Existence and nonexistence of global solutions for doubly nonlinear diffusion equations

Existence and nonexistence of global solutions for doubly nonlinear diffusion equations

with logarithmic nonlinearity

Nhan Cong Le

and Truong Xuan Le

1VNUHCM - University of Science, 227, Nguyen Van Cu Str, District 5, Ho Chi Minh City, Vietnam

2HCMC University of Technology and Education, 1 Vo Van Ngan Str, District Thu Duc, Ho Chi Minh City, Vietnam

3University of Economics Ho Chi Minh City, 59C, Nguyen Dinh Chieu Str, District 3, Ho Chi Minh City, Vietnam

Received 18 March 2018, appeared 2 August 2018 Communicated by Vilmos Komornik

Abstract. In this paper, we study an initial-boundary value problem for a doubly non- linear diffusion equation with logarithmic nonlinearity. By using the potential well method, we give some threshold results on existence or nonexistence of global weak so- lutions in the case of initial data with energy less than or equal to potential well depth.

In addition, the asymptotic behavior of solutions is also discussed.

Keywords: global existence, blow-up, asymptotic behavior, logarithmic nonlinearity.

2010 Mathematics Subject Classification: 35A01, 35B44, 35K55.

1 Introduction

In this paper, we will study the following doubly nonlinear diffusion equations with logarith- mic nonlinearity











ut−_∆pu^(m−1)

= fq(u), x∈ _Ω,t>0, u(x,t) =0, x∈ _∂Ω,t>0, u(x, 0) =u₀(x), x∈ _Ω,

(1.1)

where Ω ⊂ _Rⁿ (n ≥ 1) is an open bounded domain with smooth boundary ∂Ω, u^(m−1) :=

|u|^m−2u, ∆p(u) := div (∇u)^(p−1) the usual p-Laplacian operators and fq is of the form of logarithmic term fq(s) =s^(q−1)log|s|.

Let us consider the following equation which is so-called doubly nonlinear parabolic equa- tions

ut−_∆pu^(m−1)

= f(u), (1.2)

BCorresponding author. Email: lxuantruong@gmail.com

where f(u)is a source if f(u) ≥ 0, whereas f(u) is called a sink. This equation generalizes many equations such as heat equation (as m = 2 and p = 2), the porous medium equation (as m > 1 and p = 2), and the p-Laplacian equation (as m = 2 and p > 1). The equation (1.2) can be divided into three following cases which are called the degenerate, critical and singular case, respectively.

(m−1) (p−1)>1, (m−1) (p−1) =1, and (m−1) (p−1)<1 (1.3) In this paper, we merely consider the degenerate case, that is, the constantsm,p>1 satisfy

(m−1) (p−1)>1. (1.4)

The initial-boundary value problem for (1.2) has been studied by many mathematicians.

For example, Tsutsumi (see [32]) studied the existence, uniqueness, regularity and large time behavior for weak, mild and strong solutions of an equivalent equation to (1.2) (after changing variables) with absorption f(u) =−λu^γ,λ> 0 and initial valueu₀ in some certain Lebesgue spaces. In [23], Matas et al. also studied the existence of weak solution to the equation (1.2) with inhomogeneous nonlinearities f(u) in the degenerate case with initial value u₀ in Lebesgue spaces by using Galerkin method. The existence of weak solutions of Cauchy problem for equation (1.2) with f(u) = 0 has been studied by Ishige [14] for all three cases (1.3).

Regarding of the global existence and nonexistence results, there are some well-known methods to study the equation (1.2) depends on whetherΩis bounded or unbounded domain inRⁿ. For example, in the case Ω = _Rⁿ, Fujita [9] studied the initial value problem for heat equations with power nonlinearity f(u) =u^pand then Levine in the survey [18] has extended the results of Fujita for more general parabolic equations with nonlinear dissipative terms in unbounded domains. In [25], Pokhozaev and Mitidieri introduced the nonlinear capacity method in order to study of questions on the blow up of solutions of many nonlinear partial differential equations and inequalities. It is noting that although these methods are really powerful tools to treat the case of nonnegative nonlinearities f(u), it cannot be applied to the case of sign-changing nonlinear terms. And therefore, this method cannot be applied to our problem.

On the other hand, in the case of bounded subdomain ofRⁿ, we refer the seminar papers of Kaplan [16], Levine [17] and Ball [3] in which the authors proved the blow up results under condition of non-positive initial energy. In [27], Payne and Sattinger developed the potential well method which is introduced by Lions [20] and Sattinger [30] to study the existence and nonexistence of global weak solutions to heat and wave equations with power like nonlinearity under condition of positive initial energy. More precisely, in [27] the authors show that if the initial energy J(u₀) < d, then weak solutionu(t)to equation (1.2) (for m = p = 2) is global provided thatu₀ ∈ W (stable sets) and blows up in finite time provided thatu₀ ∈ U _(unstable sets). Afterward, analogous results have been studied extensively to various kind of equations.

We refer to [6,10–13,17,21,33] for many heat and wave equations and [7,19] for porous medium equations.

In the case of p-Laplacian equation, Tsustumi [31], Fujii [8] and Ishii [15] studied the initial-boundary value problem for the equation

u_t−_∆pu= f(u), (1.5)

where f(u) = |u|^q−2u, with 2 ≤ q < p^∗ = _n^np_−p. As for the existence and nonexistence of global weak solutions to (1.5), the following results are well-known:

(i) if p> q, (1.5) admits a global weak solution for anyu₀∈W₀^1,p(_Ω);

(ii) if p < q, then weak solution u(t)of (1.5) is global when initial datau₀ ∈ W₀^1,p(_Ω)is in stable sets and blows up in finite time whenu₀∈W₀^1,p(_Ω)is in unstable sets.

(iii) when p = q, Fujii [8] derived sufficient conditions on blow-up of solutions depending on first eigenvalueλ₁ of the operator−_∆p.

Although there is a lot of results of global existence and nonexistence of weak solutions to (1.2) in the case of power nonlinearities and its generalization, there is a little known about the one with logarithmic nonlinearity. We refer to [4,5,26] for a few recent papers involving logarithmic nonlinearity. In this paper, in the same spirit with previous works, we utilize the potential well method to study the existence and nonexistence of global weak solutions to (1.2) with logarithmic nonlinearities fq(u) = (u)^(q−1)log|u|, q > 2 and initial value u⁽₀^m−1) belonging to Sobolev spaceW₀^1,p(_Ω). Roughly speaking, our results are as follows:

(i) if(m−1) (p−1)> q−1, then (1.1) admits a global solution for eachu₀^(m−1) ∈W₀^1,p(_Ω); (ii) if (m−1) (p−1) ≤ q−1, then there exists a weak solution to (1.1) which is global provided that u₀ belonging to stable sets, and blows up provided that u₀ belonging to unstable sets. In addition, decay estimates are also proved for the former case.

Define ϕ: L^m0(_Ω)→L^m(_Ω)as follows

ϕ(u) =u^(m0−1),

where m⁰ > 1 is Hölder conjugate ofm satisfying _m¹ +_m¹0 = 1, then one has ϕ u^(m−1)

= u.

Hence, by changing variablew=u^(m−1), the equation (1.1) leads to the reformulated equation

∂_tϕ(w)−_∆pw= m⁰−1

f_γ(w), where γ= m⁰−1

(q−1) +1. (1.6) It is also noticed that f_γ(s) is nonhomogeneous and can change signs for s ∈ (0,+_∞). In addition, since lim_s→0⁺ fγ(s) = 0, it can be extended continuously to the function ˜fγ with

f˜γ(0) =0.

In what follows, for the sake of brevity, we still denote ˜f_γ by f_γ with noticing that f_γ(0) = fγ(1) =0. The nonlinearity with such properties can be found in the paper [2]. Hence, instead of (1.1) we consider the following initial boundary value problems











∂tϕ(u)−_∆p(u) = (m⁰−1)fγ(u), x ∈_Ω,t>0, u(x,t) =0, x ∈∂Ω,t >0, u(x, 0) =u₀(x), x ∈_Ω,

(1.7)

whereu₀∈W₀^1,p(_Ω)andγ= (m⁰−1) (q−1) +1>1,m⁰ is Hölder conjugate ofm.

The rest of this paper is organized as follows: Section 2 devotes to preliminaries in which we establish some properties of stationary problem associated to (1.7) and introduce the stable sets (potential well) and unstable sets as well as its properties; Section 3 states main results of this paper and their proofs are presented in the remaining sections.

2 Local minima and potential wells

In this section, we need the following logarithmic Gagliardo–Nirenberg inequality which was introduced by Merker [24].

Lemma 2.1([24], Logarithmic Gagliardo–Nirenberg inequalities). The inequalities Z |u|^q

kuk^q_q^log

|u|^q kuk^q_q

!

dx≤ ¹

1−q/p^∗ log C^q_n,p,qk∇uk^q_p kuk^q_q

!

(2.1) are valid for parameters1≤ p < +_∞,0 < q < p^∗ and function u ∈ L^q(_Rⁿ)with ∇u ∈ L^p(_Rⁿ). Hereby the constant C depends on n and p only in the case p<n, and on n,p and a finite upper bound of q in the case p≥n.

This inequality can be reformulated in parametric form. Here, one introduces the follow- ing parametric form of logarithmic Gagliardo–Nirenberg inequalities

Z |u|^q kuk^q_q^log

|u|^q kuk^q_q

!

dx≤µ k∇uk^r_p

kuk^r_q + ^qp

∗

(p^∗−q)r log

qp^∗C_n,p,q^r (p^∗−q)rµe

, (2.2)

for all µ > 0 where 0 < r ≤ min{p,q}. By virtue of Young’s inequality, one obtains the following proposition.

Proposition 2.2(Parametric form of logarithmic Gagliardo–Nirenberg inequality). Let us sup- pose all assumptions in Lemma2.1. Then we have

Z

|u|^qlog |u|^q kuk^q_q

!

dx+C^r_n,p,q,µkuk^q_q≤µr

pk∇uk^p_p+µp−r p kuk

(q−r) p−r p

q ,

for allµ>0where0<r≤min{p,q}and C^r_n,p,q,µis a constant given by C^r_n,p,q,µ= ^qp

∗

(p^∗−q)rlog (p^∗−q)rµe qp^∗C_n,p,q^r

! .

Proposition 2.3([28,29], Parametric form of logarithmic Sobolev inequality). Let u∈W^1,p(_Rⁿ), u6=0andµ>0be any number. Then

p Z

Rⁿ|u(x)|^plog |u(x)|

kuk_p

!

dx+ⁿ plog

pµe nL_p

_Z

Rⁿ|u(x)|^pdx ≤µ Z

Rⁿ|∇u(x)|^pdx, where

L_p = ^p n

p−1 e

p−1

π⁻

p 2





Γ ⁿ₂+1 Γ

n^p−_p¹+1





p n

and for1≤ p <+_∞.

For u∈W₀^1,p(_Ω), we can define u(x) =0for x ∈_Rⁿ\_{Ω. Then u}∈W^1,p(_Rⁿ)and therefore, we derive

p Z

Ω|u(x)|^plog |u(x)|

kuk_p

!

dx+ ⁿ plog

pµe nL_p

Z

Ω|u(x)|^pdx≤µ Z

Ω|∇u(x)|^pdx, for u∈W₀^1,p(_Ω)andµ>₀is any number.

We now define the energy functional J and Nehari functional I onW₀^1,p(_Ω)related to the problem (1.7) as follows

J(u) = ¹ p

Z

Ω|∇u|^pdx+ ^m

0−1 γ²

Z

Ω|u|^γdx− ^m

0−1 γ

Z

Ω|u|^γlog(|u|)dx, (2.3) I(u) =

Z

Ω|∇u|^pdx− m⁰−₁

Z

Ω|u|^γlog(|u|)dx. (2.4)

It is clear that the functionals I andJ are continuous onW₀^1,p(_Ω)and J(u) =

1 p− ¹

γ

k∇uk^p_p+ ^m

0−₁

γ² kuk^γ_γ+ ¹

γI(u). (2.5)

We also define the Nehari manifold as follows

N = ⁿu∈W₀^1,p(_Ω)\{0}: I(u) =J⁰(u),u

=0o

. (2.6)

We shall see below (see Lemma 2.5) that each half line starting from the origin of the phase spaceW₀^1,p(_Ω)intersects the Nehari manifoldN exactly once.

It is also useful to understand the Nehari manifoldN in terms of the critical points of the fibrering map λ7→ J(λu)forλ>0 given by

J(λu) = ^λ

p

p k∇uk^p_p+^m

0−1

γ² λ^γkuk^γ_γ− ^m

0−1 γ

Z

Ω|λu|^γlog(|λu|)dx, λ>_0.

Then we have d

dλJ(λu) =J⁰(λu),u

= ¹ λ

J⁰(λu),λu

= ¹

λI(λu), forλ>0. (2.7) This implies the following lemma immediately.

Lemma 2.4. Let u ∈ W₀^1,p(_Ω)\{0}andλ >0. Thenλu∈ N if and only ifλis a critical point of the mapλ7→ J(λu), that is, _dλ^d J(λu) =0.

Thanks to (2.7), in order to study the critical point of fibrering map, we need to study zero points of the mapλ7→ I(λu)foru∈W₀^1,p(_Ω)\{0}given by I(λu) =λ^pK(λu),λ>0 where

K(λu) =k∇uk^p_p− m⁰−1 λ^γ−p

Z

Ω|u|^γlog(|u|)dx− m⁰−1

kuk^γ_γλ^γ−plogλ.

Lemma 2.5. Let2≤ p≤γand u∈W₀^1,p(_Ω)\{0}. Then we possess

(i) there exists a uniqueλ∗ := λ∗(u)> 0such that I(λ∗u) =0and I(λu)> 0forλ∈ (0,λ∗), and I(λu)<₀forλ>λ∗;

(ii) the fibrering mapλ7→ J(λu)attains its maximizer atλ= λ∗, that is, d

dλJ(λu) λ=λ∗

=0 and d²

dλ²J(λu) λ=λ∗

<0.

Proof. In the case p =γ, it is not difficult to see that λ∗ :=λ∗(u) =expn

I(u)/ m⁰−1 kuk^p_p^o

satisfies(i)and(ii). It remains to verify for the case p<γ. Indeed, if this is the case, it is not difficult to see that the functionλ7→K(λu)is continuous on(0,+_∞)and

lim

λ→0⁺K(λu) =k∇uk^p_p >0 and lim

λ→+_∞K(λu) =−_∞, andK(λu)attains its unique maximizer at ¯λ:=λ^¯ (u)>0

λ¯ =exp

(kuk^γ_γ+ (γ−p)R

Ω|u|^γlog(|u|)dx (p−γ)kuk^γ_γ

) .

Hence, there must be a uniqueλ∗ > λ^¯ such that K(λ∗u) = 0 and K(λu)>0 for λ∈ (0,λ∗), andK(λu)<_{0 for}λ> λ^∗. As a consequence, the proof follows from I(λu) =λ^pK(λu)and (2.7).

We now define the depth of potential well d=infn

sup_λ≥0J(λu):u∈W₀^1,p(_Ω)\{0}^o, which is also characterized as

0<d= inf

u∈N J(u). (2.8)

Lemma 2.6. Let p ≥ 2 and p ≤ γ < _n^np_−p := p^∗. Then there exists an extremal of the variational problem

0<d= inf

u∈N J(u).

Proof. The caseγ = p can be proved similarly to [26]. It remains to consider the case γ> p.

Letu∈ N, then it follows by (2.5) J(u) =

1 p− ¹

γ

k∇uk^p_p+ ^m

0−1

γ² kuk^γ_γ. (2.9)

On the other hand, by logarithmic Gagliardo–Nirenberg inequality, one has 1

m⁰−1k∇uk^p_p=

Z

Ω|u|^γlog(|u|)dx

≤ ^µr

γpk∇uk^p_p+µp−r γp kuk

γ−r p−rp

γ + ¹

γC_n,p,γ,µ^r kuk^γ_γ+ ¹

γkuk^γ_γlog

kuk^γ_γ, wherer ∈ (0,p)is a constant andC_n,p,q,µ^r is a constant given by Proposition2.2. By choosing µ= _(m^γp₀₋

1)r then we get p−r (m⁰−1)rkuk

γ−r p−rp

γ + ¹

γC_n,p,q,µ^r kuk^γ_γ+ ¹

γkuk^γ_γlog

kuk^γ_γ≥0. (2.10)

It is noticed that for r ∈ (0,p) and p < γ then ^γ_p−^−r_rp > γ. And therefore, we deduce from (2.10) that there exists a positive constant Rindependent ofusuch that kuk_γ ≥ R> 0 which implies

k∇uk_p ≥ ¹

S_p,γ kuk_γ ≥ ^R

S_p,γ. (2.11)

Here Sp,γ stands for the best constant in the Sobolev embedding W₀^1,p(_Ω) ,→ L^γ(_Ω) with 0<γ≤ p^∗ = _n^np_−p (p<n). Thus, the proof follows from (2.9) and (2.11).

Denote the nontrivial stationary solution of problem (1.7) by E=ⁿu∈W₀^1,p(_Ω)\{0}:−_∆pu= m⁰−1

fγ(u):u|_∂Ω=0o , E_d={u∈ E: J(u) =d}.

Then, by virtue of critical point theory, it is not difficult to see that if u ∈ E (in weak sense) thenu is a nontrivial critical point ofJ(u). Hence, we get

E_d ={u∈ N : J(u) =d}. (2.12)

As a consequence of Lemma2.6,E_d is a nonempty set.

We now define stable setW and unstable setU _{as in [15,27].}

W = ⁿu∈W₀^1,p(_Ω): J(u)<d,I(u)>0o

∪ {0}, (2.13)

U = ⁿu∈W₀^1,p(_Ω): J(u)<d,I(u)<0o

. (2.14)

By continuity of the functionals I and J onW₀^1,p(_Ω), one has W =ⁿu∈W₀^1,p(_Ω): J(u)≤d,I(u)≥0o

and U =ⁿu∈W₀^1,p(_Ω): J(u)≤d,I(u)≤0o . Some properties ofW andU are listed below.

Lemma 2.7.

(i) W is a bounded neighborhood of 0 in W₀^1,p(_Ω), that is, there exist 0 < _r1 < _{r2 such that} B(0,r₁)⊂ W ⊂B(0,r₂);

(ii) 0 /∈ U;

(iii) E_d⊂ N andW ∩ U =_Ed.

Proof. (i)Let u∈ W withu6=0, then it follows from the definition ofW and (2.5) that k∇uk^p_p < ^pγ

γ−pd and kuk^γ_γ< ^γ

2

m⁰−1d, (2.15)

forγ> p. In the caseγ= p, we also deduce from (2.5) that kuk^p_p < ^p

2

m⁰−1d and I(u)< pd. (2.16)

On the other hand, by virtue of logarithmic Sobolev inequality, we get I(u)≥

1− ^µ(m⁰−1) p

k∇uk^p_p+ ⁿ(m⁰−1) p² log

pµe nL_p

kuk^p_p−^m

0−1

p kuk^p_plog

kuk^p_p. It follows that

1− ^µ(m⁰−1) p

k∇uk^p_p ≤ I(u) + ⁿ(m⁰−1) p² log

nL_p pµe

kuk^p_p + ^m

0−1

p kuk^p_plog

kuk^p_p. (2.17)

By choosing µ< _m0^p−1, it follows from (2.15)–(2.17) thatk∇uk^p_p < C_d, where C_d independent ofu. The remain part of(i)can be prove similar to Lions [20]. Hence, we possess(i).

(ii)By contradiction, we assume that 0∈ U. Then there exists a sequence{u_n} ∈ U such that u_n → 0 inW₀^1,p(_Ω) as n → _∞. It follows from (i)that u_n ∈ W forn sufficiently large.

This contradicts to the fact thatW ∩ U =_∅.

(iii) It is clear thatE_d ⊂ N. We now let u ∈ W∩ U, then I(u) = 0 and J(u) ≤ d. Since (ii), we get u 6= 0 and therefore u ∈ N. On the other hand, by variational characterization of d, one has J(u) = d. Thus u ∈ E_d. Conversely, if u ∈ E_d, then it follows from (2.12) that u∈W₀^1,p(_Ω)\{0}satisfying I(u) =0 and J(u) =d. This impliesu∈W∩ U. The lemma has been proven.

3 Main results

Firsly, we introduce the definitions of weak solutions to (1.7) and maximal existence time.

Definition 3.1. A function u is said to be a weak solution of problem (1.7) on [0,T) if u ∈ L^∞ 0,T;W₀^1,p(_Ω)is such that ϕ(u)∈ L^∞(0,T;L^m(_Ω))with

∂tϕ(u)∈L^p0

0,T;W^−1,p0(_Ω) and ∂t

|u|^m

0 2

∈ L²(QT),

satisfies the initial valueu(0) =u₀and the equation (1.7) in a generalized sense, that is, Z _T

0

h∂_tϕ(u),vidt+

Z _T

0

D(∇u)^(p−1),∇vE

dt= m⁰−1 Z _T

0

hfγ(u),vidt, (3.1) for allv∈ L^p 0,T;W₀^1,p(_Ω).

Definition 3.2(Maximal existence time). Let ube a weak solution to problem (1.7). Then we define the maximal existence timeT_maxof uas follows:

• if u:=u(t)exists on[0,T)for allT>0, thenT_max= +∞. In this case, we say that uis a global solution of (1.7);

• if there is T > 0 such thatu := u(t)exists on[0,T), but it does not exist att = T, then T_max= T. In this case, we say that uis blow up att= T.

We now give the existence and nonexistence of global weak solutions to (1.7) depending on parametersm, pandq.

Theorem 3.3. Let T> 0, u₀ ∈W₀^1,p(_Ω)and let m, p be constants satisfying(1.4). Then we possess the following statements.

(i) If(m⁰−1)(q−1)< p−1, then there exists a weak solution u to problem(1.7)on[0,Tmax)for T_max=T which satisfies∂_t |u|^m

0

2

∈ L²(Q_T)and the energy inequality Z _t

0

kUτ(τ)k²₂dτ+J(u(t))≤ J(u₀), a.e. in [0,T_max), (3.2) where U(t) = ²

√m⁰−1

m⁰ |u(t)|^m

0 2 .

(ii) If(m⁰−1)(q−1)≥ p−1and q>2such that m⁰−₁(q−₁) +1< p

1+ ^m

0

n

if p<n.

then there exists a weak solution u satisfying(3.2)on[0,T_max)with0<T_max <T.

Next, we give similar results as in [15,27,31] on the existence and nonexistence of global solution when the initial datau0is in stable set W and unstable setU.

Theorem 3.4(Global existence forJ(u₀)<d). Let m,p satisfy(1.4)and q>2such that p≤(m⁰−1)(q−1) +1< p^∗ = ^np

n−p as p <n, (3.3)

and u₀ ∈ W. Then the problem (1.7) admits a global weak solution u ∈ L^∞ 0,T;W₀^1,p(_Ω) with

∂_tU∈ L²(Q_T)and u(t)∈ W for t∈ [0,T)for any T >0. In addition, we have the following decay estimates:

k∇u(t)k_p ≤ k∇u₀k_p

p

m⁰+ω(p−m⁰)t ¹

p−m0

for t≥0, (3.4)

for some ω>0.

Theorem 3.5(Blow up for J(u₀)<d). Let m,p satisfy(1.4)and q>₂such that

p≤ (m⁰−1)(q−1) +1< p^∗ as p<n, (3.5) and u0∈ U. Then weak solution u of the problem(1.7)blows up in finite time, that is, there is T∗such that

tlim→T∗

k∇u(t)k^p_p = +_∞.

Remark 3.6. The results of Theorem 3.4 and3.5 are still valid if we replace the initial value u₀ byu(t₀)for somet₀ ∈ [0,T_max). In Theorem3.4, by assumption u₀ ∈ W, we can relax the constraint on γ< p 1+^m_n⁰ byγ< p^∗.

Remark 3.7(Sharp condition for J(u₀)<d). Letm, psatisfy (1.4) andq>2 such that p≤ m⁰−1

(q−1) +1< p^∗,

and u₀ ∈ W₀^1,p(_Ω)\{0} with J(u₀) < d. Then problem 1.7 admits a global weak solution provided thatI(u₀)>0 and does not admit any global weak solution provided thatI(u₀)<_0.

Finally, we have a threshold result on the existence and non-existence of global weak solution to (1.7) in the case J(u0) =d.

Theorem 3.8. Let m, p satisfy(1.4)and q>2such that p≤(m⁰−1)(q−1) +1< p^∗ = ^np

n−p as p<n, (3.6)

and u₀ ∈ W₀^1,p(_Ω)\{0}with J(u₀) = d. Then the local weak solution u of (1.7) is global provided that I(u₀)>₀and blows up in finite time provided that I(u₀)<0. Moreover, in the former case, there exists a positive constantω₁ such that

k∇u(t)k_p≤ k∇u(t₁)k_p

p

m⁰+ω₁(p−m⁰)t _p−¹_m0

, for t ≥t₁, (3.7) for some t₁ >0.

4 Proof of Theorem 3.3

In this section we prove the existence of weak solutions by Faedo–Galerkin method. The proof comprises of several steps in which we use the following well-known Gronwall–Bellman–

Bihari integral inequality [1, p. 53].

Lemma 4.1(Gronwall–Bellman–Bihari). Let S(t)be a nonnegative continuous function such that S(t)≤C₁+C₂

Z _t

0 S^κ(s)ds, where C₁,C₂are positive constants. Then we get

(i) S(t)≤^hC¹₁^−κ+ (1−κ)C₂ti₁₋¹

κ for0<κ<1;

(ii) S(t)≤C₁exp{C₂t}forκ=1;

(iii) S(t)≤C₁h

1−(κ−1)C2C^κ₁⁻¹ti−_κ−¹₁

forκ >1.

Step 1: Finite-dimensional approximations Let

w_j be a system of basis functions inW₀^1,p(_Ω)and define V_k ={w₁,w₂, . . . ,w_k}. Letu_0k be an element ofV_k such that

u_0k =

∑

a_jkw_j →u₀, in W₀^1,p(_Ω), (4.1) as k → ∞. We shall construct the approximate solutionsu_k(x,t)of the problem (1.1) by the form

u_k(t) =

∑

α_kj(t)w_j, fork =1, 2, . . . , (4.2)

where the coefficientsα_kj (1≤ j≤k)satisfies the system of integro-differential equations D

m⁰−1

|u_k(t)|^m0−2u_kt(t),w_iE

+^D(∇u_k(t))^(p−1),∇w_iE

= m⁰−1D

(u_k(t))^(γ−1)log(|u_k(t)|),w_iE

, (4.3)

fori=1, 2, . . . ,k, with the initial conditions

α_kj(₀) =a_kj, j=1, 2, . . . ,k. (4.4) In order to recognize that the system (4.3)–(4.4) has a local solution, forα= (α₁,α₂, . . . ,α_k)∈ R^k, we set

• ψ(α) = (ψ₁(α), . . . ,ψ_k(α))^T, with

ψ_i(α) =

Z

Ω







∑

α_jw_j

!(m⁰−1)

w_i





 dx;

• B(α) = (B₁(α)_{, . . . ,}B_k(α))^T_{, with}

B_i(α) =

Z

Ω







∑

α_j∇w_j

!(p−1)

∇w_i





 dx;

• F(α) = (F₁(α)_{, . . . ,}F_k(α))^T_{, with}

F_i(α) = m⁰−1 Z

Ω







∑

α_jw_j

!(γ−1)

log

∑

α_jw_j

!



 w_idx.

Then it is obvious that the system (4.3)–(4.4) can be rewritten as d

dtψ(α_k(t)) +B(α_k(t)) =F(α_k(t)), (4.5) which is also equivalent to the integral equation

ψ(α_k(t)) =ψ(α_k(0))−

Z _t

0

[−B(α_k(s)) +F (α_k(s))]ds, (4.6) where α_k(t) = (α_k1(t),α_k2(t), . . . ,α_kk(t))^T. The standard theory of ordinary differential and integral equations yields that there exists a positive 0 < T_k ≤ T such that α_kj ∈ C¹([0,T_k])_, and thereforeu_k ∈C¹ [0,T_k];W₀^1,p(_Ω).

Step 2: The fundamental priori estimates

In order to obtain the boundedness of the approximate solutions{u_k}, we need the following inequality.

Lemma 4.2. Let1<m⁰ < p<_∞and r be a constant such that p ≤r< p

1+ ^m

0

n

if p<n and p ≤r if p≥n.

Then for eachε>0, there exists a positive constant Cε such that kvk^r_r≤ εk∇vk^p_p+Cε

kvk^m_m⁰0

κ

, (4.7)

for all v∈W₀^1,p(_Ω), where κ= (1−θ)r

(1−α)m⁰ >1, θ= 1

m⁰ −¹ r

1 m⁰ − ¹

p^∗ −1

, α= ^θr p. Proof. By virtue of Gagliardo–Nirenberg inequality, we have

kvk_r ≤Ck∇vk^θ_pkvk¹_m⁻0^θ, ∀v∈W₀^1,p(_Ω), where

θ = 1

m⁰ −¹ r

1 m⁰ − ¹

p^∗ −1

. This implies

kvk^r_r≤C

k∇vk^p_p^αkvk^m_m⁰0

^(1−θ)r

m0

, withα= ^θr p.

Since p ≤ r < p(1+m⁰/n), we getθ ≤ α = θr/p < 1. By virtue of Young’s inequality, one has

kvk^r_r≤ εk∇vk^p_p+Cε

kvk^m_m⁰0

κ

, whereκ= ₍^(1−θ)r

1−α)m⁰ >1. The proof is complete.

Multiplying both sides of (4.3) byα_ki(t)and taking the sum over i= 1, 2, . . . ,k, and then integrating with respect to time variable from 0 tot, one has

ku_k(t)k^m_m⁰0 =ku_0kk^m_m⁰0−

Z _t

0 I(u_k(τ))dτ, (4.8)

where

I(u_k(t)) =k∇u_k(t)k^p_p− m⁰−1 Z

Ω|u_k(t)|^γlog(|u_k(t)|)dx. (4.9) We now estimate I(u_k(t)). By elementary inequality, we get the following estimate forβ> 0 sufficiently small

Z

Ω|u_k(t)|^γlog|u_k(t)|dx=

Z

{|u_k(t)|≤1}

|u_k(t)|^γlog|u_k(t)|dx+

Z

{|u_k(t)|>1}

|u_k(t)|^γlog|u_k(t)|dx

≤e⁻¹|_Ω|+ ¹ β

Z

Ω|u_k(t)|^γ+βdx. (4.10)

We now consider the two following cases:

Case1: (m⁰−1)(q−1) < p−1. In this case, we haveγ < p. By virtue of Young inequality and Poincaré inequality, we get

Z

Ω|u_k(t)|^γlog|u_k(t)|dx≤εk∇u_k(t)k^p_p+C(_Ω,ε), (4.11) with ε>0. It follows from (4.9) and (4.11) that

I(u_k(t))≥ 1−(m⁰−1)ε

k∇u_k(t)k^p_p−C(_Ω,ε). (4.12) By choosingε= _p(m^p−0−¹1), we deduce from (4.1), (4.8) and above inequality that

S_k(t)_:=ku_k(t)k^m_m⁰0+ ¹ p

Z _t

0

k∇u_k(τ)k^p_pdτ≤C_T, ∀t ∈[_0,T]_, ∀k∈_{N. (4.13)} Case2: (m⁰−1)(q−1)≥ p−1 and(m⁰−1) (q−1) +1< p 1+ ^m_n⁰. If this is the case, then we have p≤ γ< p 1+^m_n⁰_{. By Lemma4.2} and (4.10), we derive that

Z

Ω|u_k(t)|^γlog|u_k(t)|dx≤ εk∇u_k(t)k^p_p+C(ε)ku_k(t)k^m_m⁰0

κ

+C(_Ω,ε), (4.14) whereκ >1 andε>0, which implies

I(u_k(_t))≥ 1−(_m⁰−1)ε

k∇u_k(_t)k^p_p−C(ε)ku_k(_t)k^m_m⁰0

κ

−C(_Ω,ε). (4.15) By choosingε= _p(m^p−0−¹1), it follows from (4.1), (4.8) and (4.15) that

S_k(t)≤C₁+C₂ Z _t

0

S^κ_k(τ)dτ, ∀t∈ [0,T], (4.16) whereκ >1 andC₁,C₂ are positive constants independence ofk, and

S_k(t) =ku_k(t)k^m_m⁰0+ ¹ p

Z _t

0

k∇u_k(τ)k^p_pdτ. (4.17) By virtue of Gronwall–Bellman–Bihari integral inequality, Lemma4.1, there exists a constant T^∗ =1/2(κ−1)C2C₁^κ−1 ∈(0,T)such that

S_k(t)≤C_T^∗, ∀t∈[0,T^∗], ∀k∈_N. (4.18) Now, by multiplying thei^th equation of (4.3) by α⁰_ki(t), summing up with respect toiand integrating with respect to time variable from 0 tot, we obtain

Z _t

0

kU_kτ(τ)k²_L2(_Ω)dτ+J(u_k(t))≤ J(u_0k), ∀t ∈[0,T], (4.19) where U_k(t) = ²

√m⁰−1

m⁰ |u_k(t)|^m

0

2 . Thanks to (4.1) and the continuity of J, there is a positive constantCsuch that

J(u_0k)≤C, ∀k∈ _N. (4.20)

We now estimateJ(u_k(t)). It is worth noting that J(u_k(t)) =

1 p− ¹

γ

k∇u_k(t)k^p_p+ ^m

0−₁

γ² ku_k(t)k^γ_γ+ ¹

γI(u_k(t)). On the other hand, it follows from (4.12)–(4.13) and (4.15)–(4.18) that

I(u_k(t))≥ 1−ε(m⁰−1)k∇u_k(t)k^p_p−C, for sufficiently smallε >0. Hence, we get

J(u_k(t))≥ 1

p −^ε(m⁰−1) γ

k∇u_k(t)k^p_p+ ^m

0−1

γ² ku_k(t)k^γ_γ−C. (4.21) It follows from (4.19)-(4.21) that

Z _t

0 kU_kτ(τ)k²₂dτ+ 1

p −^ε(m⁰−1) γ

k∇u_k(t)k^p_p+ ^m

0−1

γ² ku_k(t)k^γ_γ ≤C, (4.22) for sufficiently smallε >0.

Step 3: Passage to the limit

In this section, we use some compactness results which is given by Matas and Merker [23].

Lemma 4.3([23]). Let m, p satisfy(1.4), then we have

(i) the map ϕ : W₀^1,p(_Ω)∩L^m0(_Ω) → L^m(_Ω⁰) defined by ϕ(u) = u^(m0−1) is compact for any arbitrary bounded subdomainΩ⁰ ⊂_Ω.

(ii) Let {u_k} ⊂ L^p 0,T;W₀^1,p(_Ω)∩L^∞ 0,T;L^m0(_Ω) be the sequence of weak solutions of pro- jected equations. Then{ϕ(u_k)}is relatively compact in L¹(0,T;L^m(_Ω)).

From the priori estimates devired above (see (4.13), (4.18) and (4.22)), we deduce a subse- quence that still denotes as{u_k}such that

u_k →u weakly in L^p

0,T;W₀^1,p(_Ω), (4.23)

u_k →u weakly star in L^∞

0,T;W₀^1,p(_Ω)∩L^m0(_Ω), (4.24) d

dtU_k → d

dtU

ex

weakly in L² 0,T;L²(_Ω), (4.25) ϕ(u_k)→(ϕ(u))_ex weakly star in L^∞(0,T;L^m(_Ω)), (4.26)

∆pu_k → _∆pu

ex weakly in L^p0

0,T;W^−1,p0(_Ω). (4.27) It is obviously to deduce from (4.23)–(4.25) that _dt^dU

ex = _dt^dU. By virtue of Lemma 4.3, it follows from (4.23)–(4.24) that ϕ(u_k)is bounded inL^∞(0,T;L^m(_Ω))and is relatively compact in L¹(0,T;L^m(_Ω)). By monotone operator theory, using similar arguments as in [23], we get (ϕ(u))_ex = ϕ(u)_and

ϕ(u_k)−→ϕ(u) strongly in L¹(0,T;L^m(_Ω)) and a.e. in Q_T = _Ω×[0,T]. (4.28)

This implies

u_k(x,t)→u(x,t)a.e. in Q_T which implies f_γ(u_k(x,t))→ f_γ(u(x,t))a.e. in Q_T. (4.29) On the other hand, direct computation gives us

Z

Ω|f_γ(u_k(t))|^γ0dx=

Z

|uk(t)|≤1

|f_γ(u_k(t))|^γ0dx+

Z

|uk(t)|>1

|f_γ(u_k(t))|^γ0dx

≤ e^−γ0|_Ω|+C Z

Ω|u_k(t)|^γ+εγ0dx.

By the Poincaré inequality and Lemma4.2, it is not difficult to see that Z

Ω|f_γ(u_k(t))|^γ0dx≤C_T,Ω forγ>1. (4.30) Combining (4.29) and (4.30), we get

fγ(u_k)−→ fγ(u) weakly star in L^∞

0,T;L^γ0(_Ω). (4.31) Letk →_∞in (4.2)–(4.3), we obtain

Z _T

0

h∂_tϕ(u)_,widt+

Z _T

0

D(∇u)^p−1,wE

dt= m⁰−₁

Z _T

0

hf_γ(u),widt, (4.32) for all w∈ L^p 0,T;W₀^1,p(_Ω).

Moreover, ifm⁰ ≥2, then we have 1< _m^m0−⁰1 ≤2 and∂tϕ(u)∈ L ^m

0

m0 −₁ (Q_T), since Z

Ω|∂_tϕ(u(t))| ^m

0 m0 −1 dx=

Z

Ω

m⁰−1

|u(t)|^m0−2u_t(t)

m0 m0 −1

dx

= ^m

0√ m⁰−1

2

! ^m

0 m0 −1 Z

Ω|U_t(t)| ^m

0

m0 −1 |u(t)|^m

0(m0 −2) 2(m0 −1) dx

≤ ^m

0√ m⁰−1

2

! ^m

0 m0 −₁ Z

Ω|Ut(t)|²dx ^m

0 2(m0 −1)Z

Ω|u(t)|^m0dx ₂₍^m_m^{0 −}_{0 −}²₁₎

= ^m

0√ m⁰−1

2

! ^m

0 m0 −1

kU_t(t)k²₂

m0 2(m0 −1)

ku(t)k^m_m⁰0

^{m0 −2}

2(m0 −1)

≤ C

kU_t(t)k²₂+ku(t)k^m_m⁰0

.

Here we use the well-known Hölder and Young inequalities. If 1 < m⁰ < 2, then we have u_t ∈L^m0(Q_T),

Z

Ω|u_t(t)|^m0dx=

m⁰ 2√

m⁰−1 m⁰Z

Ω(|U_t(t)|)^m0|u(t)|⁽

2−m0)m0

2 dx

≤

m⁰ 2√

m⁰−1

m⁰_Z

Ω|U_t(t)|²dx ^m

0

2 _Z

Ω|u(t)|^m0dx ^2−m

0 2

≤

m⁰ 2√

m⁰−1 m⁰

m⁰ 2

Z

Ω|Ut(t)|²dx+²−m⁰ 2

Z

Ω|u(t)|^m0dx

.