see for the properties of such spaces, and for applications of variable exponent spaces on partial differential equations

Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 113, pp. 1–19.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF WEAK SOLUTIONS FOR QUASILINEAR PARABOLIC SYSTEMS IN DIVERGENCE FORM WITH

VARIABLE GROWTH

MIAOMIAO YANG, YONGQIANG FU Communicated by Vicentiu D. Radulescu

Abstract. In this article we study the existence of weak solutions for quasi- linear parabolic system in divergence form with variable growth. By means of Young measures, Galerkin’s approximation method and the theory of variable exponents spaces, we obtain the existence of weak solutions.

1. Introduction and statement of main result

The spacesL^p(x)(Ω) andW^m,p(x)(Ω) were first discussed by Kov´aˇcik and R´akosn´ık in [24]. Lately, a lot of attention has been paid to the study of various mathemat- ical problems with variable exponent growth conditions; see [9, 11, 12, 14, 16] for the properties of such spaces, and [6, 15, 20] for applications of variable exponent spaces on partial differential equations. The theory of variable exponent spaces has been driven by various problems in nonlinear elastic mechanics, imaging pro- cessing, electrorheological fluids and other physics phenomena; see for example [1, 2, 3, 7, 27, 39].

Whenp(x) is a constant function, Norbert Hungerb¨uhler studied the following problem in [22]:

−divσ x, u(x), Du(x)

=f, x∈Ω

u(x) = 0, x∈∂Ω (1.1)

The classical monotone operator methods developed by [5, 25, 29, 36] cannot be applied here. Norbert Hungerb¨uhler obtain the existence of weak solutions for (1.1) by Young measures which were proposed by Young in [38]. Many applications and developments of Young measures to the calculus of variations, optimal con- trol theory and nonlinear partial differential equations are presented by MacShane, Gamkrelidze and Tarter in [21, 26, 35, 37]. Inspired by the works mentioned above, results from [22] were extended in [18] to the case that σsatisfies variable growth conditions, by Young measures generated by sequences in variable exponent spaces;

see [18, 19] for the basic theorems and properties.

2010Mathematics Subject Classification. 35D30, 35K59, 46E35.

Key words and phrases. Variable exponent; Young measures; weak solutions;

quasilinear parabolic system; Galerkin’s approximation.

c

2018 Texas State University.

Submitted November 16, 2017. Published May 10, 2018.

1

In this article, we consider the initial and boundary value problem for the quasi- linear parabolic system:

∂u

∂t −divσ x, t, Du(x, t)

=−divf, (x, t)∈Q u(x, t) = 0, (x, t)∈∂Ω×(0, T)

u(x,0) =u0(x), x∈Ω,

(1.2)

where Ω ⊂R^N (N ≥ 2) is a bounded open domain, u: Ω×(0, T) → R^m, 0 <

T <∞, Q= Ω×(0, T),p(x) is Lipschitz continuous and 1< p₋:= inf_x∈Ω¯p(x)≤ p(x)≤p+:= sup_x∈Ω¯p(x)<∞,f ∈L^p0(x)(Q;M^m×N),u0∈L²(Ω;R^m), σsatisfies the conditions (H1)–(H3) below. Inspired by [8], we consider thatp(x) only depends xin this problem. The related definition and properties will be given in section 2.

In our paper, we denote byM^m×n the real vector space ofm×nmatrices equipped with the inner product

M◦N := X

1≤i≤m,1≤j≤n

M_ijN_ij.

Now we give conditions required forσin (1.2).

(H1) (Continuity)σ: Ω×(0, T)×M^m×N →M^m×N is a Carath´eodory function, i.e. (x, t)7→σ(x, t, ξ) is measurable for everyξ∈M^m×N andξ7→σ(x, t, ξ) is continuous for almost every (x, t)∈Q.

(H2) (Growth and coercivity) There exist c₁ ≥0, c₂ >0,0< a ∈L^p0(x)(Q), b∈ L¹(Q), such that

|σ(x, t, ξ)| ≤a(x, t) +c1|ξ|^p(x)−1, σ(x, t, ξ)◦ξ≥ −b(x, t) +c2|ξ|^p(x). (H3) (Monotonicity)σsatisfies one of the following conditions:

(i) For all (x, t)∈Q, ξ7→σ(x, t, ξ) is aC¹-function and is monotone, i.e.

for all (x, t)∈Qandξ, η∈M^m×N, we have σ(x, t, ξ)−σ(x, t, η)

◦(ξ−η)≥0.

(ii) There exists a function W : Ω×(0, T)×M^m×N → R such that σ(x, t, ξ) = DξW(x, t, ξ), and ξ → W(x, t, ξ) is convex and C¹ for all (x, t)∈Q.

(iii)σis strictly monotone, i.e. σis monotone and σ(x, t, ξ)−σ(x, t, η)

◦ (ξ−η) = 0 impliesξ=η.

(iv) Z

Q

Z

M^m×n

σ(x, t, λ)−σ(x, t, λ)

◦(λ−λ) dν_(x,t)(λ) dxdt >0

where λ=hν(x,t), Ii, ν ={ν(x,t)}_(x,t)∈Q is any family of Young measures generated by a bounded sequence inL^p(x)(Q) and not a Dirac measure for a.e. (x, t)∈Q.

Our main result is as follows:

Theorem 1.1. Ifσ satisfies conditions(H1)–(H3), then problem (1.2)has a weak solution for everyf ∈L^p(x)(Q;M^m×N)and every u₀∈L²(Ω;R^m).

Condition (H2) states the variable growth and coercivity condition. (H3)(iv) is weaker than typical strictly monotone condition, even than the p-quasimonotone condition introduced by Norbert Hungerb¨uhler in [22] whenp(x) is a constant.

This article is organized as the following: In Section 2, several important prop- erties on variable exponent spaces and the theory of Young measures will be pre- sented. In Section 3, we will give the Galerkin approximation and necessary priori estimates. In section 4, the existence of weak solutions for problem (1.2) will be proved; the conclusions will be given in section 5.

2. Preliminaries

In this section, we recall some facts on variable exponent spaces L^p(x)(Ω) and W^k,p(x)(Ω).

LetP(Ω) be the set of all Lebesgue measurable functionsp: Ω→[1,+∞), where Ω⊂Rⁿ(n≥2) is a nonempty open subset. Denote

ρ_p(x)(u) = Z

Ω

|u(x)|^p(x)dx, (2.1) kuk_p(x)= inf{t >0 :ρ_p(x)(u

t)≤1}. (2.2)

The variable exponent Lebesgue spaceL^p(x)(Ω) is the class of all functions usuch thatρ_p(x)(t0u)<∞for somet0>0. L^p(x)(Ω) is a Banach space endowed with the norm (2.2). (2.1) is called the modular ofuin L^p(x)(Ω).

For a givenp(x)∈P(Ω), we define the conjugate functionp⁰(x) as p⁰(x) =

(∞, ifx∈Ω1={x∈Ω :p(x) = 1};

p(x)

p(x)−1, for otherx∈Ω.

Lemma 2.1 ([9]). Let p∈P(Ω), then Z

Ω

|u(x)·v(x)|dx≤2kuk_p(x)kvk_p⁰_(x) for everyu∈L^p(x)(Ω) and every v∈L^p0(x)(Ω).

In the rest of this section, for everyp∈P(Ω), we assume that 1≤p₋ ≤p(x)≤ p₊<∞.

Lemma 2.2 ([16]). For everyu∈L^p(x)(Ω), we have:

(1) If kukp(x)≥1, thenkuk^p_p(x)⁻ ≤ρp(x)(u)≤ kuk^p_p(x)⁺ . (2) If kuk_p(x)<1, thenkuk^p_p(x)⁺ ≤ρ_p(x)(u)≤ kuk^p_p(x)⁻ .

Lemma 2.3([16]). Ifp₋>1,L^p(x)(Ω)is reflexive, and the dual space ofL^p(x)(Ω) isL^p0(x)(Ω).

Lemma 2.4 ([24]). Let |Ω| < ∞, where |Ω| denotes the Lebesgue measure of Ω, p1(x), p2(x) ∈ P(Ω), then a necessary and sufficient condition for L^p2(x)(Ω) ⊂ L^p1(x)(Ω) is that p1(x) ≤ p2(x) for almost every x ∈ Ω, and in this case the embedding is continuous.

We assume that Ω ⊂ R^d is a bounded domain, (0, T) ⊂ R, T is a fixed real number,Q= Ω×(0, T),pis a Lipschitz function. We define

X(Q) :=n

u∈L²(Q)^d:∇u∈L^p(·,·)(Q)^d×d, u(τ,·)∈Vτ(Ω) a.e. τ ∈(0, T)o

The norm onX(Q) is given by

kuk_X(Q):=kuk_L2(Q)^d+k∇uk_Lp(·,·)(Q)^d×d

where for allτ∈(0, T), we have V_τ(Ω) :=

u∈L²(Ω)^d∩W₀^1,1(Ω)^d:∇u∈L^p(τ,·)(Ω)^d×d The norm onVτ(Ω) is defined by

kuk_Vτ(Ω):=kuk_L2(Ω)^d+k∇uk_Lp(τ,·)(Ω)^d×d

Lemma 2.5 ([10]). The spaceX(Q)is a Banach space under the normk · kX(Q); andC₀^∞(Q)is density in X(Q).

Lemma 2.6 ([10]). The spaceX(Q) is reflexible.

Lemma 2.7 ([10]). The dual spaceX⁰(Q)is isomorphic to the subspace ofD⁰(Q) consisting of distributions T of the form

kTk:= infn

kgk_L2(Q)^d+kGk_Lp(·,·)(Q)^d×d:T =g−divGo , whereg∈L²(Q)^d.

Lemma 2.8 ([4]). Let Ω⊂Rⁿ be Lebesgue measurable (not necessarily bounded) and zj : Ω → R^m, j = 1,2, . . ., be a sequence of Lebesgue measurable functions.

Then there exists a subsequence zk and a family {νx}_x∈Ω of nonnegative Radon measures onRⁿ, such that

(i) kνxk:=R

dνx≤1 for almost everyx∈Ω.

(ii) ϕ(zk)*^∗ϕ¯ weakly* in L^∞(Ω) for any ϕ∈C0(R^m), where ϕ(x) =¯ hνx, ϕi andC0(R^m) ={ϕ∈C(R^m) : lim_|z|→∞|ϕ(z)|= 0}.

(iii) If for anyR >0

L→∞lim sup

k∈N

meas{x∈Ω∩B(0, R) :|zk(x)| ≥L}= 0,

thenkνxk= 1for almost everyx∈Ω, and for any measurableA⊂Ωthere holds ϕ(zk)*ϕ¯=hνx, ϕi weakly in L¹(A)for continuous ϕ provided the sequence ϕ(zk)is weakly precompact inL¹(A).

Lemma 2.9 ([4]). If meas Ω < ∞ and νx is a Young measure generated by the sequence {uj} , then u_j converges by measures tou if and only if for a.e. x∈Ω we have ν_x=δ_u(x).

Lemma 2.10 ([5]). Let {f_j} be a uniformly boundedness inL¹(Ω), sup

j

kfjkL¹(Ω)=C <∞.

There exists a subsequence, not relabled, a nonincreasing sequence of measurable setsΩ_n,Ω_n+1⊂Ω_n, andf ∈L¹(Ω) such that

fj * f in L¹(Ω\Ωn) for alln.

Lemma 2.11 ([5]). If {zj} is a sequence of measurable functions with associated Young measureν ={νx}_x∈Ω,

lim inf

j→∞

Z

E

ψ x, zj(x) dx≥

Z

E

Z

R^m

ψ(x, λ) dνx(λ) dx,

for every nonnegative, Carath´eodory functionψand every measurable subsetE⊂Ω.

The above theorem is obtained in [31] by proving a complicated lemma. We will give a much easier proof by using the contradiction method.

Proof. Assume that

lim inf

j→∞

Z

E

ψ x, zj(x)

dx <∞.

Thenψ(x, zj(x)) is a bounded sequence inL¹(E). Let ψ(x) =¯

Z

R^m

ψ(x, λ) dνx(λ).

By Lemmas 2.8 and 2.10, there existsE_n⊂Ω,E_n+1⊂E_n, measE_n→0 asn→ ∞, such that

Z

E\En

ψ

x, zj(x) dx→

Z

E\En

ψ¯dx (2.3)

asj → ∞for alln. On the other hand, it is apparent that Z

E\E_n

ψ¯dx→ Z

E

ψ¯dx

as n → ∞. Now we show the proof of our conclusion by contradiction. Assume that

lim inf

j→∞

Z

E

ψ x, zj(x) dx <

Z

E

Z

R^m

ψ(x, λ) dνx(λ) dx.

Let

a:=

Z

E

Z

R^m

ψ(x, λ) dν_x(λ) dx−lim inf

j→∞

Z

E

ψ x, z_j(x)

dx >0.

Since lim_n→∞R

E\En

ψ¯dx=R

Eψ¯dx, fora >0, there existn₀which is large enough, such that

Z

E

ψ¯dx− Z

E\En0

ψ¯dx < a= Z

E

Z

R^m

ψ(x, λ) dνx(λ) dx−lim inf

j→∞

Z

E

ψ x, zj(x) dx.

Therefore

Z

E\En0

ψ¯dx >lim inf

j→∞

Z

E

ψ x, zj(x) dx.

Combining this with (2.3) leads to a contradiction.

Lemma 2.12 ([18]). If {uj} is bounded in L^p(x)(Ω,R^m), then{uj} can generate Young measureνx satisfied thatkνxk= 1and there is a subsequence of{uj}weakly convergent toR

R^mλdνx(λ)inL¹(Ω,R^m).

3. Galerkin approximation and a priori estimates Let

X:=n

u∈L²(Q;R^m) :Du∈L^p(x)(Q;M^m×N), u(·, τ)∈V_τ(Ω) a.e. τ∈(0, T)o , where forτ∈(0, T),

Vτ(Ω) :=n

u∈L²(Ω;R^m)∩W₀^1,1(Ω;R^m) :Du(·, τ)∈L^p(x)(Ω;M^m×N)o . The norm onX is defined by

kukX:=kukL²(Q;R^m)+kDuk_Lp(x)(Q;M^m×N).

According to Lemmas 2.5–2.7, it is easy to show that X is a Banach space and C₀^∞(Q;R^m) is dense inX. X⁰ denotes the dual space ofX. For allg∈X⁰,u∈X, there existsg0∈L²(Q;R^m), g1∈L^p0(x)(Q;M^m×N), such that

hg, ui= Z

Q

g0·udxdt+ Z

Q

g1◦Dudxdt.

Based on the above notes, we will show the definition of weak solutions for (1.2).

Definition 3.1. A functionu∈L^∞(0, T;L²(Ω))T

Xis called as the weak solution of problem (1.2), if

− Z

Q

u∂ϕ

∂t dxdt+

Z

Ω

u(x, t)ϕ(x, t) dx

T 0+

Z

Q

σ(x, t, Du)◦Dϕdxdt= Z

Q

f◦Dϕdxdt holds for allϕ∈C¹(0, T;C₀^∞(Ω)).

We choose anL²(Ω;R^m)-orthonormal base{ωj}^∞_j=1, such that {ωj}^∞_j=1⊂C₀^∞(Ω;R^m), C₀^∞(Ω;R^m)⊂ ∪^∞_n=1V_n^C

1( ¯Ω;R^m)

. HereVn= span{ω1, ω2, . . . , ωn}.

Sincef ∈L^p0(x)(Q;M^m×N) andC₀^∞(Q;M^m×N) is identity inL^p0(x)(Q;M^m×N), there exists a sequence{f_n} ⊂C₀^∞(Q;M^m×N) such that

fn→f in L^p(x)(Q;M^m×N)

as n → ∞. For every u0 ∈ L²(Ω;R^m), there is a sequence {ψn}^∞_n=1, such that ψn∈ ∪^∞_n=1Vn and

ψn →u0 inL²(Ω;R^m) asn→ ∞.

Definition 3.2. u_n ∈ C¹(0, T;V_n) is called by the Galerkin solution of problem (1.2), if

− Z

Q_τ

∂u_n

∂t φdxdt+ Z

Q_τ

σ(x, t, Du_n)◦Dφdxdt= Z

Q_τ

f_n◦Dφdxdt holds for allτ∈(0, T] andφ∈C¹(0, T;Vk)(k≤n), whereQτ = Ω×(0, τ).

Now we construct the Galerkin solution of problem (1.2). DefineP_n(t, η) : [0, T]×

Rⁿ→Rⁿ

(P_n(t, η))_i= Z

Ω

σ(x, t,

n

X

j=1

η_jDω_j)◦Dω_idx

where η= (η₁,· · ·, η_n). Sinceσis a Carath´eodory function,P_n(t, η) is continuous int, η.

Consider the ordinary differential equation η⁰(t) +Pn t, η(t)

=Fn

η(0) =Un(0) (3.1)

where

(Fn)i= Z

Ω

fn◦Dωidx, (Un(0))i= Z

Ω

ψn(x)ωidx.

From (3.1) we haveη⁰η+Pn(t, η)η=Fnη. Furthermore, Pn(t, η)η=

Z

Q

σ(x, t,

n

X

j=1

ηjDωj)◦

n

X

i=1

ηiDωidxdt

≥ − Z

Q

b(x, t) dxdt+c₂ Z

Q

n

X

i=1

η_iDω_i

p(x)

dxdt≥C.

It is apparent that

η⁰η+C≤F_nη≤ 1

2|F_n|²+1 2|η(t)|². Consequently,

1 2

∂|η(t)|²

∂t ≤ 1

2|Fn|²+1

2|η(t)|²+C After integrating the both sides of this inequality , we obtain

|η(t)|²≤Cn+ Z T

0

|η(s)|²ds Then by Gronwall’s inequality,|η(t)| ≤Cn(T). Let

Mn= max

(t,η)∈[0,T]×B(η(0),2Cn(T))|Fn−Pn(t, η)|, Tn = min

T,2Cn(T) Mn

whereB

η(0),2Cn(T)

is a ball of radius 2Cn(T) with the center at the pointη(0) inRⁿ.

By Peano’s theorem, (3.1) has aC¹ solution on [0, T_n]. Let t₁ =T_n andη(t₁) be a initial value, then we can repeat the above process and get aC¹ solution on [t₁, t₂], where t₂ =t₁+T_n. Thus there is a interval [t_i−1, t_i−2]⊂[0, T], such that (3.1) admits a solution on [t_i−1, t_i−2], wheret_i=t_i−1+T_n, i= 1,2, . . . , l−1, t_l=T. Moreover we can get a solutionηn(t)∈C¹([0, T]).

From the definition ofPn, it is easy to know thatun(x, t) =Pn

j=1 ηn(t)

jωj(x) is the Galerkin solution of (1.2).

Now we study the boundedness and convergence of some function sequences.

Lemma 3.3. The sequence {un} is bounded inX, and {σ(x, t, Dun)} is bounded inL^p0(x)(Q;M^m×N).

Proof. Letφ=un. By Definition 3.2, for everyτ ∈[0, T], one has Z

Q_τ

∂u_n

∂t u_ndxdt+ Z

Q_τ

σ(x, t, Du_n)◦Du_ndxdt= Z

Q_τ

f_n◦Du_ndxdt, which is denoted asI+II=III. By integration and (H2),

I= 1

2kun(·, τ)k²_L2(Ω)−1

2kun(·,0)k²_L2(Ω)

and

II ≥ − Z

Q_τ

b(x, t) dxdt+c₂ Z

Q_τ

|Du_n|^p(x)dxdt Sincef_n ∈L^p0(x)(Q;M^m×N), we have

III ≤Ckfnk_Lp(x)(Qτ;M^m×N)kDunk_Lp(x)(Qτ;M^m×N)

We know thatun(x,0) =ψn(x)→u0 inL²(Ω) . As a result, Z

Ω

u²_n(x,0) dx= Z

Ω

|ψn(x)|²dx≤C foralln.

Consequently, 1

2kun(·, τ)k²_L2(Ω)+c2

Z

Qτ

|Dun|^p(x)dxdt

≤1

2kun(·,0)k²_L2(Ω)+kbk_L1(Q_τ)

+Ckfnk_Lp0(x)(Qτ;M^m×N)kDunk_Lp(x)(Q_τ;M^m×N)

≤C+CkDunk_Lp(x)(Q_τ;M^m×N)

(3.2)

By Lemma 2.2, it follows that kDu_nk_Lp(x)(Qτ)≤maxnZ

Qτ

|Du_n|^p(x)dxdt1/p−

,Z

Qτ

|Du_n|^p(x)dxdt1/p+o .

IfkDunk_Lp(x)(Qτ;M^m×N)is unbounded, thenR

Q_τ|Dun|^p(x)dxdtis unbounded. This contradict (3.2). Thus

kDu_nk_Lp(x)(Q;M^m×N)≤C.

Moreover

kun(·, τ)k²_L2(Ω)≤C (3.3) Then we can get the conclusion that{u_n}is bounded inX. By Lemma 2.6, there is a subsequence of{u_n} (also denoted by{u_n}) satisfying u_n* uinX, asn→ ∞.

Owing to (H2), we obtain Z

Q

|σ(x, t, Dun)|^p0(x)dxdt

≤CZ

Q

|a(x, t)|^p0(x)dxdt+c1

Z

Q

|Dun|^p(x)dxdt .

Sincea∈L^p0(x)(Q) andkDunk_Lp(x)(Q;M^m×N)≤C, it follows that Z

Q

|σ(x, t, Dun)|^p0(x)dxdt≤C.

From Lemma 2.2 we have

kσ(x, t, Dun)k_Lp0(x)(Q;M^m×N)≤C. (3.4) Then σ(x, t, Du_n) * χ in L^p0(x)(Q;M^m×N) as n → ∞ (we can choose a proper

subsequence if necessary).

Lemma 3.4. For function sequences {u_n} constructed above, we have un(·, T)* u(·, T) inL²(Ω),

u(·,0) =u₀.

Proof. Thanks to (3.3), the sequence {un} is bounded in L^∞ 0, T;L²(Ω) . Thus there exists a subsequence (also denoted by{un}) such that

un(·, T)* z in L²(Ω)

as n→ ∞. We will prove thatz=u(·, T), andu(·,0) =u₀. We denote u(·, T) as u(T), and denoteu(·,0) asu(0).

For everyψ∈C^∞([0, T]),v∈Vk,k≤n, we have Z T

0

Z

Ω

∂tunvψdxdt+ Z T

0

Z

Ω

σ(x, t, Dun)◦Dvψdxdt= Z T

0

Z

Ω

fn◦Dvψdxdt.

After integrating, one gets Z

Ω

un(T)ψ(T)vdx− Z

Ω

un(0)ψ(0)vdx

=− Z T

0

Z

Ω

σ(x, t, Dun)◦Dvψdxdt+ Z T

0

Z

Ω

fn◦Dvψdxdt +

Z T

0

Z

Ω

unvψ⁰dxdt.

Ifn→ ∞, then Z

Ω

zψ(T)vdx− Z

Ω

u₀ψ(0)vdx

= Z T

0

Z

Ω

f ◦Dvψdxdt− Z T

0

Z

Ω

χ◦Dψvdxdt+ Z T

0

Z

Ω

ψ⁰vudxdt.

(3.5)

Letψ(0) =ψ(T) = 0. Then Z T

0

Z

Ω

f◦Dvψdxdt− Z T

0

Z

Ω

χ◦Dψvdx=− Z T

0

Z

Ω

ψ⁰vudx= Z T

0

Z

Ω

ψvu⁰dx.

Thus by (3.5), we can obtain Z

Ω

zψ(T)vdx− Z

Ω

u₀ψ(0)vdx= Z T

0

Z

Ω

ψvu⁰dx+ Z T

0

Z

Ω

ψ⁰vudx

= Z

Ω

uψvdx

T 0

= Z

Ω

u(T)ψ(T)vdx− Z

Ω

u(0)ψ(0)vdx Letk→ ∞, if we takeψ(T) = 0 andψ(0) = 1, then we haveu(0) =u₀; if we take

ψ(T) = 1 andψ(0) = 0, then we have u(T) =z.

4. Existence of weak solutions

The proof of Lemma 3.3 implies that{Dun}is bounded inL^p(x)(Q;M^m×N). By Lemma 2.12,{Dun}can generate a family of Young measuresν_(x,t), andhν(x,t), Ii= Du(x, t). By Lemmas 2.3 and 2.6, we can choose a proper subsequence if necessary such that

un * u in X, n→ ∞, Dun* Du in L^p(x)(Q;M^m×N).

Lemma 4.1. Suppose that σ satisfies(H1)–(H3), then the Young measures ν_(x,t) generated by {Dun}, which is the gradient of Galerkin sequence {un} constructed before, satisfy

Z

Q

Z

M^m×N

σ(x, t, λ)◦λdν_(x,t)(λ) dxdt

≤ Z

Q

Z

M^m×N

σ(x, t, λ)◦Dudν_(x,t)(λ) dxdt

(4.1)

Proof. Consider the sequence

I_n:= σ(x, t, Du_n)−σ(x, t, Du)

◦(Du_n−Du)

=σ(x, t, Dun)◦(Dun−Du)−σ(x, t, Du)◦(Dun−Du)

=:I_n,1+I_n,2 Assumption (H2) implies

Z

Q

|σ(x, t, Du)|^p0(x)dxdt

≤CZ

Q

|a(x, t)|^p0(x)dxdt+c₁ Z

Q

|Du|^p(x)dxdt .

SinceDu∈L^p(x)(Q;M^m×N), it follow that σ∈L^p0(x)(Q;M^m×N). Because of the weak convergence of{Dun}, we obtainI_n,2→0 asn→ ∞. It follows from Lemma 2.11 that

I:= lim inf

n→∞

Z

Q

Indxdt

= lim inf

n→∞

Z

Q

In,1dxdt

= lim inf

n→∞

Z

Q

σ(x, t, Dun)◦(Dun−Du) dxdt

≥ Z

Q

Z

M^m×N

σ(x, t, λ)◦(λ−Du)dν_(x,t)(λ) dxdt

(4.2)

Now we prove thatI≤0. Using Z

Q

∂un

∂t undxdt+ Z

Q

σ(x, t, Dun)◦Dundxdt= Z

Q

fn◦Dundxdt, we find that

I= lim inf

n→∞

Z

Q

σ(x, t, Dun)◦(Dun−Du) dxdt

= lim inf

n→∞

Z

Q

σ(x, t, Dun)◦Dundx− Z

Q

σ(x, t, Dun)◦Dudxdt

= lim inf

n→∞

Z

Q

fn◦Dundxdt− Z

Q

un∂tundxdt− Z

Q

σ(x, t, Dun)◦Dudxdt . Obviously,

Z

Q

fn◦Dundxdt− Z

Q

f◦Dudxdt= Z

Q

fn◦Dundxdt− Z

Q

f◦Dundxdt

− Z

Q

f◦Dundxdt+ Z

Q

f◦Dudxdt.

Since

kfn−fk_Lp0(x)(Q;M^m×N)→0, as n→ ∞, it is easy to see that

Z

Q

fn◦Dundxdt− Z

Q

f ◦Dundxdt

≤Ckfn−fk_Lp0(x)(Q;M^m×N)kDunk_Lp(x)(Q;M^m×N)→0, as n→ ∞.

Because of the weak convergence ofDun, Z

Q

f◦Dundxdt− Z

Q

f◦Dudxdt→0, as n→ ∞.

Consequently, Z

Q

f_n◦Du_ndxdt− Z

Q

f ◦Dudxdt→0, asn→ ∞.

From the weak convergence ofσ(x, t, Dun), Z

Q

σ(x, t, Dun)◦Dudxdt→ Z

Q

χ◦Dudxdt, asn→ ∞.

For everyψ∈C¹(0, T;Vk), k≤n, Z

Q

ψ∂tundxdt− Z

Q

σ(x, t, Dun)◦Dψdxdt= Z

Q

fn◦Dψdxdt.

After integrating, we have Z

Ω

u_n(·, T)ψ(T)vdx− Z

Ω

u_n(·,0)ψ(0)vdx− Z

Q

u_n∂_tψdxdt +

Z

Q

σ(x, t, Du_n)◦Dψdxdt

= Z

Q

f_n◦Dψdxdt.

Lettingn→ ∞, we have Z

Ω

u(·, T)ψ(T)vdx− Z

Ω

u(·,0)ψ(0)vdx− Z

Q

u∂tψdxdt+ Z

Q

χ◦Dψdxdt

= Z

Q

f◦Dψdxdt.

Letk→ ∞, for allψ∈C¹ 0, T;C¹( ¯Ω)

. The the above equality is valid. Then for allψ∈C₀^∞(Q), the above equality also holds. Thus

− Z

Q

u∂tψdxdt=− Z

Q

χ◦Dψdxdt+ Z

Q

f◦Dψdxdt=hdiv(χ−f), ψi.

Obviously∂_tu= div(χ−f). Foru∈X, we can derive that Z

Q

u∂tudxdt=− Z

Q

χ◦Dudxdt+ Z

Q

f◦Dudxdt.

On the other hand, Z

Q

u∂tudxdt=1

2ku(·, T)k²_L2(Ω)−1

2ku(·,0)k²_L2(Ω), Z

Q

un∂tundxdt=1

2kun(·, T)k²_L2(Ω)−1

2kun(·,0)k²_L2(Ω). From the structure ofu_n, we obtain

kun(·,0)kL²(Ω)→ ku(·,0)kL²(Ω).

Using Lemma 3.4, we haveun(·, T)* u(·, T) inL²(Ω). Owing to the weakly lower semicontinuity of the norm,

ku(·, T)kL²(Ω)≤lim inf

n→∞ kun(·, T)kL²(Ω). Clearly,

lim inf

n→∞

− Z

Q

un∂tundxdt

≤ −1

2ku(·, T)k²_L2(Ω)+1

2ku(·,0)k²_L2(Ω).

Thus we arrived at the conclusion thatI≤0.

Lemma 4.2. For a.e. (x, t)∈Q, we have σ(x, t, λ)−σ(x, t, Du)

◦(λ−Du) = 0 on suppν_(x,t). Proof. Since

Z

M^m×N

λdν_(x,t)(λ) =hν_(x,t), Ii=Du(x, t), andν_(x,t)is a family of probability measures,R

M^m×N1dν_(x,t)= 1. Consequently Z

Q

Z

M^m×N

σ(x, t, Du)◦(λ−Du) dν(x,t)(λ) dxdt

= Z

Q

Z

M^m×N

σ(x, t, Du)◦λdν(x,t)(λ) dxdt

− Z

Ω

Z

M^m×N

σ(x, t, Du)◦Dudν(x,t)(λ) dxdt

= Z

Q

σ(x, t, Du)◦ Z

M^m×N

λdν_(x,t)(λ) dxdt

− Z

Ω

σ(x, t, Du)◦Du Z

M^m×N

1 dν_(x,t)(λ) dxdt

= Z

Q

σ(x, t, Du)◦Dudxdt− Z

Ω

σ(x, t, Du)◦Du Z

M^m×N

1 dν_(x,t)(λ) dxdt= 0.

From Lemma 4.1, we obtain Z

Ω

Z

M^m×N

σ(x, t, λ)◦(λ−Du) dν_(x,t)(λ) dxdt≤0.

Thus Z

Ω

Z

M^m×N

(σ(x, t, λ)−σ(x, t, Du))◦(λ−Du)dν_(x,t)(λ) dxdt≤0.

By the monotonicity of σ, the integrand in the above inequality is nonnegative.

Then for a.e. (x, t)∈Q, we can obtain that

(σ(x, t, λ)−σ(x, t, Du))◦(λ−Du) = 0 in suppνx.

We are now in a position to show the existence of solutions of (1.2).

Proof of Theorem 1.1. We consider 4 cases which correspond to the 4 cases in (H3).

Case (i). We prove that for a.e. (x, t)∈ Q and every µ∈ M^m×N the following equation holds on suppνx,

σ(x, t, λ)◦µ=σ(x, t, Du)◦µ+

∇σ(x, t, Du)µ

◦(Du−λ), (4.3) where∇ is the derivative with respect to the third variable ofσ. Actually, by the monotonicity ofσ, for allα∈R, we have

σ(x, t, λ)−σ(x, t, Du+αµ)

◦(λ−Du−αµ)≥0.

From Lemma 4.2 on suppν_x we obtain

σ(x, t, λ)−σ(x, t, Du+αµ)

◦(λ−Du−αµ)

=σ(x, t, λ)◦(λ−Du)−σ(x, t, λ)◦αµ−σ(x, t, Du+αµ)◦(λ−Du−αµ)

=σ(x, t, Du)◦(λ−Du)−σ(x, t, λ)◦αµ−σ(x, t, Du+αµ)◦(λ−Du−αµ).

It can be easily seen that

−σ(x, t, λ)◦αµ≥ −σ(x, t, Du)◦(λ−Du) +σ(x, t, Du+αµ)◦(λ−Du−αµ), and

σ(x, t, Du+αµ) =σ(x, t, Du) +∇σ(x, t, Du)αµ+o(α).

Then we infer that

σ(x, t, Du+αµ)◦(λ−Du−αµ)

=σ(x, t, Du+αµ)◦(λ−Du)−σ(x, t, Du+αµ)◦αµ

=σ(x, t, Du)◦(λ−Du) +∇σ(x, t, Du)αµ◦(λ−Du)

−σ(x, t, Du)◦αµ+∇σ(x, t, Du)αµ◦αµ+o(α)

=σ(x, t, Du)◦(λ−Du) +α

∇σ(x, t, Du)µ◦(λ−Du)−σ(x, t, Du)◦µ +o(α).

Moreover

−σ(x, t, λ)◦αµ≥α

∇σ(x, t, Du)µ

◦(λ−Du)−σ(x, t, Du)◦µ +o(α) Since the sign of α is arbitrary, the above equation implies (4.2). Set µ = E_ij, whereE_ij is the matrix whose entry in the ith row and jth column is 1 and others are 0. Then by (4.2),

σ(x, t, λ)_ij =σ(x, t, Du)_ij+

∇σ(x, t, Du)Eij

◦(Du−λ).

Furthermore, Z

suppν_(x,t)

σ(x, t, λ)_ijdν_(x,t)(λ)

= Z

suppν_(x,t)

σ(x, t, Du)_ijdν_(x,t)(λ) +

∇σ(x, t, Du)Eij

◦ Z

suppν_(x,t)

(Du−λ) dν_(x,t)(λ).

Note that Z

suppν_(x,t)

(Du−λ) dν_(x,t)(λ) =Du(x, t)− Z

suppν_(x,t)

λdν_(x,t)(λ) = 0.

Thus we can derived that Z

suppν_(x,t)

σ(x, u, λ) dν_(x,t)(λ) = Z

suppν_(x,t)

σ(x, t, Du) dν_(x,t)(λ)

=σ(x, t, Du) Z

suppν_(x,t)

dν_(x,t)(λ)

=σ(x, t, Du).

Since{σ(x, t, Dun)} is weakly convergent inL^p0(x)(Q;M^m×N). By Dunford-Pettis criterion and Lemma 2.9,{σ(x, t, Dun)} has aL¹-weak limit:

σ:=

Z

suppν_(x,t)

σ(x, t, λ) dν(x,t)(λ) =σ(x, t, Du).

Evidently,

σ(x, t, Dun)* σ(x, t, Du) inL^p0(x)(Q;M^m×N).

For allφ∈C¹(0, T;Vk), k ≤n, one has Z

Q

φ∂_tu_ndxdt+ Z

Q

σ(x, t, Du_n)◦Dφdxdt= Z

Q

f_n◦Dφdxdt, where

Z

Q

φ∂_tu_ndxdt= Z

Ω

u_n(·, T)φ(T) dx− Z

Ω

u_n(·,0)φ(0)vdx− Z

Q

u_n∂_tφdxdt.

Lettingn→ ∞we obtain Z

Ω

u(·, T)φ(T) dx− Z

Ω

u(·,0)φ(0)vdx− Z

Q

u∂tφdxdt+ Z

Q

σ(x, t, Du)◦Dφdxdt

= Z

Q

f◦Dφdxdt.

Letk→ ∞, then forφ∈C¹(0, T;C₀^∞(Ω)), we are led to the conclusion that

− Z

Q

u∂φ

∂t dxdt+

Z

Ω

u(x, t)φ(x, t) dx

T 0+

Z

Q

σ(x, t, Du)◦Dφdxdt= Z

Q

f◦Dφdxdt.

Case (ii). We prove that for all (x, t)∈Qwe have suppν_(x,t)⊂K_(x,t)

=

λ∈M^m×N :W(x, t, λ) =W(x, t, Du) +σ(x, t, Du)◦(λ−Du) . Ifλ∈suppν(x,t), by Lemma 4.2, for everyβ ∈[0,1],

1−β

σ(x, t, λ)−σ(x, t, Du)

◦ λ−Du

= 0.

By monotonicity, forβ∈[0,1], we have 1−β

σ x, t, Du+β(λ−Du)

−σ(x, t, λ)

◦ Du−λ

≥0.

Thus for allβ ∈[0,1], 1−β

σ x, t, Du+t(λ−Du)

−σ(x, t, Du)

◦ Du−λ

≥0.

In view of the monotonicity condition,

σ x, t, Du+β(λ−Du)

−σ(x, t, Du)

◦β λ−Du

≥0.

Sinceβ∈[0,1], we have

σ x, t, Du+β(λ−Du)

−σ(x, t, Du)

◦(1−β) λ−Du

≥0.

For allβ∈[0,1], if λ∈suppν_(x,t), then

σ x, t, Du+β(λ−Du)

−σ(x, t, Du)

◦ λ−Du

= 0. (4.4)

It follows that

W(x, t, λ) =W(x, t, Du) + Z 1

0

σ x, t, Du+β(λ−Du)

◦(λ−Du) dβ

=W(x, t, Du) +σ(x, t, Du)◦(λ−Du).

So we can getλ∈K_(x,t), i.e. suppν_(x,t)⊂K_(x,t). On account of the convexity ofW, for allξ∈M^m×N,

W(x, t, ξ)≥W(x, u, Du) +σ(x, t, Du)◦(ξ−Du).

For allλ∈K_(x,t), put

P(λ) =W(x, t, λ), Q(λ) =W(x, t, Du) +σ(x, t, Du)◦(λ−Du).

Asλ→W(x, u, λ) is continuous and differentiable, for everyϕ∈M^m×N, γ ∈R, P(λ+γϕ)−P(λ)

γ ≥Q(λ+γϕ)−Q(λ)

γ (γ >0), P(λ+γϕ)−P(λ)

γ ≤Q(λ+γϕ)−Q(λ)

γ (γ <0).

ThusDP =DQ, and

σ(x, t, λ) =σ(x, t, Du) ∀λ∈K(x,t)⊃suppν(x,t). (4.5) Consequently,

σ(x, t) :=

Z

M^m×N

σ(x, t, λ) dν(x,t)(λ)

= Z

suppν_(x,t)

σ(x, t, λ) dν(x,t)(λ) =σ(x, t, Du).

(4.6)

Now we consider the Carath´eodory function

g(x, t, λ) =|σ(x, t, λ)−σ(x, t)|, λ∈M^m×N.

Since σ(x, t, Du_n) is weakly convergent in L^p0(x)(Q;M^m×N), then σ(x, t, Du_n) is equi-integrable. Thusg_n(x, t) =g(x, t, Du_n) is equi-integrable, and

g_n* g in L¹(Q).

Taking (4.4) and (4.5) into consideration, we obtain g(x, t) =

Z

M^m×N

σ(x, t, λ)−σ(x, t)

dν_(x,t)(λ)

= Z

suppν_(x,t)

σ(x, t, λ)−σ(x, t)

dν_(x,t)(λ)

= Z

suppν_(x,t)

σ(x, t, λ)−σ(x, t, Du(x, t))

dν_(x,t)(λ) = 0.

It turns out that Z

Q

σ(x, t, Du_n)−σ(x, t, Du)

dxdt→0.

The remainder of the argument is similar to that in case (i) and so is omitted.

Case (iii). By the strict monotonicity and Lemma 4.2, we have suppν_(x,t)={Du(x, t)}.

Thus for a.e. (x, t)∈Q,ν_(x,t)=δ_Du(x,t). Using Lemma 2.9, we findDun→Du in measure. For a proper subsequence, we assert thatDun →Dua.e. inQ. It follows that σ(x, t, Dun)→σ(x, t, Du) a.e. inQ. Moreover σ(x, t, Dun)→σ(x, t, Du) in measure.

From a similar analysis in case (i), we obtain the existence of (1.2) for case (ii).

Case (iv). Suppose thatν_(x,t) is not a Dirac measure, for a.e. (x, t)∈Q, then we have

0<

Z

Q

Z

M^m×N

σ(x, t, λ)−σ(x, t,λ)¯

◦(λ−¯λ) dν_(x,t)(λ) dxdt

= Z

Q

Z

M^m×N

σ(x, t, λ)◦λ−σ(x, t, λ)◦¯λ

−σ(x, t,¯λ)◦λ+σ(x, t,λ)¯ ◦λ)¯

dν(x,t)(λ) dxdt.

Since Z

M^m×N

1 dν_(x,t)(λ) = 1 and Z

M^m×N

λdν_(x,t)(λ) = ¯λ=Du(x, t), we obtain

Z

Q

Z

M^m×N

σ(x, t, λ)◦λdν_(x,t)(λ) dxdt

>

Z

Q

Z

M^m×N

σ(x, t, λ)◦λ¯+σ(x, t,¯λ)◦λ−σ(x, t,λ)¯ ◦λ)¯

dν_(x,t)(λ) dxdt

= Z

Q

Z

M^m×N

σ(x, t, λ) dν_(x,t)(λ)◦λ¯+σ(x, t,¯λ)◦ Z

M^m×N

λdν_(x,t)(λ)

−σ(x, t,λ)¯ ◦¯λ· Z

M^m×N

1 dν_(x,t)(λ) dxdt

= Z

Q

Z

M^m×N

σ(x, t, λ) dν_(x,t)(λ)◦λ¯dxdt

= Z

Q

Z

M^m×N

σ(x, t, λ) dν_(x,t)(λ)◦Du(x, t) dxdt.

By Lemma 4.1, Z

Q

Z

M^m×N

σ(x, t, λ)◦Dudν_(x,t)(λ) dxdt

≥ Z

Q

Z

M^m×N

σ(x, t, λ)◦λdν_(x,t)(λ) dxdt

>

Z

Q

Z

M^m×N

σ(x, t, λ) dν_(x,t)(λ)◦Dudxdt.

This is a contradiction. Henceν(x,t)is a Dirac measure. Assume thatν(x,t)=δh(x,t). Then

h(x, t) = Z

M^m×N

λdδ_h(x,t)(λ) = Z

M^m×N

λdν_(x,t)(λ) =Du(x, t)

Thusν_(x,t)=δ_Du(x,t). Lemma 2.10 implies thatDun →Du asn→ ∞. Moreover σ(x, t, Dun)→σ(x, t, Du) in measure asn→ ∞. An argument similar to the one in case (iii) shows the conclusion we want. The proof is complete.

Conclusions. In this article, we study the existence of weak solutions for quasilin- ear parabolic system in divergence form with variable growth by means of Young measures generated by sequences in variable exponent spaces. We can conclude that problem (1.2) has a weak solution under four kinds of monotonicity conditions in (H3). We need notice that (H3)(iii) requires σ is strictly monotone. Actually classical monotonicity operator method can get our result under (H3)(iii). We give the other method to obtain the main theorem by Young measures in our paper under (H3)(iii). But conventional method can not prove the main result under the other monotonicity conditions. And in H3(iv), we define a new monotonicity condition. If σ is strictly monotone, then (H3)(iv) holds. Obviously, (H3)(iv) is weaker than typical strictly monotone condition.

Currently, the research on Young measures generated by sequences in variable exponent Lebesgue and Sobolev spaces is still in exploration. Our results enrich and perfect the theory of variable exponent spaces and Young measures.

For related results on nonlinear problems with variable growth we refer to the monograph by R˘adulescu and Repovˇs [33] and the survey paper by R˘adulescu [32].

Recent contributions to this field may be found in the papers [28, 29, 30, 34].

Acknowledgements. This work was supported by the National Natural Science Foundation of China (No. 11771107, No. 11601251), and by the Natural Science Foundation of Shandong Province (ZR2016AM13, ZR2018PA003).

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Miaomiao Yang

School of Science, Qilu Institute of Technology, Jinan 250001, China E-mail address:[email protected]

Yongqiang Fu

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China E-mail address:[email protected]