# Lower Bounds Estimate for the Blow-Up Time of a Slow Diffusion Equation with Nonlocal Source and Inner Absorption

Authors: Zhong Bo Fang; Rui Yang; Yan Chai
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

We investigate a slow diffusion equation with nonlocal source and inner absorption subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. Based on an auxiliary function method and a differential inequality technique, lower bounds for the blow-up time are given if the blow-up occurs in finite time.


---

## Body

## 1. Introduction

Our main interest lies in the following slow diffusion equation with nonlocal source term and inner absorption term:(1)ut=Δum+up∫Ω‍uqdx-kus,(x,t)∈Ω×(0,t*),(2)u(x,0)=u0(x)≥0,x∈Ω¯,
subject to homogeneous Dirichlet boundary condition(3a)u=0,(x,t)∈∂Ω×(0,t*),
or homogeneous Neumann boundary condition
(3b)∂u∂v=0,(x,t)∈∂Ω×(0,t*),where Ω⊂ℝ3 is a bounded domain with smooth boundary ∂Ω, Ω¯ is the closure of Ω, m>1, p≥0, q>0, s>1, p+q>max{m,s}, k>0, v is the unit outer normal vector on ∂Ω, and t* is the possible blow-up time. By the maximum principle, it follows that u(x,t)≥0 in the time interval of existence. In the present investigation we derive a lower bound for the blow-up time t* when Ω⊂ℝ3 for the solutions that blow up.Equation (1) describes the slow diffusion of concentration of some Newtonian fluids through porous medium or the density of some biological species in many physical phenomena and biological species theories. It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [1–3]. In the nonlinear diffusion theory, there exist obvious differences among the situations of slow (m>1), fast (0<m<1), and linear (m=1) diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situation, whereas an infinite speed propagation exists in the fast diffusion situation.The bounds for the blow-up time of the blow-up solutions to nonlinear diffusion equations have been widely studied in recent years. Indeed, most of the works have dealt with the upper bounds for the blow-up time when blow-up occurs. For example, Levine [4] introduced the concavity method, Gao et al. [5] employed the way of combining an auxiliary function method and comparison method with upper-lower solutions method, and Wang et al. [6] used the regularization method and an auxiliary function method. However, the lower bounds for the blow-up time are more difficult in general. Recently, Payne and Schaefer in [7, 8] used a differential inequality technique and an auxiliary function method to obtain a lower bound on blow-up time for solution of the heat equation with local source term under boundary condition (3a) or (3b). Specially, Song [9] considered the lower bounds for the blow-up time of the blow-up solution to the nonlocal problem (1)-(2) when m=1 and p=0, subject to homogeneous boundary condition (3a) or (3b); for the case k=0, we refer to [10].Motivated by the works above, we investigate the lower bounds for the blow-up time of the blow-up solutions to the nonlocal problem (1)-(2) with homogeneous boundary condition (3a) or (3b). Actually, it is well known that if p+q>max{m,s} and the initial value is large enough, then the solutions of our problem blow up in a finite time; one can see [11]. Unfortunately, our results are restricted in ℝ3 because of the best constant of a Sobolev type inequality (see [12]).This paper is organized as follows. In Section2, we establish problem (1)-(2) with homogeneous Dirichlet boundary condition (3a). Problem (1)-(2) with homogeneous Neumann boundary condition (3b) is considered in Section 3.


## 2. Blow-Up Time for Dirichlet Boundary Condition

In this section, we derive a lower bound fort* if the solution u(x,t)≥0 of (1)–(3a) blows up in finite time t*.Theorem 1.
Letu(x,t) be a classical solution of (1)–(3a) with p+q>max{m,s}; then a lower bound of the blow-up time for any solution which blows up in Ln(p+q-1) norm (n>max{2,(1/(p+q-1))}) is t*≥1/(2A[η(0)]2), where A is a suitable positive constant given later and η(0)=∫Ω‍u0n(p+q-1)dx.Proof.
Define an auxiliary function of the form(4)η(t)=∫Ω‍un(p+q-1)dx,
with
(5)n>max{2,1p+q-1}.
Taking the derivative ofη(t) with respect to t gives
(6)η′(t)=n(p+q-1)∫Ω‍un(p+q-1)-1utdx=n(p+q-1)×∫Ω‍un(p+q-1)-1×(Δum+up∫Ω‍uqdx-kus)dx=n(p+q-1)∫Ω‍un(p+q-1)-1Δumdx+n(p+q-1)∫Ω‍un(p+q-1)+p-1dx∫Ω‍uqdx-nk(p+q-1)∫Ω‍un(p+q-1)+s-1dx=-mn(p+q-1)[n(p+q-1)-1]×∫Ω‍un(p+q-1)+m-3|∇u|2dx+n(p+q-1)∫Ω‍un(p+q-1)+p-1dx∫Ω‍uqdx-nk(p+q-1)∫Ω‍un(p+q-1)+s-1dx=-4mn(p+q-1)[n(p+q-1)-1][n(p+q-1)+m-1]2×∫Ω‍|∇u(n(p+q-1)+m-1)/2|2dx+n(p+q-1)∫Ω‍un(p+q-1)+p-1dx∫Ω‍uqdx-nk(p+q-1)∫Ω‍un(p+q-1)+s-1dx,
where ∇ is the gradient operator.
The application of Hölder inequality to the second term on the right hand side of (6) yields
(7)∫Ω‍uqdx≤|Ω|1-(q/(n+1)(p+q-1))×(∫Ω‍u(n+1)(p+q-1)dx)q/(n+1)(p+q-1),∫Ω‍un(p+q-1)+p-1dx≤|Ω|q/(n+1)(p+q-1)×(∫Ω‍u(n+1)(p+q-1)dx)(n(p+q-1)+p-1)/((n+1)(p+q-1)),
where |Ω| denotes the volume of Ω.
By (7), it follows from (6) that
(8)η′(t)≤-4mn(p+q-1)[n(p+q-1)-1][n(p+q-1)+m-1]2×∫Ω‍|∇u(n(p+q-1)+m-1)/2|2dx+n(p+q-1)|Ω|∫Ω‍u(n+1)(p+q-1)dx-nk(p+q-1)∫Ω‍un(p+q-1)+s-1dx.
Let(9)v=up+q-1,m1=m-1p+q-1,δ=s-1p+q-1;
then
(10)η(t)=∫Ω‍vndx,
and (8) can be written in the from
(11)η′(t)≤-4mn(p+q-1)[n(p+q-1)-1][n(p+q-1)+m-1]2×∫Ω‍|∇v(n+m1)/2|2dx+n(p+q-1)|Ω|×∫Ω‍v(n+1)dx-nk(p+q-1)∫Ω‍vn+δdx.
Now we seek a bound for∫Ω‍vn+1dx in terms of η and the first and third terms on the right in (11). First, the application of Hölder inequality yields
(12)∫Ω‍vn+1dx≤(∫Ω‍vn+δdx)(2n+3m1-4)/(2n+3m1-4δ)×(∫Ω‍v(6n+3m1)/4dx)(4-4δ)/(2n+3m1-4δ).
Using the following Sobolev type inequality (see [12]):
(13)(∫Ω|∅|β‍dx)1/β≤c(∫Ω|∇∅|‍γdx)1/γ,
with β=6, γ=2, and c=41/33-1/2π-2/3, we obtain
(14)∫Ω‍vn+1dx≤(∫Ω‍vn+δdx)(2n+3m1-4)/(2n+3m1-4δ)×[c3/2(∫Ω‍vndx∫Ω‍|∇v(n+m1)/2|2dx)3/4](4-4δ)/(2n+3m1-4δ).
Then for some positive constantμ1 to be determined it follows that
(15)∫Ω‍vn+1dx≤c6(1-δ)/(2n+3m1-4δ)×(μ14(1-δ)/(2n+3m1-4)∫Ω‍vn+δdx)(2n+3m1-4)/(2n+3m1-4δ)×[μ1(∫Ω‍vndx∫Ω‍|∇v(n+m1)/2|2dx)3/4](4-4δ)/(2n+3m1-4δ).
Next, we use the fundamental inequality
(16)a1r1a2r2≤r1a1+r2a2,a1,a2>0,r1,r2>0,r1+r2=1,
to obtain
(17)∫Ω‍vn+1dx≤c6(1-δ)/(2n+3m1-4δ)×[2n+3m1-42n+3m1-4δμ1-(4(1-δ)/(2n+3m1-4))(∫Ω‍vndx∫Ω‍|∇v(n+m1)/2|2dx)3/4×∫Ω‍vn+δdx+4(1-δ)μ12n+3m1-4δ×(∫Ω‍vndx∫Ω‍|∇v(n+m1)/2|2dx)3/4].
Note the fact that, for some positive constant μ2,
(18)[(∫Ω‍vndx)3]1/4(∫Ω|∇v(n+m1)/2|2dx‍)3/4≤14μ23(∫Ω‍vndx)3+3μ24∫Ω‍|∇v(n+m1)/2|2dx.
Substituting inequality (18) into (17) gives
(19)∫Ω‍vn+1dx≤c6(1-δ)/(2n+3m1-4δ)×{2n+3m1-42n+3m1-4δμ1-(4(1-δ)/(2n+3m1-4δ))[(∫Ω‍vndx)3]×∫Ω‍vn+δdx+4(1-δ)μ12n+3m1-4δ×[14μ23(∫Ω‍vndx)3[(∫Ω‍vndx)3]+3μ24∫Ω‍|∇v(n+m1)/2|2dx]}.
Then, by applying inequality (19), it follows from (11) that
(20)η′(t)≤{4mn(p+q-1)[n(p+q-1)-1][n(p+q-1)+m-1]23μ2c6(1-δ)/(2n+3m1-4δ)×(1-δ)μ1n(p+q-1)|Ω|2n+3m1-4δ-4mn(p+q-1)[n(p+q-1)-1][n(p+q-1)+m-1]2}×∫Ω‍|∇v(n+m1)/2|2dx+c6(1-δ)/(2n+3m1-4δ)×(1-δ)μ1n(p+q-1)|Ω|(2n+3m1-4δ)μ23(∫Ω‍vndx)3+[c6(1-δ)/(2n+3m1-4δ)μ1-4(1-δ)/(2n+3m1-4)(2n+3m1-4)n(p+q-1)|Ω|2n+3m1-4δ×(2n+3m1-4)n(p+q-1)|Ω|2n+3m1-4δ(2n+3m1-4)n(p+q-1)|Ω|2n+3m1-4δ-nk(p+q-1)c(6(1-δ))/(2n+3m1-4δ)]∫Ω‍vn+δdx.
We next chooseμ1 to make the coefficient of ∫Ω‍vn+δdx vanish and then choose μ2 to make the coefficient of ∫Ω‍|∇v(n+m1)/2|2dx vanish. It follows that
(21)η′(t)≤A[η(t)]3,
with
(22)A=c6(1-δ)/(2n+3m1-4δ)μ1n|Ω|(1-δ)(p+q-1)(2n+3m1-4δ)μ23.
Integrating inequality (21) from 0 to t gives
(23)1[η(0)]2-1[η(t)]2≤2At,
from which we derive a lower bound fort*:
(24)t*≥12A[η(0)]2.
This completes the proof of Theorem1.


## 3. Blow-Up Time for Neumann Boundary Condition

In this final section, we discuss a lower bound fort* if the solution u(x,t) of (1), (2), and (3b) is blow-up in finite time t*.Theorem 2.
Letu(x,t) be a classical solution of (1), (2), and (3b) with p+q>max{m,s}; then a lower bound of the blow-up time for any solution which blows up in Ln(p+q-1) norm is t*≥∫η(0)∞‍(dξ/(K2ξ(3(n+1))/(n+4-3m1)+K3ξ3)), where K2 and K3 are suitable positive constants given later, respectively, and η(0)=∫Ω‍u0n(p+q-1)dx.Proof.
We estimate∫Ω‍v(6n+3m1)/4dx in inequality (14). In a similar way to the process of the derivation of (3.3) in [10], we have
(25)∫Ω‍v3/2((2n+m1)/2)dx≤133/4[(∫Ω|∇v(n+m1)/2|2‍)1/232ρ0∫Ω‍v(2n+m1)/2dx+(2n+m1)(d+ρ0)4ρ0(∫Ω‍vndx)1/2×(∫Ω|∇v(n+m1)/2|2‍)1/2]3/2,
where ρ0=min∂Ωxivi, d2=maxΩ¯xixi, i=1,2,3, and vi is the ith component of the unit outer normal vector v on ∂Ω. By virtue of Hölder inequality, we get
(26)∫Ω‍v(2n+m1)/2dx≤(∫Ω‍vndx)1/2(∫Ω‍vn+m1dx)1/2≤(|Ω|(1-m1)/(n+1)(∫Ω‍vn+1dx)(n+m1)/(n+1))1/2×(∫Ω‍vndx)1/2.
Substituting inequality (26) into (25) yields
(27)∫Ω‍v3/2((2n+m1)/2)dx≤133/4[32ρ0(|Ω|(1-m1)/(n+1)(∫Ω‍vn+1dx)(n+m1)/(n+1))1/2×(∫Ω‍vndx)1/2+(2n+m1)(d+ρ0)4ρ0(∫Ω‍vndx)1/2×(∫Ω‍|∇v(n+m1)/2|2dx)1/2]3/2.
Applying the following inequality:(28)(a1+a2)s≤2s(a1s+a2s),a1,a2>0,s>1,
we conclude that
(29)∫Ω‍v3/2((2n+m1)/2)dx≤23/233/4(32ρ0)3/2|Ω|3(1-m1)/4(n+1)×(∫Ω‍vn+1dx)3(n+m1)/4(n+1)×(∫Ω‍vndx)3/4+23/233/4((2n+m1)(d+ρ0)4ρ0)3/2×(∫Ω‍vn)3/4(∫Ω‍|∇v(n+m1)/2|2dx)3/4.
Applying inequality (16), we obtain
(30)∫Ω‍v3/2((2n+m1)/2)dx≤23/233/4(32ρ0)3/2|Ω|3(1-m1)/4(n+1)×3(n+m1)4(n+1)θ1∫Ω‍vn+1dx+23/233/4(32ρ0)3/2|Ω|3(1-m1)/4(n+1)×n+4-3m14(n+1)θ1-3(n+m1)/(n+4-3m1)×(∫Ω‍vndx)3(n+1)/(n+4-3m1)+23/24×33/4×((2n+m1)(d+ρ0)4ρ0)3/2×θ2-3(∫Ω‍vndx)3+3×23/24×33/4((2n+m1)(d+ρ0)4ρ0)3/2×θ2∫Ω‍|∇v(n+m1)/2|2dx,
where θ1 and θ2 are arbitrary positive constants.
Recalling (12) and applying inequality (16) again, for a suitable constant μ3, we obtain
(31)∫Ω‍vn+1dx≤2n+3m1-42n+3m1-4δμ3-(4(1-δ)/(2n+3m1-4))×∫Ω‍vn+δdx+4-4δ2n+3m1-4δμ3×∫Ω‍v(6n+3m1)/4dx.
By applying (30), it follows from (31) that
(32)∫Ω‍vn+1dx≤2n+3m1-42n+3m1-4δμ3-(4(1-δ)/(2n+3m1-4))×∫Ω‍vn+δdx+1-δ2n+3m1-4δ23/233/4(32ρ0)3/2×|Ω|3(1-m1)/4(n+1)3(n+m1)n+1θ1μ3∫Ω‍vn+1dx+1-δ2n+3m1-4δ23/233/4(32ρ0)3/2|Ω|3(1-m1)/4(n+1)×n+4-3m1n+1θ1-3(n+m1)/(n+4-3m1)×μ3(∫Ω‍vndx)3(n+1)/(n+4-3m1)+1-δ2n+3m1-4δ23/233/4((2n+m1)(d+ρ0)4ρ0)3/2×μ3θ2-3(∫Ω‍vndx)3+3(1-δ)2n+3m1-4δ23/233/4×((2n+m1)(d+ρ0)4ρ0)3/2×μ3θ2∫Ω‍|∇v(n+m1)/2|2dx.
Taking(33)K0=1-1-δ2n+3m1-4δ23/233/4(32ρ0)3/2×|Ω|3(1-m1)/4(n+1)3(n+m1)n+1θ1μ3>0,
then combining (32) with (11) gives
(34)η′(t)≤K1∫Ω‍|∇v(n+m1)/2|2dx+K2(∫Ω‍vndx)3(n+1)/(n+4-3m1)+K3(∫Ω‍vndx)3+K4∫Ω‍vn+δdx,
where
(35)K1=1K03(1-δ)2n+3m1-4δ23/233/4×((2n+m1)(d+ρ0)4ρ0)3/2×μ3θ2n(p+q-1)|Ω|-4mn(p+q-1)[n(p+q-1)-1][n(p+q-1)+m-1]2,K2=1K03(1-δ)2n+3m1-4δ23/233/4(32ρ0)3/2×|Ω|1+(3(1-m1)/4(n+1))n+4-3m1n+1×θ1-3(n+m1)/(n+4-3m1)μ3n(p+q-1),K3=1K01-δ2n+3m1-4δ23/233/4((2n+m1)(d+ρ0)4ρ0)3/2×μ3θ2-3n(p+q-1)|Ω|,K4=1K02n+3m1-42n+3m1-4δμ3-(4(1-δ)/(2n+3m1-4))×n(p+q-1)|Ω|-nk(p+q-1).
We can make K1 and K4 vanish by taking suitable μ3, θ1, and θ2; then we have
(36)η′(t)≤K2η3(n+1)/(n+4-3m1)+K3η3.
Integrating inequality above from 0 to t gives
(37)t≥∫η(0)η(t)‍dξK2ξ3(n+1)/(n+4-3m1)+K3ξ3,
from which we derive a lower bound for t<t*; namely,
(38)t*≥∫η(0)∞‍dξk2ξ3(n+1)/(n+4-3m1)+K3ξ3.
This completes the proof of Theorem 2.



---

*Source: 764248-2014-01-02.xml*