Blow-up phenomena for a pseudo-parabolic system with variable exponents

Blow-up phenomena for a pseudo-parabolic system with variable exponents

Qi Qi, Yujuan Chen

and Qingshan Wang

School of Science, Nantong University, Nantong, 226019, P.R. China Received 16 March 2017, appeared 11 May 2017

Communicated by Maria Alessandra Ragusa

Abstract. In this paper, we consider a pseudo-parabolic system with nonlinearities of variable exponent type

(ut−_∆ut−div(|∇u|^m(x)−2∇u) =|uv|^p(x)−2uv² inΩ×(0,T), vt−_∆vt−div(|∇v|^n(x)−2∇v) =|uv|^p(x)−2u²v inΩ×(0,T)

associated with initial and Dirichlet boundary conditions, where the variable exponents p(·),m(·),n(·)are continuous functions onΩ. We obtain an upper bound and a lower bound for blow-up time if variable exponents p(·),m(·),n(·)and the initial data satisfy some conditions.

Keywords: pseudo-parabolic system, blow-up, upper bound, lower bound, variable exponent.

2010 Mathematics Subject Classification: 35B44, 35K55, 35K57.

1 Introduction

In this paper, we consider the initial-boundary value problem















ut−_∆ut−div(|∇u|^m(x)−2∇u) =|uv|^p(x)−2uv² inΩ×(0,T), v_t−_∆v_t−div(|∇v|^n(x)−2∇v) =|uv|^p(x)−2u²v inΩ×(0,T),

u=0, v=0 on∂Ω×(0,T),

u(x, 0) =u0(x), v(x, 0) =v0(x) inΩ,

(1.1)

whereΩis a bounded domain ofR^N (N≥1)with smooth boundary∂Ω, the nonlinear term div(|∇u|^m(x)−2∇u)is calledm(x)-Laplace operator, and the variable exponents p(·),m(·),n(·) are continuous functions onΩ, later specified.

It is well known that nonlinear pseudo-parabolic equations appear in the study of var- ious problems of the hydrodynamics, filtration theory, electrorheological fluids and others

BCorresponding author. Email: nttccyj@ntu.edu.cn

(see [1,4,6]). Recently, Di et al. [2] has been studied the following initial-boundary value problem

ut−ν∆ut−div(|∇u|^m(x)−2∇u) =|u|^p(x)−2u inΩ×(0,T) (1.2) with Dirichlet boundary condition. By means of a differential inequality technique, they obtained an upper bound and a lower bound for blow-up time if variable exponents p(·), m(·) and the initial data satisfy some conditions. Obviously, if ν = 1, m(x) = 2, p(x) = p, (1.2) reduces to the following pseudo-parabolic equation

ut−_∆u−_∆ut =|u|^p−2u inΩ×(0,T). (1.3) As for (1.3), there are many results concerning asymptotic behavior [7,14], the existence and uniqueness [1,13] of solutions, blow-up [8,14] property and so on. Especially, Xu [14] prove that the solutions blow up in finite time inH¹₀(_Ω)-norm. Luo [8] obtain an upper bound and a lower bound of the blow-up rate. More generally, Peng et al. [10] considered the following initial-boundary value problem

u_t−_ν∆ut−div(ρ(|∇u|²)∇u) = f(u) inΩ×(0,T).

A lower bound for blow-up time is determined if blow-up does occur. Furthermore, they establish an upper bound for blow-up time to a special class.

As we know, on the bounds, has been less studied the case of blow-up time to the system (1.1). Our objective in this paper is to study the blow-up phenomenon of solutions of the system (1.1) in the framework of the Lebesgue and Sobolev spaces with variable exponents.

In details, this paper is organized as follows: in Section 2, we introduce the function spaces of Orlicz–Sobolev type and present a brief description of their main properties. In Section 3, a criterion for blow-up to the system (1.1) that leads to the upper bound for blow-up time is obtained. In Section 4, we give the lower bound of blow-up time to the system (1.1).

2 Function spaces

As in [2], we first recall some known results about the Lebesgue and Sobolev spaces with variable exponents (see [3,5,11,12]) which will be needed in this paper.

Letr(·)_:Ω→[_1,∞)be a measurable function, whereΩis a domain ofRⁿ. We denote by r− =ess inf_x∈_Ωr(x)andr+=ess sup_x∈Ωr(x). The variable exponent Lebesgue spaceL^r(·)(_Ω) consists of all measurable functionsudefined onΩfor which

ρ_r(·)=

Z

Ω|u(x)|^r(x)dx<_∞.

The setL^r(·)(_Ω)equipped with the Luxembourg normkuk_r(·)=inf{λ>0 :ρ_r(·)(u/λ)≤1} is a Banach space (see [3]). The variable exponent Sobolev spaceW^1,r(·)(_Ω)is defined by

(W^1,r(·)(_Ω) ={u∈ L^r(·)(_Ω): |∇u(x)|^r(x)∈ L¹(_Ω)}, kuk_W1,r(·)(_Ω) =kuk_1,r(·) =k∇uk_r(·)+kuk_r(·).

W₀^1,r(·)(_Ω) is defined as the closure inW^1,r(·)(_Ω) of C₀^∞(_Ω). W^1,r0(·)(_Ω) is the dual space of W^1,r(·)(_Ω)wherer⁰(·)is the function such that _r(·)¹ + ¹

r⁰(·) =1.

Let the variable exponent p(·)satisfy the Zhikov–Fan conditions:

|p(x)−p(y)| ≤ ^A

log(_|x−¹_y|)^{, for allx,y}∈_Ωwith|x−y|<δ, (2.1) where A>0 and 0< δ<1.

Now, we present some useful lemmas which will be used later.

Lemma 2.1(see [3,5]). We have the following results.

(1) If Ωhas a finite measure and q₁(·), q₂(·)are variable exponents satisfying q₁(x) ≤q₂(x)almost everywhere inΩ, then there is a continuous embedding from L^q2(·)(_Ω),→ L^q1(·)(_Ω).

(2) Let the variable exponent p(·)satisfy (2.1), then kuk_p(·) ≤ Ck∇uk_p(·) for all u ∈ W₀^1,p(·)(_Ω), whereΩis bounded.

(3) Let the variable exponents q₁(·)∈C(_Ω), q₂: Ω→[1,∞)be a measurable function and satisfy

ess inf

x∈_Ω

(q₁^∗(x)−q₂(x))>0, where q^∗₁ =







nq₁(x)

n−q₁(x)^{, if q1}(x)<n, +_∞, if q₁(x)≥n.

Then, the Sobolev embedding W₀^1,q1(·)(_Ω),→L^q2(·)(_Ω)is continuous and compact.

3 Upper bound for blow-up time

Since p(·)_,m(·)_,n(·)are continuous functions onΩ, we denote by

`₊=_max

Ω¯ `(x)<sub>,</sub> `₋=_min

Ω¯ `(x) where` stands forp(·), m(·)andn(·)respectively. Assume that

p−>max{m+,n+}, min{m−,n−} ≥2, (3.1) and

m+≥ n−, n+≥m−. (3.2)

Firstly, we start with the following local existence theorem for the solutions of system (1.1) which can be obtained by Faedo–Galerkin method.

Theorem 3.1. Let the variable exponent p(·)satisfy the Zhikov–Fan conditions(2.1) and(3.1)hold.

Then for any u₀ ∈ W₀^1,m(·)(_Ω)∩L^p(·)(_Ω), v₀ ∈ W₀^1,n(·)(_Ω)∩L^p(·)(_Ω), there exists a number T₀ ∈ (0,T]such that the system(1.1)has a unique solution

u∈ L^∞([0,T0];W₀^1,m(·)(_Ω)∩L^p(·)(_Ω)), ut ∈ L²([0,T0];W₀^1,2(_Ω)), v∈ L^∞([0,T₀];W₀^1,n(·)(_Ω)∩L^p(·)(_Ω)), v_t∈ L²([0,T₀];W₀^1,2(_Ω)),

satisfying

(u_t,ϕ) + (∇u_t,∇ϕ) + (|∇u|^m(x)−2∇u,∇ϕ) = (|uv|^p(x)−2uv²,ϕ),

∀ϕ∈W₀^1,m(·)(_Ω)∩L^p(·)(_Ω),

(vt,ψ) + (∇vt,∇ψ) + (|∇v|^n(x)−2∇v,∇ψ) = (|uv|^p(x)−2u²v,ψ),

∀ψ∈W₀^1,n(·)(_Ω)∩L^p(·)(_Ω),

(3.3)

where(u_t,ϕ) =R

Ωu_tϕdx.

Next, we seek the upper bound for the blow-up time of the system (1.1).

Theorem 3.2. Assume that (2.1), (3.1) and (3.2) hold. Let u₀ ∈ W₀^1,m(·)(_Ω)∩ L^p(·)(_Ω), v₀ ∈ W₀^1,n(·)(_Ω)∩L^p(·)(_Ω)such thatku₀k_H1

0,kv₀k_H1

0 >₀and Z

Ω

"

|u₀v₀|^p(x)

p(x) − |∇u₀|^m(x)

m(x) + |∇v₀|^n(x) n(x)

!#

dx≥0. (3.4)

Then, the solution(u,v)of the system(1.1)blows up in finite time T^∗ in H₀¹(_Ω)-norm. Moreover, an upper bound for blow-up time is given by

T^∗ ≤ ^b(F(0))^1−1b

(b−1)β , (3.5)

whereβand b are suitable positive constants given later and F(0) =ku₀k²

H₀¹+kv₀k²

H₀¹. Proof. Replacing ϕbyu_t,ψbyv_t in (3.3) respectively, and adding, we have

Z

Ω(|ut|²+|∇ut|²+|vt|²+|∇vt|²)dx+ ^d dt

Z

Ω

1

m(x)|∇u|^m(x)+ ¹

n(x)|∇v|^n(x)

dx

= ^d dt

Z

Ω

1

p(x)|uv|^p(x)dx. (3.6)

Let us define the energy as follows E(t) =

Z

Ω

1

m(x)|∇u|^m(x)+ ¹

n(x)|∇v|^n(x)− ¹

p(x)|uv|^p(x)

dx. (3.7)

Hence, by (3.6) and (3.7), we have E⁰(t) =−

Z

Ω(|u_t|²+|∇u_t|²+|v_t|²+|∇v_t|²)dx ≤0. (3.8) We define an auxiliary function

F(t) =

Z

Ωu²dx+

Z

Ω|∇u|²dx+

Z

Ωv²dx+

Z

Ω|∇v|²dx. (3.9) Multiplyingu andv on two sides of two equations of the system (1.1) respectively, and inte- grating by part, we have

Z

Ωuutdx+

Z

Ω∇u· ∇utdx+

Z

Ω|∇u|^m(x)dx=

Z

Ω|uv|^p(x)dx (3.10)

and

Z

Ωvvtdx+

Z

Ω∇v· ∇vtdx+

Z

Ω|∇v|^n(x)dx=

Z

Ω|uv|^p(x)dx. (3.11) Adding (3.10) and (3.11), we get

Z

Ωuu_tdx+

Z

Ω∇u· ∇u_tdx+

Z

Ωvv_tdx+

Z

Ω∇v· ∇v_tdx

=−

Z

Ω(|∇u|^m(x)+|∇v|^n(x))dx+2 Z

Ω|uv|^p(x)dx. (3.12) By differentiatingF(t)with respect tot, we have

F⁰(t) =2 Z

Ωuutdx+2 Z

Ω∇u· ∇utdx+2 Z

Ωvvtdx+2 Z

Ω∇v· ∇vtdx

=4 Z

Ω|uv|^p(x)dx−2 Z

Ω(|∇u|^m(x)+|∇v|^n(x))dx

=₄

Z

Ωp(x)

"

|uv|^p(x)

p(x) − |∇u|^m(x)

m(x) +|∇v|^n(x) n(x)

!#

dx+₄

Z

Ωp(x) 1

m(x)− ¹ p(x)

|∇u|^m(x)dx +4

Z

Ωp(x) 1

n(x)− ¹ p(x)

|∇v|^n(x)dx+2 Z

Ω(|∇u|^m(x)+|∇v|^n(x))dx. (3.13) Thanks to E⁰(t)≤0, we have

Z

Ωp(x)

"

|uv|^p(x)

p(_x) − |∇u|^m(x)

m(_x) + |∇v|^n(x) n(_x)

!#

dx

≥

Z

Ωp(x)

"

|u₀v₀|^p(x)

p(x) − |∇u₀|^m(x)

m(x) + |∇v₀|^n(x) n(x)

!#

dx

≥

Z

Ωp−

"

|u₀v₀|^p(x)

p(x) − |∇u₀|^m(x)

m(x) +|∇v₀|^n(x) n(x)

!#

dx

≥0. (3.14)

By (3.13) and (3.14), we see F⁰(t)≥4

Z

Ωp−

1 m+

− ¹ p−

|∇u|^m(x)dx+4 Z

Ωp−

1 n+

− ¹ p−

|∇v|^n(x)dx+2 Z

Ω(|∇u|^m(x)+|∇v|^n(x))dx

=C₁ Z

Ω|∇u|^m(x)dx+C₂ Z

Ω|∇v|^n(x)dx, whereC₁=2+4p−(_m¹

+ − _p¹

−),C2 =2+4p−(_n¹

+ − _p¹

−). Define the setsΩ+={x ∈_Ω| |∇u| ≥ 1,|∇v| ≥1}andΩ− = {x ∈ _Ω| |∇u|< 1,|∇v| <1}. By the fact that k∇uk₂ ≤ Ck∇uk_r for allr≥2, it follows

F⁰(t)≥ C_{1 Z}

Ω−

|∇u|^m+dx+

Z

Ω⁺|∇u|^m−dx

+C_{2 Z}

Ω−

|∇v|ⁿ⁺dx+

Z

Ω⁺|∇v|ⁿ⁻dx

≥ C₃

"

_Z

Ω−

|∇u|²dx ^m₂⁺

+ _Z

Ω+

|∇u|²dx ^m₂⁻#

+C₄

"

_Z

Ω−

|∇v|²dx ⁿ₂⁺

+ _Z

Ω+

|∇v|²dx ⁿ₂⁻#

.

This implies that

(F⁰(t))^a ≥C5

Z

Ω−

(|∇u|²+|∇v|²)dx≥0, (F⁰(t))^b ≥C6

Z

Ω+

(|∇u|²+|∇v|²)dx≥0,

(3.15)

where a = max(_m²

+,_n²

+), b = max(_m²

−,_n²₋). The Poincaré inequality gives k∇uk²₂ ≥ λ₁kuk²₂, whereλ₁is the first eigenvalue of the problem

(4ω+λω =0, in Ω, ω =0, on ∂Ω.

Thus, the follow relations

k∇uk²₂ = ¹

1+λ₁k∇uk²₂+ ^λ1

1+λ₁k∇uk²₂

≥ ^λ1

1+λ₁kuk²₂+ ^λ1

1+λ₁k∇uk²₂ = ^λ1 1+λ₁kuk²

H¹₀, k∇vk²₂ = ¹

1+λ₁k∇vk²₂+ ^λ1

1+λ₁k∇vk²₂

≥ ^λ1

1+λ₁kvk²₂+ ^λ1

1+λ₁k∇vk²₂ = ^λ1

1+λ₁kvk²_H1 0

(3.16)

hold, wherekuk_p = (R

Ωu^pdx)^1p _andkuk²

H¹₀ =kuk²₂+k∇uk²₂. Combining (3.15) and (3.16), we conclude

(F⁰(t))^a ≥ ^C5λ1 1+λ₁(kuk²

H₀¹+kvk²

H¹₀), (F⁰(t))^b≥ ^C6λ1

1+λ₁(kuk²

H₀¹+kvk²

H¹₀). Consequently,

(F⁰(t))^a+ (F⁰(t))^b≥ ^λ1(C₅+C₆) 1+λ₁ (kuk²

H¹₀ +kvk²

H¹₀) =C₇F(t), (3.17) which implies

(F⁰(t))^b1+ (F⁰(t))^a−b≥C₇F(t). (3.18) By (3.17) and the fact thatF(t)≥F(0)>0(F⁰(t)≥0), we have

(F⁰(t))^a ≥ ^C7

2 F(t)≥ ^C7 2 F(0) or

(F⁰(t))^b≥ ^C7

2 F(t)≥ ^C7 2 F(₀)_, which implies that

F⁰(t)≥C₈(F(0))^1a or

F⁰(t)≥C₉(F(₀))^1b_.

Therefore, we have that F⁰(t) ≥ α, where α = min{C₈(F(0))^1a,C₉(F(0))^1b}. From (3.2), it is easy to see a−b≤0. So, combining with (3.18), we get

F⁰(t)≥β(F(t))^1b_{, (3.19)} where the constant β= ^C7

1+α^a−b

¹_b

. By (3.19), we receive F⁰(t)

(F(t))^1b

≥ β. (3.20)

Integrating the inequality (3.20) from 0 to t, we see

(F(t))^1−1b ≤(F(0))^1−1b + (b−1)βt

b , (3.21)

which implies that

F(t)≥ ¹

[(F(₀))^1−1b +^(b−_b^1)βt]^{1−bb. (3.22)} Thus, (3.22) shows thatF(t)blows up at some finite timeT^∗ such that

T^∗ ≤ ^b(F(0))^1−1b

(b−1)β . (3.23)

Finally, we get the solution (u,v)blows up in H¹₀(_Ω)-norm in finite time.

Remark 3.3. From (3.23), we see that the larger F(0)is, the smaller the blow-up timeT^∗ is.

4 Lower bound for blow-up time

In this section, our aim is to determine a lower bound for blow-up time of the system (1.1).

The technique is the same as [2].

Theorem 4.1. Suppose that (2.1) and(3.1) hold. Furthermore assume that2 < p+ < _∞ if n ≤ 2, 2 < p+ ≤ _n²ⁿ₋₂ if n ≥ 3, u0 ∈ W₀^1,m(·)(_Ω)∩L^p(·)(_Ω), v0 ∈W₀^1,n(·)(_Ω)∩L^p(·)(_Ω)and the solution (u,v)of the system(1.1)becomes unbounded at finite time T^∗ in H₀¹(_Ω)-norm, then a lower bounded T^∗ for blow-up time is given by

T^∗ ≥

Z _∞

F(0)

dη

Mη^p++Nη^p−, (4.1)

where M and N are suitable positive constants given later and F(0) =ku₀k²

H₀¹+kv₀k²

H¹₀. Proof. We define the function F(t)the same as (3.9). By (3.13), it is easy to get

F⁰(t) =2 Z

Ωuu_tdx+2 Z

Ω∇u· ∇u_tdx+2 Z

Ωvv_tdx+2 Z

Ω∇v· ∇v_tdx

≤4 Z

Ω|uv|^p(x)dx. (4.2)

Let us denote the setsΩ+ = {x ∈ _Ω | |uv| ≥ 1} andΩ− = {x ∈ _Ω | |uv| < 1}. Using the Cauchy–Schwarz inequality and the Sobolev embedding inequalities , we get

Z

Ω|uv|^p(x)dx ≤

Z

Ω+

|uv|^p+dx+

Z

Ω−

|uv|^p−dx

≤

Z

Ω|uv|^p+dx+

Z

Ω|uv|^p−dx

≤ _Z

Ω|u|^{2p+ 1}₂

· _Z

Ω|v|^{2p+ 1}₂

+ _Z

Ω|u|^{2p− 1}₂

· _Z

Ω|v|^{2p− 1}₂

≤(B₊^p+)²k∇uk₂^p+· k∇vk₂^p++ (B₋^p−)²k∇uk₂^p−· k∇vk₂^p−, (4.3) where B+,B− are the Sobolev embedding constants for H₀¹(_Ω) ,→ L^p+(_Ω) and H₀¹(_Ω) ,→ L^p−(_Ω), respectively. From the Cauchy–Schwarz inequality, we have

F⁰(t)²≥ _Z

Ω|∇u|²dx 2

+ _Z

Ω|∇v|²dx 2

≥ 2 Z

Ω|∇u|²dx·

Z

Ω|∇v|²dx.

Then

(F⁰(t))^p+ ≥2^p2+ Z

Ω|∇u|²dx ^p₂⁺

· Z

Ω|∇v|²dx ^p₂⁺

and

(F⁰(t))^p− ≥2^p2− _Z

Ω|∇u|²dx ^p₂⁻

· _Z

Ω|∇v|²dx ^p₂⁻

, which implies that

(F⁰(t))^p+·2^−p+2 ≥ k∇uk₂^p+· k∇vk₂^p+ (4.4) and

(F⁰(t))^p−·2^−p−2 ≥ k∇uk₂^p−· k∇vk₂^p−. (4.5) Thus, the combination of (4.2)–(4.5) implies that

F⁰(t)≤ M(F(t))^p++N(F(t))^p−, where M=2^−p2+(B₊^P+)², N=2^−p2−(B₊^P−)². Therefore

F⁰(t)

M(F(t))^p++N(F(t))^p− ≤1. (4.6) Integrating the inequality (4.6) from 0 to t, we get

Z _F(t) F(0)

dη

Mη^p++Nη^p− ≤t.

If(u,v)blows up in H¹₀(_Ω)-norm, then we obtain a lower boundT^∗ given by T^∗ ≥

Z _∞

F(0)

dη Mη^p+ +Nη^p−. Clearly, the integral is bound since exponentsp+≥ p−>2.

Acknowledgements

This work was supported by 2016’s College Students’ innovative projects in P. R. China. And the authors would also like to express their gratitude to the editor and anonymous reviewers for their helpful comments to greatly improve the readability of the paper.

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