# Blow-Up Results for a Class of Quasilinear Parabolic Equation with Power Nonlinearity and Nonlocal Source

Authors: Xiaorong Zhang; Zhoujin Cui
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper deals with a class of quasilinear parabolic equation with power nonlinearity and nonlocal source under homogeneous Dirichlet boundary condition in a smooth bounded domain; we obtain the blow-up condition and blow-up results under the condition of nonpositive initial energy.


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## Body

## 1. Introduction

In this paper, we consider the following quasilinear parabolic equation with power nonlinearity and nonlocal source term:(1)ut=Δpu+μup∫Ωup+1y,tdy−kup−2u,x,t∈Ω×0,T,ux,t=0,x,t∈∂Ω×0,T,ux,0=u0x,x∈Ω,where Ω⊂RNN≥1 is a bounded domain with smooth boundary ∂Ω and Δpu=div∇up−2∇u is the standard p-Laplace operator with p>2, μ,k>0, u0x∈W01,pΩ\0.In the past decades, many physical phenomena have been expressed as nonlocal mathematical models (see [1, 2]). It is also suggested that the nonlocal growth term provides a more realistic model for the physical model of compressible reaction gas. Problem (1) appears in the study of fluid flow through porous media with integral source (see [3, 4]) and population dynamics (see [5, 6]). Actually, equations of the above form are mathematical models occurring in studies of the p-Laplace equation ([7–15] and references therein), generalized reaction-diffusion theory [16], non-Newtonian fluid theory [17, 18], non-Newtonian filtration theory [19, 20], and the turbulent flow of a gas in porous medium [7]. Media with p>2 are called dilatant fluids and those with p<2 are called pseudoplastics. If p=2, they are Newtonian fluids. When p≠2, the problem becomes more complicated since certain nice properties inherent to the case p=2 seem to be lose or at least difficult to verify.Blow-up results of parabolic equations with nonlocal sources have been studied as well. For example, the problem of the form(2)ut=div∇up−2∇u+∫Ωuqdx,x,t∈Ω×0,Tux,t=0,x,t∈∂Ω×0,Tux,0=u0x,x∈Ω,was studied by Li and Xie [7]. They established global existence of solutions and discussed the blow-up properties of solutions.The authors in [8] studied the following p-Laplacian equation with power nonlinearity(3)ut−div∇up−2∇u+up−2u=up−2ulnu,x∈Ω,t>0,ux,t=0,x,t∈∂Ω×0,T,ux,0=u0x,x∈Ω.By using an efficient technique and according to some sufficient conditions, the global existence and decay estimates of solutions under some sufficient conditions are discussed.The authors in [9] considered the Neumann problem to the following initial parabolic equation with logarithmic source:(4)ut−div∇up−2∇u=up−2ulogu−∮Ωup−2ulogudx,x∈Ω,t>0,∂ux,t∂η=0,x∈∂Ω,t>0,ux,0=u0x,x∈Ω,in a bounded domain with smooth boundary, p>2. By using the logarithmic Sobolev inequality and potential wells method, they obtain the decay, blow-up, and nonextinction of solutions under some conditions.In [10], the following model of a quasilinear diffusion equation with interior logarithmic source has been studied:(5)ut−div∇up−2∇u=up−2ulogu,x∈Ω,t>0,ux,t=0,x∈∂Ω,t>0,ux,0=u0x,x∈Ω,in which p>2, u0x∈W01,pΩ\0. By using the potential well method and a logarithmic Sobolev inequality, the authors obtained results of existence or nonexistence of global weak solution. They also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions. Among some other interesting results, they showed that the weak solution ux,t of problem (5) blows up at finite time under the condition Ju0≤M and Iu0<0, where M>0 is a constant; the energy functional Ju and Nehari functional Iu are defined as follows:(6)Ju=1p∇upp−1p∫Ωuplogudx+1p2upp,Iu=∇upp−∫Ωuplogudx,in which ·p=∫Ω·pdx1/p.Motivated by the above studies, in this paper, we investigate blow-up results of problem (1). We will give the conditions for the blow-up results and establish the lower bounds for the blow-up rate. Our main results are as follows.Theorem 1.
Assume thatJ1u0<0. Then, the weak solution u=ux,t of problem (1) blows up at finite time.Theorem 2.
Assume thatu0∈H01Ω and J1u0<0, let u=ux,t be the nonnegative solution of problem (1), then u blows up in finite time(7)T≔u022−p4p+4J1u0.Theorem 3.
Assume thatu0∈H01Ω, J1u0≤0, ∫0t0us·,s22ds>0 for any t0>0, then, the weak solution u=ux,t of problem (1) blows up at infinity. Moreover, if u02≤−Ju02/p, the lower bound for blow-up rate can be estimated by(8)u22≥u022.


## 2. Criterions of Blow-Up

### 2.1. Preliminaries

In this section, we start with the definition of weak solution and blow-up at infinity of (1).Definition 4 (weak solution).
LetT>0. A function u=ux,t∈L∞0,T;X0 with ut∈Lp′0,T;W−1,p′Ω∩L20,T;L2Ω is called a weak solution to problem (1) in Ω×0,T, if ux,0=u0x∈X0 and ux,t satisfies (1) in the sense of distribution, i.e.,(9)ut,ω+∇up−2∇u,∇ω+kup−2u,ω=μup∫Ωup+1y,tdy,ω,for all ω∈W01,pΩ,t∈0,T, where X0=W01,pΩ\0, W−1,p~Ω to denote the dual space of W01,pΩ, and p~ is Holder conjugate exponent of p>1.Definition 5 (blow-up at infinity).
Letux,t be a weak solution of (1), we call ux,t blow-up at +∞ if the maximal existence time T=+∞ and(10)limt⟶+∞u·,t2=∞.
To obtain the blow-up results, define the potential energy functional and the Nehari’s functional as follows:(11)J1u=1p∇upp+kpupp−μ2p+2∫Ω∫Ωup+1x,tup+1y,tdxdy,I1u=∇upp+kupp−μ∫Ω∫Ωup+1x,tup+1y,tdxdy.
To prove the main result, we need the following lemmas.Lemma 6.
Assume thatux,t is a weak solution of (1). Then, J1u is nonincreasing with respect to t and satisfies the energy inequality(12)∫0tus·,s22ds+J1u≤J1u0,t∈0,T0.Proof.
Similar to the proof in [8, 10, 11], we can get the result.Lemma 7.
[8] J1u is a nonincreasing function, for t≥0,(13)J1′u=−ut2≤0.Lemma 8 [12].
LetΦ be a positive, twice differentiable function satisfying the following conditions:(14)Φt¯>0,Φ′t¯>0,for some t¯∈0,T, and the inequality(15)ΦtΦ′′t−αΦ′t2≥0,∀t∈t¯,T,where α>1. Then, we have(16)Φt≥1Φ1−αt¯−Φ~t−t¯1/α−1,t∈t,T∗,in which Φ~ is a positive constant, and(17)T∗=t¯+Φt¯α−1Φ′t¯.
This implies(18)limt⟶T∗Φt=+∞.Lemma 9 [9].
Suppose thatθ>0, α>0, β>0, and ht is a nonnegative and absolutely continuous function satisfying h′t+αhθt≥β, then for 0<t<+∞, it holds(19)ht≥minh0,βα1/θ.


### 2.2. Proof of the Main Results

We will consider the finite time blow-up results of problem (1) under the condition of nonpositive initial energy. The theorems are proved as follows.Proof of Theorem1.
Assume thatux,t is the weak solution of problem (1), for any T>0, we define the functional(20)Γt=∫0tu·,s22ds+T−tu022,t∈0,T.
It is obvious thatΓt>0 for all t∈0,T. Since Γ is continuous, there exists ρ>0 (independent of T) such that Γt>ρ for all t∈0,T.
Then, we have(21)Γ′t=u·,t22−u022,(22)Γ′′t=2∫Ωutudx=−2I1u≥−2pJ1u.
By using (12) in Lemma 7, we have(23)−2pJ1u≥2p∫0tus·,s22ds−2pJ1u0.
FromJ1u0<0, (22) and (23), we get(24)Γ′′t≥2p∫0tus·,s22ds−2pJ1u0>2p∫0tus·,s22ds.
Now, multiplying (24) by Γt, we have(25)Γ′′tΓt>2p∫0tus·,s22dsΓt=2p∫0tus·,s22ds∫0tu·,s22ds+2pT−tu022∫0tus·,s22ds.
Noticing that(26)Γ′t=u·,t22−u022=∫0tddsu·,s22ds=2∫0t∫Ωusudxds,then(27)Γ′t2=4∫0t∫Ωusudxds2.
With the help of Cauchy-Schwarz inequality, we have(28)Γ′t2≤4∫0tus·,s22ds∫0tu·,s22ds.
Using (25) and (28), we further get(29)Γ′′tΓt−p2Γ′t2>2pT−tu022∫0tus·,s22ds>0,for all t∈0,T.
By Lemma8, there exists T∗>0 such that(30)limt⟶T∗Γt=+∞,which implies(31)limt⟶T∗∫0tu·,s22ds=+∞.
As a consequence, we get(32)limt⟶T∗u·,t22ds=+∞,this means ux,t blows up at finite time T∗.Proof of Theorem2.
SetGt=u·,s22, then(33)G′t=2∫Ωutudx=2−∇upp+μ∫Ω∫Ωup+1x,tup+1y,tdxdy−kupp=−2I1u≥Ft,in which(34)Ft≔−4p+4J1u=−4p+4p∇upp−k4p+4pupp+2μ∫Ω∫Ωup+1x,tup+1y,tdxdy.
By Lemma6, we can get(35)F′t=−4p+4ddtJ1u=4p+4∫Ωut2dx>0.
From (35) and the condition J1u0<0, we can get Ft>0. Then, by Cauchy-Schwarz inequality and (21), the following inequality can be obtained:(36)GtF′t=4p+4∫Ωu2dx∫Ωut2dx≥4p+4∫Ωuutdx2=p+1G′t2≥p+1G′tFt,that is,(37)F′tFt≥p+1G′tGt.
Integrating (37) on 0,t, we can get(38)lnFt−lnF0≥p+1lnGt−p+1lnG0,that is,(39)FtGtp+1≥F0G0p+1.
By (33), we have(40)G′tGtp+1≥F0G0p+1.
By integrating (40) on 0,t, we can get(41)1Gtp≤1G0p−pF0G0p+1t=G0−pF0tG0p+1.
Take the reciprocal of (41) to get(42)Gtp≥G0p+1G0−pF0t.
Let(43)T=G0pF0=u022−p4p+4J1u0,when t⟶T−, we can get Gtp⟶+∞; this means that u blows up in a finite time.Proof of Theorem3.
Assume thatux,t is the weak solution of problem (1). Set Gt=u·,s22, then(44)G′t=2∫Ωutudx=−2I1u≥−2pJ1u,
By (12) in Lemma 6 and the condition J1u0≤0, we can get(45)−2pJ1u≥2p∫0tus·,s22ds.
Then, by (44) and (45), we have(46)G′t≥2p∫0tus·,s22ds>0.
Fixt0>0 and let κ=∫0tu22ds. By the condition ∫0t0us·,s22ds>0, we can get κ, which is a positive constant. Integrating (46) over t0,t, we can get(47)Gt≥Gt0+2p∫t0t∫0tus·,s22dsdτ≥Gt0+2p∫t0tκdτ≥2pκt−t0.
Hence, we have(48)limt⟶+∞Gt=∞.
This means that the weak solutionu=ux,t of problem (1) blows up at infinity.
From (12) and (44), we have(49)G′t≥−2pJ1u≥−2pJ1u0,G′t+Gp/2t≥−2pJ1u0≥−J1u0,for p>2.
By Lemma9, consider J1u0<0 and u022≤−J1u02/p, we have(50)Gt≥minu022,−J1u02/p≥u022,this ends the proof.


## 2.1. Preliminaries

In this section, we start with the definition of weak solution and blow-up at infinity of (1).Definition 4 (weak solution).
LetT>0. A function u=ux,t∈L∞0,T;X0 with ut∈Lp′0,T;W−1,p′Ω∩L20,T;L2Ω is called a weak solution to problem (1) in Ω×0,T, if ux,0=u0x∈X0 and ux,t satisfies (1) in the sense of distribution, i.e.,(9)ut,ω+∇up−2∇u,∇ω+kup−2u,ω=μup∫Ωup+1y,tdy,ω,for all ω∈W01,pΩ,t∈0,T, where X0=W01,pΩ\0, W−1,p~Ω to denote the dual space of W01,pΩ, and p~ is Holder conjugate exponent of p>1.Definition 5 (blow-up at infinity).
Letux,t be a weak solution of (1), we call ux,t blow-up at +∞ if the maximal existence time T=+∞ and(10)limt⟶+∞u·,t2=∞.
To obtain the blow-up results, define the potential energy functional and the Nehari’s functional as follows:(11)J1u=1p∇upp+kpupp−μ2p+2∫Ω∫Ωup+1x,tup+1y,tdxdy,I1u=∇upp+kupp−μ∫Ω∫Ωup+1x,tup+1y,tdxdy.
To prove the main result, we need the following lemmas.Lemma 6.
Assume thatux,t is a weak solution of (1). Then, J1u is nonincreasing with respect to t and satisfies the energy inequality(12)∫0tus·,s22ds+J1u≤J1u0,t∈0,T0.Proof.
Similar to the proof in [8, 10, 11], we can get the result.Lemma 7.
[8] J1u is a nonincreasing function, for t≥0,(13)J1′u=−ut2≤0.Lemma 8 [12].
LetΦ be a positive, twice differentiable function satisfying the following conditions:(14)Φt¯>0,Φ′t¯>0,for some t¯∈0,T, and the inequality(15)ΦtΦ′′t−αΦ′t2≥0,∀t∈t¯,T,where α>1. Then, we have(16)Φt≥1Φ1−αt¯−Φ~t−t¯1/α−1,t∈t,T∗,in which Φ~ is a positive constant, and(17)T∗=t¯+Φt¯α−1Φ′t¯.
This implies(18)limt⟶T∗Φt=+∞.Lemma 9 [9].
Suppose thatθ>0, α>0, β>0, and ht is a nonnegative and absolutely continuous function satisfying h′t+αhθt≥β, then for 0<t<+∞, it holds(19)ht≥minh0,βα1/θ.


## 2.2. Proof of the Main Results

We will consider the finite time blow-up results of problem (1) under the condition of nonpositive initial energy. The theorems are proved as follows.Proof of Theorem1.
Assume thatux,t is the weak solution of problem (1), for any T>0, we define the functional(20)Γt=∫0tu·,s22ds+T−tu022,t∈0,T.
It is obvious thatΓt>0 for all t∈0,T. Since Γ is continuous, there exists ρ>0 (independent of T) such that Γt>ρ for all t∈0,T.
Then, we have(21)Γ′t=u·,t22−u022,(22)Γ′′t=2∫Ωutudx=−2I1u≥−2pJ1u.
By using (12) in Lemma 7, we have(23)−2pJ1u≥2p∫0tus·,s22ds−2pJ1u0.
FromJ1u0<0, (22) and (23), we get(24)Γ′′t≥2p∫0tus·,s22ds−2pJ1u0>2p∫0tus·,s22ds.
Now, multiplying (24) by Γt, we have(25)Γ′′tΓt>2p∫0tus·,s22dsΓt=2p∫0tus·,s22ds∫0tu·,s22ds+2pT−tu022∫0tus·,s22ds.
Noticing that(26)Γ′t=u·,t22−u022=∫0tddsu·,s22ds=2∫0t∫Ωusudxds,then(27)Γ′t2=4∫0t∫Ωusudxds2.
With the help of Cauchy-Schwarz inequality, we have(28)Γ′t2≤4∫0tus·,s22ds∫0tu·,s22ds.
Using (25) and (28), we further get(29)Γ′′tΓt−p2Γ′t2>2pT−tu022∫0tus·,s22ds>0,for all t∈0,T.
By Lemma8, there exists T∗>0 such that(30)limt⟶T∗Γt=+∞,which implies(31)limt⟶T∗∫0tu·,s22ds=+∞.
As a consequence, we get(32)limt⟶T∗u·,t22ds=+∞,this means ux,t blows up at finite time T∗.Proof of Theorem2.
SetGt=u·,s22, then(33)G′t=2∫Ωutudx=2−∇upp+μ∫Ω∫Ωup+1x,tup+1y,tdxdy−kupp=−2I1u≥Ft,in which(34)Ft≔−4p+4J1u=−4p+4p∇upp−k4p+4pupp+2μ∫Ω∫Ωup+1x,tup+1y,tdxdy.
By Lemma6, we can get(35)F′t=−4p+4ddtJ1u=4p+4∫Ωut2dx>0.
From (35) and the condition J1u0<0, we can get Ft>0. Then, by Cauchy-Schwarz inequality and (21), the following inequality can be obtained:(36)GtF′t=4p+4∫Ωu2dx∫Ωut2dx≥4p+4∫Ωuutdx2=p+1G′t2≥p+1G′tFt,that is,(37)F′tFt≥p+1G′tGt.
Integrating (37) on 0,t, we can get(38)lnFt−lnF0≥p+1lnGt−p+1lnG0,that is,(39)FtGtp+1≥F0G0p+1.
By (33), we have(40)G′tGtp+1≥F0G0p+1.
By integrating (40) on 0,t, we can get(41)1Gtp≤1G0p−pF0G0p+1t=G0−pF0tG0p+1.
Take the reciprocal of (41) to get(42)Gtp≥G0p+1G0−pF0t.
Let(43)T=G0pF0=u022−p4p+4J1u0,when t⟶T−, we can get Gtp⟶+∞; this means that u blows up in a finite time.Proof of Theorem3.
Assume thatux,t is the weak solution of problem (1). Set Gt=u·,s22, then(44)G′t=2∫Ωutudx=−2I1u≥−2pJ1u,
By (12) in Lemma 6 and the condition J1u0≤0, we can get(45)−2pJ1u≥2p∫0tus·,s22ds.
Then, by (44) and (45), we have(46)G′t≥2p∫0tus·,s22ds>0.
Fixt0>0 and let κ=∫0tu22ds. By the condition ∫0t0us·,s22ds>0, we can get κ, which is a positive constant. Integrating (46) over t0,t, we can get(47)Gt≥Gt0+2p∫t0t∫0tus·,s22dsdτ≥Gt0+2p∫t0tκdτ≥2pκt−t0.
Hence, we have(48)limt⟶+∞Gt=∞.
This means that the weak solutionu=ux,t of problem (1) blows up at infinity.
From (12) and (44), we have(49)G′t≥−2pJ1u≥−2pJ1u0,G′t+Gp/2t≥−2pJ1u0≥−J1u0,for p>2.
By Lemma9, consider J1u0<0 and u022≤−J1u02/p, we have(50)Gt≥minu022,−J1u02/p≥u022,this ends the proof.


## 3. Conclusions

In this work, we consider the initial boundary value problem for a class of quasilinear parabolic equation with power nonlinearity and nonlocal source under homogeneous Dirichlet boundary condition in a smooth bounded domain. Some new results of blow-up and blow-up time under the condition of nonpositive initial energy are obtained. The blow-up results of problem (1) with arbitrary initial energy will be the direction of further research.



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