposition 2.4. Given an N-function M

andafunctionψ : Q → R

such that ψ ∈L(Q) the sequence {M(

∗ ψ)} is

uniformly integrable.

3. Closures of compactly supported smooth functions

In the present section we concentrate on the issue of closures. We consider the

closure of C

∞

((−∞,T) ×Ω) in the three diﬀerent topologies:

1. strong (norm) topology of L

(Q) and we denote this space by X

,namely

={ϕ ∈ L

∞

(0,T; L

(Ω)), ∇ϕ ∈ L

(Q) |∃{ϕ

}

∞

j=1

⊂ C

∞

((−∞,T) × Ω) :

∗

ϕin L

∞

(0,T; L

(Ω)) and ∇ϕ

→∇ϕ strongly in L

(Q)},

2. modular topology of L

(Q),whichwedenotebyY

,namely

={ϕ ∈ L

∞

(0,T; L

(Ω)), ∇ϕ ∈ L

(Q) |∃{ϕ

}

∞

j=1

⊂ C

∞

((−∞,T) × Ω) :

∗

ϕin L

∞

(0,T; L

(Ω)) and ∇ϕ

−→ ∇ϕ modularly in L

(Q)},

3. weak-star topology of L

(Q), which we denote by Z

,namely

={ϕ ∈ L

∞

(0,T; L

(Ω)), ∇ϕ ∈ L

(Q) |∃{ϕ

}

∞

j=1

⊂ C

∞

((−∞,T) × Ω) :

∗

ϕin L

∞

(0,T; L

(Ω)) and ∇ϕ

∗

 ∇ϕ weakly star in L

(Q)}.

The closures in the strong and weak topology of the ∇u in L

(Q)areequal

if and only if M satisﬁes the Δ

-condition, cf. [4]. Hence as we do not require

-condition, this will not hold and we only concentrate on the relation between

modular and weak-star closures.

Lemma 3.1. Let M be an N-function and Y

be the function spaces deﬁned

in 2.and3. above. Then Y

= Z

Proof. Clearly Y

⊂ Z

(modular topology is stronger than weak-star). The

proof of an opposite inclusion we will split into two steps. In the ﬁrst step we will

assume that Ω is a star-shaped domain, whereas in the second step we extend the

idea for arbitrary Lipschitz domains.

Parabolic Equations in Anisotropic Orlicz Spaces 305

Step 1. Star-shaped domains. We are aiming to show that

⊂ Y

. (3.1)

For readability of the proof we introduce the notation

:= {u ∈ L

(0,T; W

1,1

(Ω)) ∩ L

∞

(0,T; L

(Ω)) |∇u ∈ L

(Q)}.

The starting point is extending the function u by zero outside of Ω to the whole R

to mollify it. To assume the extension is properly deﬁned we split (3.1) as follows:

⊂ V

(3.2)

and

⊂ Y

. (3.3)

Inclusion (3.2) is obvious and to prove (3.3) we deﬁne

(t, x):=u(t, λ(x − x

)+x

)

where x

is a vantage point of Ω. Let ε

dist (∂Ω,λΩ) where λΩ:={y =

λ · (x − x

)+x

| x ∈ Ω}. Deﬁne then

λ,ε

(t, x):=

∗ u

(t, x)

where 

is a standard molliﬁer, the convolution is done w.r.t. x and t and ε<ε

This approximation has the property that if u ∈ V

,thenu

λ,ε

∈ V

for

ε<ε

. First we pass to the limit with ε → 0 and hence u

λ,ε

ε→0

−−−→ u

(0,T; W

1,1

(Ω)) and ∇u

λ,ε

ε→0

−−−→∇u

modularly in L

(Q). Then we pass with

λ → 1 and obtain that u

λ→1

−−−→ u in L

(0,T; W

1,1

(Ω)) and ∇u

λ→1

−−−→∇u modu-

larly in L

(Q).

Step 2. Arbitrary Lipschitz domains. Since we consider the domain with Lipschitz

boundary, then there exists a countable family of star-shaped Lipschitz domains

{Ω

} such that (cf. [8])

Ω=

;

i∈J

We introduce the partition of unity θ

with 0 ≤ θ

≤ 1,θ

∈ C

∞

(Ω

), supp θ

i∈J

(x)=1forx ∈ Ω. The proof of (3.1) we split also into two parts. First

we show that

∩ L

∞

(Q)

= V

(3.4)

where the closure above is meant w.r.t. the modular topology. Indeed, deﬁne T

(u)

– the truncation of the function u,namely

(u)=

⎧

⎨

⎩

u if |u|≤n,

n if u>n,

−n if u<−n.

(3.5)

Note that if u ∈ V

then T

(u) ∈ V

. Moreover, as n →∞we observe the

convergence

(u) → u strongly in L

(0,T; W

1,1

(Ω)).

306 P. Gwiazda and A.

Swierczewska-Gwiazda

Additionally it holds that M(∇T

(u(t, x))) ≤ M (∇u(t, x)) a.e. in Q. Indeed, this

inequality can be easily concluded if we consider separately three subsets: the set

where T

(u)andu coincide and the remaining two sets, where T

(u)isequalton

or −n.SinceT

(u) ∈ L

(0,T; W

1,1

(Ω)), then ∇T

(u) is equal to zero a.e. in these

two sets. Consequently M(∇T

(u(t, x))) is uniformly integrable, which combined

with pointwise convergence provides

∇T

(u) →∇u modularly in L

(Q).

In the next step we will show that

∩ L

∞

(Q)) ⊂ Y

which together with (3.4) and (3.2) will prove (3.1). If u ∈ V

∩L

∞

(Q)then

u · θ

∈ L

(0,T; W

1,1

(Ω

)) ∩ L

∞

(0,T; L

(Ω

)) ∩ L

∞

((0,T) × Ω

))

and

∇u · θ

+ u ·∇θ

= ∇(u ·θ

) ∈ L

((0,T) × Ω

where Ω

= supp θ

. Now we follow the case of star-shaped domains to complete

the proof. 

Remark 3.2. Note that in the classical case M = M (|·|) (i.e., the modular does

not depend on the direction) the proof is simpler. The problem which arises here

is the lack of proper Poincar´e inequality, cf. [1], which in the isotropic case has the

form



M(|∇u(t, x)|)dxdt ≥ c



M(|u|)dxdt.

4. Existence result

In the current section we will prove Theorem 1.1. The ﬁnite-dimensional approxi-

mate problem is constructed by means of the Galerkin method. The basis consist-

ing of eigenvectors of the Laplace operator is chosen and by u

we mean the solu-

tion to the problem projected to n vectors of the chosen basis. Let Q

:= (0,s) ×Ω

with 0 <s<T. In the standard manner we conclude that, for 0 <s<T,



A(t, x, ∇u

) ·∇u

dxdt =

u

(0)

−

u

(s)

(4.1)

holds, the energy estimates are derived and convergence of appropriate sequences

is concluded, namely

∇u

∗

 ∇u weakly-star in L

(Q),

u weakly in L

(0,T; W

1,1

(Ω)),

A(·, ∇u

)

∗

χ weakly-star in L

∗

(Q),

∗

u in L

∞

(0,T; L

(Ω)),

∗

u

weakly-star in W

−1,∞

(0,T; L

(Ω).

(4.2)

Parabolic Equations in Anisotropic Orlicz Spaces 307

After passing to the limit we conclude the limit identity

−



uϕ

dxdt +



χ ·∇ϕdxdt =



(x)ϕ(0,x)dx (4.3)

for each compactly supported and smooth function ϕ. In the remaining part of

the proof we will concentrate on characterizing the limit χ.Weareaimingto

integrate by parts in (4.3) although the solution is not an admissible test function.

According to the results of Section 3 a function of the form

= 

∗ 

∗ u (4.4)

is already a proper test function with  ∈ C

∞

(R),having a compact support,

(τ)=(−τ),



(τ)dτ = 1 and deﬁning 

(t)=j(jt). Indeed, if we approx-

imate the function of the form (4.4) by a sequence of smooth functions, then in

the case χ ∈ E

∗

(Q) we pass to the limit with weak-star convergence. But if

χ/∈ E

∗

(Q), which indeed is the case when E

∗

(Q) = L

∗

(Q), we will pass to

the limit by means of modular convergence. This is the reason why in Section 3

we concentrated on showing that weak-star and modular limits coincide.

Then, we can observe that for 0 <s

<s<T it follows that



u

,ϕ

 dt =



u

, (

∗ 

∗ u) dt =



(

∗ u

), (

∗ u) dt





∗ u

dt =



∗ u(s)

−



∗ u(s

)

Next, we pass to the limit with j →∞and obtain for almost all s

,s, namely for

all Lebesgue points of the function u(t), that

lim

j→∞



u

 dt =

u(s)

−

u(s

)

. (4.5)

Let us pass now with j →∞in other terms. First we concentrate for 0 <

<T on the term



χ · (

∗ 

∗∇u)dxdt =



(

∗ χ) · (

∗∇u)dxdt.

Obviously, for any ψ ∈ L

(Q)itholdsthat(

∗ ψ) → ψ in measure. Hence



∗ χ → χ in measure

and



∗∇u →∇u in measure.

Since M and M

∗

are convex and nonnegative, then weak lower semicontinuity and

a priori estimates provide that the integrals are ﬁnite,



M(∇u)dxdt and



∗

(χ)dxdt