# Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations

Authors: Yusuke Yamauchi
Journal: International Journal of Differential Equations
 (2012)

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## Abstract

Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author.


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## Body

## 1. Introduction

### 1.1. Fujita Type Results

We first recall the result on the Cauchy problem for a semilinear heat equation:(1.1)∂u∂t=Δu+up,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where n∈N, Δ is the n-dimensional Laplacian, and p>1. Let ϕ be a bounded continuous function on Rn.In pioneer work [1], Fujita showed that the exponent pF=1+2/n plays the crucial role for the existence and nonexistence of the solutions of (1.1). Let G denote the Gaussian heat kernel: Gt(x)=(4πt)-n/2exp(-|x|2/4t).Theorem 1.1 (see [1]).
Suppose thatϕ∈C2(Rn) and that its all derivatives are bounded.(i)
Letp<pF. Then there is no global solution of (1.1) satisfying that
(1.2)u(x,t)≤Mexp⁡(|x|β),
for  M>0 and 0<β<2.(ii)
Letp>pF. Then for any τ>0 there exists δ>0 with the following property: if
(1.3)ϕ(x)≤δGτ(x),
then there exists a global solution of (1.1) satisfying
(1.4)u(x,t)≤Mexp⁡(|x|β)
for M>0 and 0<β<2.In [2], Hayakawa showed first that there is no global solution of (1.1) in the critical case p=pF when n=1 or 2.Theorem 1.2 (see [2]).
In case ofn=2, p=pF=2 or n=1, p=pF=3, (1.1) has no global solutions for any nontrivial initial data.In genaral space dimensions, Kobayashi et al. [3] consider the following problem:(1.5)∂u∂t=Δu+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where n∈N and p>1. Let ϕ be a bounded continuous function on Rn.Theorem 1.3 (see [3]).
Suppose thatf satisfies the following three conditions:(a)
f is a locally Lipschitz continuous and nondecreasing function in [0,∞) with  f(0)=0 and f(λ)>0 for λ>0,(b)
∫0+εf(λ)/λ2+(2+n)dλ for some ε>0,(c)
there exists a positive constantc≤1 such that(1.6)f(λμ)≥μ1+(2+n)f(λ)(0<λ≤μ,λ<c),f(λμ)≥μ2+(2+n)f(λ)(0<μ≤λ<c).
Then each positive solution of (1.5) blows up in finite time.Remark 1.4.
(i) We remark that the proofs of the theorems in [2, 3] are mainly based on the iterated estimate from below obtained by the following integral equation:
(1.7)u(x,t)=(Gt*ϕ)(x)+∫0t(Gt-s*up(s))(x)ds.
(ii) The critical nonlinearity of power typef(u)=u1+2/n satisfies the assumptions (a), (b), and (c) in [3].Weissler proved the nonexistence of global solution inLp-framework in [4]. The proof is quite short and elegant.Theorem 1.5 (see [4]).
Supposep=pF and that ϕ≥0 in Lq(Rn) (q≥1) is not identically zero. Then there is no nonnegative global solution u:[0,∞)→Lq to the integral (1.7) with initial value ϕ.The outline of the proof is as follows. First we assume that there is a global solution. From the fact that the solutionu(ε,x)≥kGα(x)(k,α>0) for some ε>0, we can obtain that ∥u(t)∥L1(Rn)≥C(k,p,n)∫0t(s+α)-1ds. This contradicts the boundedness of ∥u(t)∥L1(Rn) for large t>0. Hence the solution is not global.Existence and nonexistence results for time-global solutions of (1.1) are summarized as follows.(i)
Letp∈(1,pF]. Then every nontrivial solution of (1.1) blows up in finite time.(ii)
Letp∈(pF,∞). Then (1.1) has a time-global classical solution for small initial data ϕ and has a blowing-up solution for large initial data ϕ.In several papers [5–7], the results for the critical exponent are extended to the more general equations.In [6], Qi consider the following Cauchy problem of porous medium equation:(1.8)∂u∂t=Δum+ts|x|σup,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where m>max{(n-2)/n,0}, σ>max{-n,-2} and p>max{m,1}. They showed that the critical exponent is pc1:=m+(m-1)s+(2+2s+σ)/n.Theorem 1.6 (see [6]).
(i)  If1<p≤pc1, then every nontrivial solution of (1.8) blows up in finite time.
(ii)  Ifp>pc1, then (1.8) has global classical solutions for small initial data.Another extension is the following quasilinear parabolic equation:(1.9)∂u∂t=div(|∇u|m-1∇u)+ts|x|σup,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where (n-1)(n+1)<m<1, s≥0, p>1 and σ>n(1-m)-(1+m+2s). In [7], the authors showed that the critical exponent is pc2:=m+(1+m+2s+σ)/n.Theorem 1.7 (see [7]).
(i)  If1<p≤pc2, then every nontrivial solution of the (1.9) blows up in finite time.
(ii)  Ifp>pc2, then (1.9) has global classical solutions for small initial data.Remark 1.8.
(i) (Sublinear case). In [8], Aguirre and Escobedo proved that if 0<p<1, then every solution of (1.1) is global.
(ii) (System of equations). There are various extensions of Fujita type results to the system of equations. See, for example, papers [9–13], survays [14, 15], and references therein. In [9, 13], the authors investigated the following system:
(1.10)∂u∂t=Δu+|x|σ1up11vp12,(x,t)∈Rn×(0,∞),∂v∂t=Δv+|x|σ1up21vp22,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,v(x,0)=ψ(x)≥0,x∈Rn,
where σj>max{-2,-n} and pjk≥0   (j,k=1,2).


### 1.2. Blow-up Results for Slowly Decaying Initial Data

Especially for slowly decaying initial data, it was shown that the solution of  (1.1) blows up in finite time for any p>1. In [16] Lee and Ni showed that a sufficient condition for finite time blows up on the decay order of initial data. We note that the slow decay of initial data in all directions was assumed. Here, let μR be the first Dirichlet eigenvalue of -Δ in the ball BR.Theorem 1.9 (see [16]).
The solution of (1.1) blows up in finite time if
(1.11)lim⁡inf⁡x→∞|x|2/(p-1)ϕ(x)>μ11/(p-1).We putΩ={(r,ω)∈(0,∞)×Sn-1;r>R,d(ω,ω0)<cr-μ} for some R>0, c>0, ω0∈Sn-1, and 0≤μ<1, where d(·,·) denotes the usual distance on the unit sphere Sn-1. Mizoguchi and Yanagida [17] showed that a sufficient condition for finite time blows up on the decay order of initial data in Ω. The authors consider the following problem:(1.12)∂u∂t=Δu+|u|p-1u,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x),x∈Rn,
where n∈N and p>1. The following results indicate that the slow decay of initial data in all directions is not necessary for finite time blow-up.Theorem 1.10 (see [17]).
(i) Assume that initial dataϕ may change sign. Suppose that ϕ∈W1,∞(Rn) satisfies the following:
(1.13)ϕ≥K1r-αinΩforsomeα>0,K1>0,|∇ϕ|≥K2r-α-1+μinΩforsomeα>0,K2>0,
with 0<α<2(1-μ)/(p-1). Then the solution of (1.12) blows up in finite time.
(ii) (Positive solutions). Assume that initial dataϕ is nonnegative. Suppose that ϕ∈L∞(Rn) satisfies that
(1.14)ϕ≥K1r-αinΩforsomeα>0K1>0,
with 0<α<2(1-μ)/(p-1). Then the solution of (1.1) blows up in finite time.Remark 1.11.
(i) From the theorem, in particular, for nondecaying initial data the solution of (1.1) blows up in finite time.
(ii) For sign changing solutions, Fujita type results are discussed in [18–22], for instance.
(iii) Many blow-up results, for instance [1, 5–7, 12, 13], are based on Kaplan’s method (the eigenfunction method). See [23].In the remainder of this paper, we discuss various studies for the lifespan of the positive solutions of the parabolic problems. In Sections2 and 3, we introduce the results of asymptotics of life span with respect to the size of initial data and to the size of diffusion constant, and the results of minimal time blow up, respectively. In Section 4, we shall show an upper bound of the life span of the solution for (1.1).


## 1.1. Fujita Type Results

We first recall the result on the Cauchy problem for a semilinear heat equation:(1.1)∂u∂t=Δu+up,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where n∈N, Δ is the n-dimensional Laplacian, and p>1. Let ϕ be a bounded continuous function on Rn.In pioneer work [1], Fujita showed that the exponent pF=1+2/n plays the crucial role for the existence and nonexistence of the solutions of (1.1). Let G denote the Gaussian heat kernel: Gt(x)=(4πt)-n/2exp(-|x|2/4t).Theorem 1.1 (see [1]).
Suppose thatϕ∈C2(Rn) and that its all derivatives are bounded.(i)
Letp<pF. Then there is no global solution of (1.1) satisfying that
(1.2)u(x,t)≤Mexp⁡(|x|β),
for  M>0 and 0<β<2.(ii)
Letp>pF. Then for any τ>0 there exists δ>0 with the following property: if
(1.3)ϕ(x)≤δGτ(x),
then there exists a global solution of (1.1) satisfying
(1.4)u(x,t)≤Mexp⁡(|x|β)
for M>0 and 0<β<2.In [2], Hayakawa showed first that there is no global solution of (1.1) in the critical case p=pF when n=1 or 2.Theorem 1.2 (see [2]).
In case ofn=2, p=pF=2 or n=1, p=pF=3, (1.1) has no global solutions for any nontrivial initial data.In genaral space dimensions, Kobayashi et al. [3] consider the following problem:(1.5)∂u∂t=Δu+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where n∈N and p>1. Let ϕ be a bounded continuous function on Rn.Theorem 1.3 (see [3]).
Suppose thatf satisfies the following three conditions:(a)
f is a locally Lipschitz continuous and nondecreasing function in [0,∞) with  f(0)=0 and f(λ)>0 for λ>0,(b)
∫0+εf(λ)/λ2+(2+n)dλ for some ε>0,(c)
there exists a positive constantc≤1 such that(1.6)f(λμ)≥μ1+(2+n)f(λ)(0<λ≤μ,λ<c),f(λμ)≥μ2+(2+n)f(λ)(0<μ≤λ<c).
Then each positive solution of (1.5) blows up in finite time.Remark 1.4.
(i) We remark that the proofs of the theorems in [2, 3] are mainly based on the iterated estimate from below obtained by the following integral equation:
(1.7)u(x,t)=(Gt*ϕ)(x)+∫0t(Gt-s*up(s))(x)ds.
(ii) The critical nonlinearity of power typef(u)=u1+2/n satisfies the assumptions (a), (b), and (c) in [3].Weissler proved the nonexistence of global solution inLp-framework in [4]. The proof is quite short and elegant.Theorem 1.5 (see [4]).
Supposep=pF and that ϕ≥0 in Lq(Rn) (q≥1) is not identically zero. Then there is no nonnegative global solution u:[0,∞)→Lq to the integral (1.7) with initial value ϕ.The outline of the proof is as follows. First we assume that there is a global solution. From the fact that the solutionu(ε,x)≥kGα(x)(k,α>0) for some ε>0, we can obtain that ∥u(t)∥L1(Rn)≥C(k,p,n)∫0t(s+α)-1ds. This contradicts the boundedness of ∥u(t)∥L1(Rn) for large t>0. Hence the solution is not global.Existence and nonexistence results for time-global solutions of (1.1) are summarized as follows.(i)
Letp∈(1,pF]. Then every nontrivial solution of (1.1) blows up in finite time.(ii)
Letp∈(pF,∞). Then (1.1) has a time-global classical solution for small initial data ϕ and has a blowing-up solution for large initial data ϕ.In several papers [5–7], the results for the critical exponent are extended to the more general equations.In [6], Qi consider the following Cauchy problem of porous medium equation:(1.8)∂u∂t=Δum+ts|x|σup,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where m>max{(n-2)/n,0}, σ>max{-n,-2} and p>max{m,1}. They showed that the critical exponent is pc1:=m+(m-1)s+(2+2s+σ)/n.Theorem 1.6 (see [6]).
(i)  If1<p≤pc1, then every nontrivial solution of (1.8) blows up in finite time.
(ii)  Ifp>pc1, then (1.8) has global classical solutions for small initial data.Another extension is the following quasilinear parabolic equation:(1.9)∂u∂t=div(|∇u|m-1∇u)+ts|x|σup,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where (n-1)(n+1)<m<1, s≥0, p>1 and σ>n(1-m)-(1+m+2s). In [7], the authors showed that the critical exponent is pc2:=m+(1+m+2s+σ)/n.Theorem 1.7 (see [7]).
(i)  If1<p≤pc2, then every nontrivial solution of the (1.9) blows up in finite time.
(ii)  Ifp>pc2, then (1.9) has global classical solutions for small initial data.Remark 1.8.
(i) (Sublinear case). In [8], Aguirre and Escobedo proved that if 0<p<1, then every solution of (1.1) is global.
(ii) (System of equations). There are various extensions of Fujita type results to the system of equations. See, for example, papers [9–13], survays [14, 15], and references therein. In [9, 13], the authors investigated the following system:
(1.10)∂u∂t=Δu+|x|σ1up11vp12,(x,t)∈Rn×(0,∞),∂v∂t=Δv+|x|σ1up21vp22,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,v(x,0)=ψ(x)≥0,x∈Rn,
where σj>max{-2,-n} and pjk≥0   (j,k=1,2).


## 1.2. Blow-up Results for Slowly Decaying Initial Data

Especially for slowly decaying initial data, it was shown that the solution of  (1.1) blows up in finite time for any p>1. In [16] Lee and Ni showed that a sufficient condition for finite time blows up on the decay order of initial data. We note that the slow decay of initial data in all directions was assumed. Here, let μR be the first Dirichlet eigenvalue of -Δ in the ball BR.Theorem 1.9 (see [16]).
The solution of (1.1) blows up in finite time if
(1.11)lim⁡inf⁡x→∞|x|2/(p-1)ϕ(x)>μ11/(p-1).We putΩ={(r,ω)∈(0,∞)×Sn-1;r>R,d(ω,ω0)<cr-μ} for some R>0, c>0, ω0∈Sn-1, and 0≤μ<1, where d(·,·) denotes the usual distance on the unit sphere Sn-1. Mizoguchi and Yanagida [17] showed that a sufficient condition for finite time blows up on the decay order of initial data in Ω. The authors consider the following problem:(1.12)∂u∂t=Δu+|u|p-1u,(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x),x∈Rn,
where n∈N and p>1. The following results indicate that the slow decay of initial data in all directions is not necessary for finite time blow-up.Theorem 1.10 (see [17]).
(i) Assume that initial dataϕ may change sign. Suppose that ϕ∈W1,∞(Rn) satisfies the following:
(1.13)ϕ≥K1r-αinΩforsomeα>0,K1>0,|∇ϕ|≥K2r-α-1+μinΩforsomeα>0,K2>0,
with 0<α<2(1-μ)/(p-1). Then the solution of (1.12) blows up in finite time.
(ii) (Positive solutions). Assume that initial dataϕ is nonnegative. Suppose that ϕ∈L∞(Rn) satisfies that
(1.14)ϕ≥K1r-αinΩforsomeα>0K1>0,
with 0<α<2(1-μ)/(p-1). Then the solution of (1.1) blows up in finite time.Remark 1.11.
(i) From the theorem, in particular, for nondecaying initial data the solution of (1.1) blows up in finite time.
(ii) For sign changing solutions, Fujita type results are discussed in [18–22], for instance.
(iii) Many blow-up results, for instance [1, 5–7, 12, 13], are based on Kaplan’s method (the eigenfunction method). See [23].In the remainder of this paper, we discuss various studies for the lifespan of the positive solutions of the parabolic problems. In Sections2 and 3, we introduce the results of asymptotics of life span with respect to the size of initial data and to the size of diffusion constant, and the results of minimal time blow up, respectively. In Section 4, we shall show an upper bound of the life span of the solution for (1.1).


## 2. Asymptotics of Life Span

### 2.1. Life Span for the Equation with Large or Small Initial Data

Recently, several studies have been made on the life span of solutions for (1.1). See [16, 19, 24–38], and references therein. In this section, we mainly consider the following Cauchy problems:(2.1)∂u∂t=Δu+up,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x)≥0,x∈Rn,(2.2)∂u∂t=Δu+|u|p-1u,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x),x∈Rn,
where n∈N, p>1. Let ψ be a bounded continuous function on Rn and λ be a positive parameter.In this paper, we define the lifespan (or blow-up time)Tmax as(2.3)Tmax⁡:=sup⁡{T>0∣.The problem possesses a unique classical solution inRn×[0,T)}.We first introduce the result about asymptotic behavior of the life spanTmax(λ) for (2.1) as large or small λ by Lee and Ni [16]. The authors showed the order of the asymptotics of the life span.Theorem 2.1 (see [16]).
Assume thatψ is nonnegative.(i)
There exist constantsC1>0 and C2>0 such that C1λ1-p≤Tmax(λ)≤C2λ1-p for large λ.(ii)
Ifliminf|x|→∞ψ(x)>0, then there exists constants C1>0 and C2>0 such that C1λ1-p≤Tmax(λ)≤C2λ1-p for small λ.In [29], Gui and Wang obtained more detailed information of the asymptotics for (2.1). They showed that for large λ the supremum of initial data ϕ is dominant in the asymptotics, and that for small λ the limiting value of ϕ at space infinity is dominant.Theorem 2.2 (see [29]).
Assume thatψ is nonnegative.(i)
We have(2.4)lim⁡λ→∞Tmax⁡(λ)⋅λp-1=1p-1‖ψ‖L∞(Rn)1-p.(ii)
Iflim|x|→∞ψ(x)=ψ∞>0, then
(2.5)lim⁡λ→0Tmax⁡(λ)⋅λp-1=1p-1ψ∞1-p.It is noteworthy that the limiting values asλ→0 and λ→∞ in the theorem are different. The proof of the theorem is also based on Kaplan’s method, and the assumption lim|x|→∞ψ(x)=ψ∞ plays an important role in the proof.Thereafter, the results in [16, 29] are extended by several authors. Pinsky considered the following problem:(2.6)∂u∂t=Δu+a(x)up,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x)≥0,x∈Rn,
where a(x)≥0 is Hölder continuous, 0≤ψ∈BC(Rn), ψ≢0, p>1 and λ>0 is a parameter. The author treated the initial data in the following two classes:(2.7)ClassS0≤ψ(x)≤δexp⁡(-γ|x|2),whereδ,γ>0.ClassLc1≤ψ(x)≤c2,wherec1,c2>0.Here, we introduce the results in [34] with a(x)~|x|m(|x|≫1) for m>max{-2,-n}.Theorem 2.3 (see [34] Class S for small λ).
Assume thatψ belongs to Class S.(i)
Letp∈(1,1+(2+m)/n). Then
(2.8)Tmax⁡(λ)~λ2(1-p)/(2+m-n(1-p)),asλ⟶0.(ii)
Letp=1+(2+m)/n. Then there exists a constant c1>0 and for every ε>0, a constant c2>0 such that
(2.9)c1λ-(2+m)/n≤log⁡Tmax⁡(λ)≤c2λ-ε-(2+m)/n,forsmallλ.Theorem 2.4 (see [34] Class L for small λ).
Assume thatψ belongs to Class L. Then
(2.10)Tmax⁡(λ)~λ2(1-p)/(2+m),asλ⟶0.Theorem 2.5 (see [34], for large λ).
Assume thata is bounded and let ψ be arbitrary initial data.(i)
If there exists anx0∈Rn such that a(x0),ψ(x0)>0, then
(2.11)Tmax⁡(λ)~λ1-p,asλ⟶∞.(ii)
Ifdist(supp(a),supp(ψ))>0, then
(2.12)Tmax⁡(λ)~(log⁡λ)-1,asλ⟶∞.Kobayashi extended the results to the following system of equations:(2.13)∂u∂t=Δu+a(x)vp,(x,t)∈Rn×(0,∞),∂v∂t=Δv+b(x)uq,(x,t)∈Rn×(0,∞),u(x,0)=λμϕ(x),x∈Rn,v(x,0)=λνψ(x),x∈Rn,
where p,q>1,   μ,ν>0, and λ>0 are parameters. See [30].On the other hand, Mukai et al. showed the results for the following equation of porous medium type in [31]:(2.14)∂u∂t=Δum+up,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x),x∈Rn,
where 1<m<p, and ψ is a nonnegative and bounded function.Theorem 2.6 (see [31]).
(i)  Letp>m. Suppose that ∥ψ∥L∞(Rn)=ψ(0)>0. Then
(2.15)lim⁡λ→∞λp-1Tmax⁡(λ)=1p-1ψ(0)1-p.
(ii)  Letp>m. Suppose that ∥ψ∥L∞(Rn)=lim|x|→∞ψ(x)>0. Then
(2.16)lim⁡λ→0λp-1Tmax⁡(λ)=1p-1‖ψ‖L∞(Rn)1-p.
(iii) Assume thata≥0. Let m<p<m+2/n or a<2/(p-m). Suppose that ψ(x)~|x|-a for large |x|. Then
(2.17)Tmax⁡(λ)~λ(1-p)/(1-min⁡{a,n}(p-m)/2)asλ⟶0.Now we turn to the problems (2.1) and (2.2). In [39], Mizoguchi and Yanagida determined the higher-order term of the life span Tmax(λ) for (2.2) as λ→∞. The authors introduced the following function space:(2.18)D={ψ∈BC(Rn):exp⁡(-c|x|2)∇ψ∈L2(Rn)forsomec>0}.Theorem 2.7 (see [39]).
Suppose thatψ∈D satisfies the following assumptions; (A1)
|ψ| attains its maximum at some point x=a, and ψ satisfies that
(2.19)|ψ(x)-ψ(a)|=|x-a|kψ̃(x)+o(|x-a|k+2),
at x=a with some k>-2, where ψ̃ is twice continuously differentiable at x=a and satisfies ψ̃(a)=0, ∇ψ̃(a)=0 and Δψ̃(a)≥0.(A2)
There existR>0 and δ>0 such that
(2.20)|ψ(x)|<‖ψ‖L∞(Rn)-δ∀|x|>R.
If|ψ| attains its maximum at only one point x=a, then
(2.21)Tmax⁡(λ)=1p-1‖ψ‖L∞(Rn)1-pλ1-p+K‖ψ‖L∞(Rn)(2+k/2)(1-p)-1Δψ̃(a)λ(2+k/2)(1-p)+o(λ(2+k/2)(1-p)),
as λ→∞ with
(2.22)K=2k+1Γ((k+n+2)/2)(p-1)1+k/2Γ((n+2)/2).
In particular, if ψ is smooth at x=a, then
(2.23)Tmax⁡(λ)=1p-1‖ψ‖L∞(Rn)1-pλ1-p+2p-1‖ψ‖L∞(Rn)2(1-p)-1Δψ̃(a)λ2(1-p)+o(λ2(1-p)),
as λ→∞.


### 2.2. Life Span for the Equation with Large Diffusion

We shall consider the following Cauchy problem:(2.24)∂u∂t=DΔu+|u|p-1u,(x,t)∈Rn×(0,∞),u(x,0)=λ+ϕ(x),x∈Rn,
where D>0, p>1, n≥3, λ>0, and ϕ∈L∞(Rn)∩L1(Rn,(1+|x|)2dx).In [24, 25] Fujishima and Ishige obtained the asymptotics of the life span Tmax(D) of the solution of (2.24) as D→∞. The situation is divided into three cases:(2.25)∫Rnϕ(x)dx>0,∫Rnϕ(x)dx=0,and∫Rnϕ(x)dx<0.
We prepare some notation. Put the following:(2.26)M(ϕ):=∫Rnϕ(x)dx,Ξ(ϕ):=∫Rnxϕ(x)dx,Sλ:=λ1-pp-1.Theorem 2.8 (see [24, 25]).
(i) Assume thatM(ϕ)>0. Then Tmax(D)≤Sλ for any D>0, and
(2.27)Sλ-Tmax⁡(D)=(4πSλ)-n/2λ-pD-n/2[M(ϕ)+O(D-1)]
as D→∞.
(ii) Assume thatM(ϕ)=0. Then Tmax(D)≤Sλ for any D>0, and
(2.28)Sλ-Tmax⁡(D)=(4πSλ)-n/2λp2eSλD(-n-1)/2[Ξ(ϕ)+O(D-1/2)],
as D→∞.
(iii) Assume thatM(ϕ)<0. Then Tmax(D)≤Sλ for any D>0, and
(2.29)Sλ-Tmax⁡(D)=O(D-n/2-1),
as D→∞.Remark 2.9.
(i) Changing variable, we can easily see that the problem with large diffusion is equivalent to the equation with small initial data (cf. [29]. Theorem 1.1(ii)).
(ii) In [24, 25], the behavior of the blow-up set of the solution as D→∞ was also studied.


## 2.1. Life Span for the Equation with Large or Small Initial Data

Recently, several studies have been made on the life span of solutions for (1.1). See [16, 19, 24–38], and references therein. In this section, we mainly consider the following Cauchy problems:(2.1)∂u∂t=Δu+up,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x)≥0,x∈Rn,(2.2)∂u∂t=Δu+|u|p-1u,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x),x∈Rn,
where n∈N, p>1. Let ψ be a bounded continuous function on Rn and λ be a positive parameter.In this paper, we define the lifespan (or blow-up time)Tmax as(2.3)Tmax⁡:=sup⁡{T>0∣.The problem possesses a unique classical solution inRn×[0,T)}.We first introduce the result about asymptotic behavior of the life spanTmax(λ) for (2.1) as large or small λ by Lee and Ni [16]. The authors showed the order of the asymptotics of the life span.Theorem 2.1 (see [16]).
Assume thatψ is nonnegative.(i)
There exist constantsC1>0 and C2>0 such that C1λ1-p≤Tmax(λ)≤C2λ1-p for large λ.(ii)
Ifliminf|x|→∞ψ(x)>0, then there exists constants C1>0 and C2>0 such that C1λ1-p≤Tmax(λ)≤C2λ1-p for small λ.In [29], Gui and Wang obtained more detailed information of the asymptotics for (2.1). They showed that for large λ the supremum of initial data ϕ is dominant in the asymptotics, and that for small λ the limiting value of ϕ at space infinity is dominant.Theorem 2.2 (see [29]).
Assume thatψ is nonnegative.(i)
We have(2.4)lim⁡λ→∞Tmax⁡(λ)⋅λp-1=1p-1‖ψ‖L∞(Rn)1-p.(ii)
Iflim|x|→∞ψ(x)=ψ∞>0, then
(2.5)lim⁡λ→0Tmax⁡(λ)⋅λp-1=1p-1ψ∞1-p.It is noteworthy that the limiting values asλ→0 and λ→∞ in the theorem are different. The proof of the theorem is also based on Kaplan’s method, and the assumption lim|x|→∞ψ(x)=ψ∞ plays an important role in the proof.Thereafter, the results in [16, 29] are extended by several authors. Pinsky considered the following problem:(2.6)∂u∂t=Δu+a(x)up,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x)≥0,x∈Rn,
where a(x)≥0 is Hölder continuous, 0≤ψ∈BC(Rn), ψ≢0, p>1 and λ>0 is a parameter. The author treated the initial data in the following two classes:(2.7)ClassS0≤ψ(x)≤δexp⁡(-γ|x|2),whereδ,γ>0.ClassLc1≤ψ(x)≤c2,wherec1,c2>0.Here, we introduce the results in [34] with a(x)~|x|m(|x|≫1) for m>max{-2,-n}.Theorem 2.3 (see [34] Class S for small λ).
Assume thatψ belongs to Class S.(i)
Letp∈(1,1+(2+m)/n). Then
(2.8)Tmax⁡(λ)~λ2(1-p)/(2+m-n(1-p)),asλ⟶0.(ii)
Letp=1+(2+m)/n. Then there exists a constant c1>0 and for every ε>0, a constant c2>0 such that
(2.9)c1λ-(2+m)/n≤log⁡Tmax⁡(λ)≤c2λ-ε-(2+m)/n,forsmallλ.Theorem 2.4 (see [34] Class L for small λ).
Assume thatψ belongs to Class L. Then
(2.10)Tmax⁡(λ)~λ2(1-p)/(2+m),asλ⟶0.Theorem 2.5 (see [34], for large λ).
Assume thata is bounded and let ψ be arbitrary initial data.(i)
If there exists anx0∈Rn such that a(x0),ψ(x0)>0, then
(2.11)Tmax⁡(λ)~λ1-p,asλ⟶∞.(ii)
Ifdist(supp(a),supp(ψ))>0, then
(2.12)Tmax⁡(λ)~(log⁡λ)-1,asλ⟶∞.Kobayashi extended the results to the following system of equations:(2.13)∂u∂t=Δu+a(x)vp,(x,t)∈Rn×(0,∞),∂v∂t=Δv+b(x)uq,(x,t)∈Rn×(0,∞),u(x,0)=λμϕ(x),x∈Rn,v(x,0)=λνψ(x),x∈Rn,
where p,q>1,   μ,ν>0, and λ>0 are parameters. See [30].On the other hand, Mukai et al. showed the results for the following equation of porous medium type in [31]:(2.14)∂u∂t=Δum+up,(x,t)∈Rn×(0,∞),u(x,0)=λψ(x),x∈Rn,
where 1<m<p, and ψ is a nonnegative and bounded function.Theorem 2.6 (see [31]).
(i)  Letp>m. Suppose that ∥ψ∥L∞(Rn)=ψ(0)>0. Then
(2.15)lim⁡λ→∞λp-1Tmax⁡(λ)=1p-1ψ(0)1-p.
(ii)  Letp>m. Suppose that ∥ψ∥L∞(Rn)=lim|x|→∞ψ(x)>0. Then
(2.16)lim⁡λ→0λp-1Tmax⁡(λ)=1p-1‖ψ‖L∞(Rn)1-p.
(iii) Assume thata≥0. Let m<p<m+2/n or a<2/(p-m). Suppose that ψ(x)~|x|-a for large |x|. Then
(2.17)Tmax⁡(λ)~λ(1-p)/(1-min⁡{a,n}(p-m)/2)asλ⟶0.Now we turn to the problems (2.1) and (2.2). In [39], Mizoguchi and Yanagida determined the higher-order term of the life span Tmax(λ) for (2.2) as λ→∞. The authors introduced the following function space:(2.18)D={ψ∈BC(Rn):exp⁡(-c|x|2)∇ψ∈L2(Rn)forsomec>0}.Theorem 2.7 (see [39]).
Suppose thatψ∈D satisfies the following assumptions; (A1)
|ψ| attains its maximum at some point x=a, and ψ satisfies that
(2.19)|ψ(x)-ψ(a)|=|x-a|kψ̃(x)+o(|x-a|k+2),
at x=a with some k>-2, where ψ̃ is twice continuously differentiable at x=a and satisfies ψ̃(a)=0, ∇ψ̃(a)=0 and Δψ̃(a)≥0.(A2)
There existR>0 and δ>0 such that
(2.20)|ψ(x)|<‖ψ‖L∞(Rn)-δ∀|x|>R.
If|ψ| attains its maximum at only one point x=a, then
(2.21)Tmax⁡(λ)=1p-1‖ψ‖L∞(Rn)1-pλ1-p+K‖ψ‖L∞(Rn)(2+k/2)(1-p)-1Δψ̃(a)λ(2+k/2)(1-p)+o(λ(2+k/2)(1-p)),
as λ→∞ with
(2.22)K=2k+1Γ((k+n+2)/2)(p-1)1+k/2Γ((n+2)/2).
In particular, if ψ is smooth at x=a, then
(2.23)Tmax⁡(λ)=1p-1‖ψ‖L∞(Rn)1-pλ1-p+2p-1‖ψ‖L∞(Rn)2(1-p)-1Δψ̃(a)λ2(1-p)+o(λ2(1-p)),
as λ→∞.


## 2.2. Life Span for the Equation with Large Diffusion

We shall consider the following Cauchy problem:(2.24)∂u∂t=DΔu+|u|p-1u,(x,t)∈Rn×(0,∞),u(x,0)=λ+ϕ(x),x∈Rn,
where D>0, p>1, n≥3, λ>0, and ϕ∈L∞(Rn)∩L1(Rn,(1+|x|)2dx).In [24, 25] Fujishima and Ishige obtained the asymptotics of the life span Tmax(D) of the solution of (2.24) as D→∞. The situation is divided into three cases:(2.25)∫Rnϕ(x)dx>0,∫Rnϕ(x)dx=0,and∫Rnϕ(x)dx<0.
We prepare some notation. Put the following:(2.26)M(ϕ):=∫Rnϕ(x)dx,Ξ(ϕ):=∫Rnxϕ(x)dx,Sλ:=λ1-pp-1.Theorem 2.8 (see [24, 25]).
(i) Assume thatM(ϕ)>0. Then Tmax(D)≤Sλ for any D>0, and
(2.27)Sλ-Tmax⁡(D)=(4πSλ)-n/2λ-pD-n/2[M(ϕ)+O(D-1)]
as D→∞.
(ii) Assume thatM(ϕ)=0. Then Tmax(D)≤Sλ for any D>0, and
(2.28)Sλ-Tmax⁡(D)=(4πSλ)-n/2λp2eSλD(-n-1)/2[Ξ(ϕ)+O(D-1/2)],
as D→∞.
(iii) Assume thatM(ϕ)<0. Then Tmax(D)≤Sλ for any D>0, and
(2.29)Sλ-Tmax⁡(D)=O(D-n/2-1),
as D→∞.Remark 2.9.
(i) Changing variable, we can easily see that the problem with large diffusion is equivalent to the equation with small initial data (cf. [29]. Theorem 1.1(ii)).
(ii) In [24, 25], the behavior of the blow-up set of the solution as D→∞ was also studied.


## 3. Minimal Time Blow-up Results

### 3.1. Minimal Time Blow-up Results

In this section, we discuss the life span for the following parabolic equations (cf. [26–28, 35]):(3.1)∂u∂t=Δu+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where ϕ is a bounded continuous function on Rn. Suppose that(3.2)fislocallyLipschitzfunctionin[0,∞),f(ξ)>0(ξ>0),∫1∞dξf(ξ)<∞.
Applying the comparison principle to (3.1), we always have(3.3)Tmax⁡≥∫‖ϕ‖L∞(Rn)∞dξf(ξ).
When f(u)=up, we always have(3.4)Tmax⁡≥1p-1‖ϕ‖L∞(Rn)1-p.
A solution u to (3.1) with initial data ϕ is said to blow up at minimal blow-up time provided that(3.5)Tmax⁡=∫‖ϕ‖L∞(Rn)∞dξf(ξ).
We put ρ(x):=e-|x|/(∫Rne-|y|dy) and Aρ(x;ϕ):=∫Rnρ(y-x)ϕ(y)dy. The necessary and sufficient conditions of initial data ϕ for blowing up at minimal blow-up time are following.Theorem 3.1 (see [26]).
Let u be a solution of (3.1). Assume that there exist constants ξ0>0 and p>1 such that f(ξ)/ξp is nondecreasing for ξ≥ξ0. Then u blows up at minimal blow-up time if and only if one of the following two conditions for initial data ϕ holds
(3.6)Thereexistsasequence{xn}⊂Rnsuchthat|xn|⟶∞andϕ(x+xn)⟶‖ϕ‖L∞(Rn)a.e.inRnasn⟶∞;(3.7)sup⁡x∈RnAρ(x;ϕ)=‖ϕ‖L∞(Rn).In [36], Seki et al. consider the following quasilinear equations:(3.8)∂u∂t=ΔΦ(u)+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where ϕ is a bounded continuous function on Rn. Suppose that(3.9)Φ(ξ),f(ξ)∈C1[0,∞)∩C∞(0,∞),Φ(ξ)>0,Φ′(ξ)>0,Φ′′(ξ)≥0(ξ>0),f(ξ)>0(ξ>0),Φ(0)=0.
The authors proved that if there exist a function Ψ(η) and constants c>0 and η1>0 such that(3.10)Ψ(η)>0,Ψ′(η)≥0,Ψ′′(η)≥0(η>η1),∫η1+1∞dηΨ(η)<∞,{f(Φ-1(η))}′Ψ(η)-f(Φ-1(η))Ψ′(η)≥cΨ(η)Ψ′(η)(η>η1),
where ξ=Φ-1(η) is the inverse function of η=Φ(ξ), then the same result as the previous theorem holds. For example, the result can be applied to the equation ut=Δum+up   (p>m).


## 3.1. Minimal Time Blow-up Results

In this section, we discuss the life span for the following parabolic equations (cf. [26–28, 35]):(3.1)∂u∂t=Δu+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where ϕ is a bounded continuous function on Rn. Suppose that(3.2)fislocallyLipschitzfunctionin[0,∞),f(ξ)>0(ξ>0),∫1∞dξf(ξ)<∞.
Applying the comparison principle to (3.1), we always have(3.3)Tmax⁡≥∫‖ϕ‖L∞(Rn)∞dξf(ξ).
When f(u)=up, we always have(3.4)Tmax⁡≥1p-1‖ϕ‖L∞(Rn)1-p.
A solution u to (3.1) with initial data ϕ is said to blow up at minimal blow-up time provided that(3.5)Tmax⁡=∫‖ϕ‖L∞(Rn)∞dξf(ξ).
We put ρ(x):=e-|x|/(∫Rne-|y|dy) and Aρ(x;ϕ):=∫Rnρ(y-x)ϕ(y)dy. The necessary and sufficient conditions of initial data ϕ for blowing up at minimal blow-up time are following.Theorem 3.1 (see [26]).
Let u be a solution of (3.1). Assume that there exist constants ξ0>0 and p>1 such that f(ξ)/ξp is nondecreasing for ξ≥ξ0. Then u blows up at minimal blow-up time if and only if one of the following two conditions for initial data ϕ holds
(3.6)Thereexistsasequence{xn}⊂Rnsuchthat|xn|⟶∞andϕ(x+xn)⟶‖ϕ‖L∞(Rn)a.e.inRnasn⟶∞;(3.7)sup⁡x∈RnAρ(x;ϕ)=‖ϕ‖L∞(Rn).In [36], Seki et al. consider the following quasilinear equations:(3.8)∂u∂t=ΔΦ(u)+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where ϕ is a bounded continuous function on Rn. Suppose that(3.9)Φ(ξ),f(ξ)∈C1[0,∞)∩C∞(0,∞),Φ(ξ)>0,Φ′(ξ)>0,Φ′′(ξ)≥0(ξ>0),f(ξ)>0(ξ>0),Φ(0)=0.
The authors proved that if there exist a function Ψ(η) and constants c>0 and η1>0 such that(3.10)Ψ(η)>0,Ψ′(η)≥0,Ψ′′(η)≥0(η>η1),∫η1+1∞dηΨ(η)<∞,{f(Φ-1(η))}′Ψ(η)-f(Φ-1(η))Ψ′(η)≥cΨ(η)Ψ′(η)(η>η1),
where ξ=Φ-1(η) is the inverse function of η=Φ(ξ), then the same result as the previous theorem holds. For example, the result can be applied to the equation ut=Δum+up   (p>m).


## 4. Estimates of Lifespan

### 4.1. Upper Bound of the Iife Span

In this section, we shall show an upper bound of the life span of positive solutions of the Cauchy problem for a semilinear heat equation:(4.1)∂u∂t=Δu+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where n∈N, and ϕ is a bounded continuous function on Rn. We assume that F(u) satisfies(4.2)f(u)≥upforu≥0,
with p>1.In order to state the results, we prepare several notations. Forξ′∈Sn-1, and δ∈(0,2), we set neighborhood Sξ′(δ):(4.3)Sξ′(δ):={η′∈Sn-1;|η′-ξ′|<δ}.
Define(4.4)M∞:=sup⁡ξ′∈Sn-1,δ>0{ess.inf⁡x′∈Sξ′(δ)(lim⁡⁡inf⁡r→+∞ϕ(rx′))}.Theorem 4.1 (cf.  [33, 37, 38]).
Letn≥2. Assume that M∞>0. Then the classical solution for (1.1) blows up in finite time, and the blow-up time is estimated as follows:
(4.5)Tmax≤1p-1M∞1-p.This result allows us to remove the assumption of Gui and Wang [29] on uniformity of initial data at space infinity; lim|x|→∞ψ(x)=ψ∞, and to give information of the life span for initial data of intermediate size.Here, we show some examples of the valuesM∞ and the initial data ϕ in space dimensions n=2. For simplicity, we employ polar coordinates.(1)ϕ(r,θ)=1-exp(-r2).Sinceliminfr→+∞ϕ(rx′)=1, we have M∞=1.(2)ϕ(r,θ)={1-exp(-r2)}(1+cosθ).Sinceliminfr→+∞ϕ(rx′)=1+cosθ, we have M∞=2.(3)ϕ(r,θ)={1-exp(-r2)}(1+cosθ)(2-cosr).Sinceliminfr→+∞ϕ(rx′)=1+cosθ, we have M∞=2.Once we admit the theorem, we can prove the following corollary immediately.Corollary 4.2.
Letn≥2. Suppose that M∞=∥ϕ∥L∞(Rn). Then the solution u blows up at minimal blow-up time.Remark 4.3.
For the examples of the initial data 1 and 2,∥ϕ∥L∞(Rn)=M∞ holds. Hence, the solutions blow up at minimal blow-up time. However, for the example 3, we cannot specify the life span Tmax.Outline of the Proof of the Theorem
The proof is based on a slight modification of Kaplan’s method. We first prepare the sequence{wj(t)}. For ξ′∈Sn-1 and δ>0, we first determine the sequences {aj}⊂Rn and {Rj}⊂(0,∞). Let {aj}⊂Rn be a sequence satisfying that |aj|→∞ as j→∞, and that aj/|aj|=ξ′ for any j∈N. Put Rj=(δ4-δ2/2)|aj|. For Rj>0, let ρRj be the first eigenfunction of -Δ on BRj(0)={x∈Rn;|x|<Rj} with zero Dirichlet boundary condition under the normalization ∫BRj(0)ρRj(x)dx=1. Moreover, let μRj be the corresponding first eigenvalue. For the solutions for (1.1), define
(4.6)wj(t):=∫BRj(0)u(x+aj,t)ρRj(x)dx.
Now we introduce the following two lemmas.Lemma 4.4 (see [13]).
The blow-up time ofwj is estimated from above as follows:
(4.7)Twj*≤log⁡(1-μRjwj1-p(0))-(p-1)μRj
for large j.Lemma 4.5 (cf.  [13]).
(i) We have(4.8)liminf⁡j→+∞wj(0)≥ess.infx′∈Sξ′(δ)ϕ∞(x′).
(ii) We have(4.9)lim⁡j→+∞log⁡(1-μRjwj1-p(0))-μRjwj1-p(0)=1.From the definition of wj(t), Tmax≤Twj* holds for large j. Using the lemmas, we obtain that
(4.10)Tmax⁡≤limsup⁡j+→∞Twj*≤limsup⁡j+→∞log⁡(1-μRjwj1-p(0))-(p-1)μRj=1p-1lim⁡j→+∞log⁡(1-μRjwj1-p(0))-μRjwj1-p(0)⋅(liminf⁡j→+∞wj(0))1-p≤1p-1(ess.infx′∈Sξ′(δ)ϕ∞(x′))1-p≤1p-1M∞1-p.
This completes the proof.Remark 4.6.
For the problem (1.1), it is well known that
(4.11)u¯(x,t)=(etΔϕ)(x)[1-(p-1)∫0t‖esΔϕ‖L∞(Rn)p-1ds]-1/(p-1),u̲(x,t)=((etΔϕ)(x)1-p-(p-1)t)-1/(p-1)
are a supersolution and a subsolution, respectively, where etΔϕ=∫RnGt(x-y)ϕ(y)dx. Hence, we note that if there exist constants T1,T2>0 satisfying
(4.12)1=(p-1)∫0T1‖esΔϕ‖L∞(Rn)p-1ds,‖eT2Δϕ‖L∞(Rn)1-p=(p-1)T2,
then we have T1≤Tmax≤T2.At last, we introduce the result inn=1. The proof does work in the same way as in n≥2.Theorem 4.7 (see [13]).
Letn=1. Assume that
(4.13)max⁡{lim⁡⁡inf⁡x→-∞ϕ(x),lim⁡⁡inf⁡x→+∞ϕ(x)}>0.
Then the classical solution for (1.1) blows up in finite time, and the blow-up time is estimated as follows:
(4.14)Tmax⁡≤1p-1(max⁡{lim⁡⁡inf⁡x→-∞ϕ(x),lim⁡⁡inf⁡x→+∞ϕ(x)})1-p.


## 4.1. Upper Bound of the Iife Span

In this section, we shall show an upper bound of the life span of positive solutions of the Cauchy problem for a semilinear heat equation:(4.1)∂u∂t=Δu+f(u),(x,t)∈Rn×(0,∞),u(x,0)=ϕ(x)≥0,x∈Rn,
where n∈N, and ϕ is a bounded continuous function on Rn. We assume that F(u) satisfies(4.2)f(u)≥upforu≥0,
with p>1.In order to state the results, we prepare several notations. Forξ′∈Sn-1, and δ∈(0,2), we set neighborhood Sξ′(δ):(4.3)Sξ′(δ):={η′∈Sn-1;|η′-ξ′|<δ}.
Define(4.4)M∞:=sup⁡ξ′∈Sn-1,δ>0{ess.inf⁡x′∈Sξ′(δ)(lim⁡⁡inf⁡r→+∞ϕ(rx′))}.Theorem 4.1 (cf.  [33, 37, 38]).
Letn≥2. Assume that M∞>0. Then the classical solution for (1.1) blows up in finite time, and the blow-up time is estimated as follows:
(4.5)Tmax≤1p-1M∞1-p.This result allows us to remove the assumption of Gui and Wang [29] on uniformity of initial data at space infinity; lim|x|→∞ψ(x)=ψ∞, and to give information of the life span for initial data of intermediate size.Here, we show some examples of the valuesM∞ and the initial data ϕ in space dimensions n=2. For simplicity, we employ polar coordinates.(1)ϕ(r,θ)=1-exp(-r2).Sinceliminfr→+∞ϕ(rx′)=1, we have M∞=1.(2)ϕ(r,θ)={1-exp(-r2)}(1+cosθ).Sinceliminfr→+∞ϕ(rx′)=1+cosθ, we have M∞=2.(3)ϕ(r,θ)={1-exp(-r2)}(1+cosθ)(2-cosr).Sinceliminfr→+∞ϕ(rx′)=1+cosθ, we have M∞=2.Once we admit the theorem, we can prove the following corollary immediately.Corollary 4.2.
Letn≥2. Suppose that M∞=∥ϕ∥L∞(Rn). Then the solution u blows up at minimal blow-up time.Remark 4.3.
For the examples of the initial data 1 and 2,∥ϕ∥L∞(Rn)=M∞ holds. Hence, the solutions blow up at minimal blow-up time. However, for the example 3, we cannot specify the life span Tmax.Outline of the Proof of the Theorem
The proof is based on a slight modification of Kaplan’s method. We first prepare the sequence{wj(t)}. For ξ′∈Sn-1 and δ>0, we first determine the sequences {aj}⊂Rn and {Rj}⊂(0,∞). Let {aj}⊂Rn be a sequence satisfying that |aj|→∞ as j→∞, and that aj/|aj|=ξ′ for any j∈N. Put Rj=(δ4-δ2/2)|aj|. For Rj>0, let ρRj be the first eigenfunction of -Δ on BRj(0)={x∈Rn;|x|<Rj} with zero Dirichlet boundary condition under the normalization ∫BRj(0)ρRj(x)dx=1. Moreover, let μRj be the corresponding first eigenvalue. For the solutions for (1.1), define
(4.6)wj(t):=∫BRj(0)u(x+aj,t)ρRj(x)dx.
Now we introduce the following two lemmas.Lemma 4.4 (see [13]).
The blow-up time ofwj is estimated from above as follows:
(4.7)Twj*≤log⁡(1-μRjwj1-p(0))-(p-1)μRj
for large j.Lemma 4.5 (cf.  [13]).
(i) We have(4.8)liminf⁡j→+∞wj(0)≥ess.infx′∈Sξ′(δ)ϕ∞(x′).
(ii) We have(4.9)lim⁡j→+∞log⁡(1-μRjwj1-p(0))-μRjwj1-p(0)=1.From the definition of wj(t), Tmax≤Twj* holds for large j. Using the lemmas, we obtain that
(4.10)Tmax⁡≤limsup⁡j+→∞Twj*≤limsup⁡j+→∞log⁡(1-μRjwj1-p(0))-(p-1)μRj=1p-1lim⁡j→+∞log⁡(1-μRjwj1-p(0))-μRjwj1-p(0)⋅(liminf⁡j→+∞wj(0))1-p≤1p-1(ess.infx′∈Sξ′(δ)ϕ∞(x′))1-p≤1p-1M∞1-p.
This completes the proof.Remark 4.6.
For the problem (1.1), it is well known that
(4.11)u¯(x,t)=(etΔϕ)(x)[1-(p-1)∫0t‖esΔϕ‖L∞(Rn)p-1ds]-1/(p-1),u̲(x,t)=((etΔϕ)(x)1-p-(p-1)t)-1/(p-1)
are a supersolution and a subsolution, respectively, where etΔϕ=∫RnGt(x-y)ϕ(y)dx. Hence, we note that if there exist constants T1,T2>0 satisfying
(4.12)1=(p-1)∫0T1‖esΔϕ‖L∞(Rn)p-1ds,‖eT2Δϕ‖L∞(Rn)1-p=(p-1)T2,
then we have T1≤Tmax≤T2.At last, we introduce the result inn=1. The proof does work in the same way as in n≥2.Theorem 4.7 (see [13]).
Letn=1. Assume that
(4.13)max⁡{lim⁡⁡inf⁡x→-∞ϕ(x),lim⁡⁡inf⁡x→+∞ϕ(x)}>0.
Then the classical solution for (1.1) blows up in finite time, and the blow-up time is estimated as follows:
(4.14)Tmax⁡≤1p-1(max⁡{lim⁡⁡inf⁡x→-∞ϕ(x),lim⁡⁡inf⁡x→+∞ϕ(x)})1-p.



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*Source: 417261-2012-04-10.xml*