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

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.

1. Introduction

1.1. Fujita Type Results

We first recall the result on the Cauchy problem for a semilinear heat equation:𝜕𝑢𝜕𝑡=Δ𝑢+𝑢^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (1.1) where 𝑛∈𝐍, Δ is the 𝑛-dimensional Laplacian, and 𝑝>1. Let 𝜙 be a bounded continuous function on 𝐑^𝑛.

In pioneer work [1], Fujita showed that the exponent 𝑝_𝐹=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: 𝐺_𝑡(𝑥)=(4𝜋𝑡)^−𝑛/2exp(−|𝑥|²/4𝑡).

Theorem 1.1 (see [1]). Suppose that 𝜙∈𝐶²(𝐑^𝑛) and that its all derivatives are bounded.(i)Let 𝑝<𝑝_𝐹. Then there is no global solution of (1.1) satisfying that   𝑢(𝑥,𝑡)≤𝑀exp|𝑥|^𝛽, (1.2) for  𝑀>0 and 0<𝛽<2.(ii)Let 𝑝>𝑝_𝐹. Then for any 𝜏>0 there exists 𝛿>0 with the following property: if 𝜙(𝑥)≤𝛿𝐺_𝜏(𝑥), (1.3) then there exists a global solution of (1.1) satisfying 𝑢(𝑥,𝑡)≤𝑀exp|𝑥|^𝛽 (1.4) 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=𝑝_𝐹 when 𝑛=1 or 2.

Theorem 1.2 (see [2]). In case of 𝑛=2, 𝑝=𝑝_𝐹=2 or 𝑛=1, 𝑝=𝑝_𝐹=3, (1.1) has no global solutions for any nontrivial initial data.

In genaral space dimensions, Kobayashi et al. [3] consider the following problem:𝜕𝑢(𝑢),()𝜕𝑡=Δ𝑢+𝑓𝑥,𝑡∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (1.5) where 𝑛∈𝐍 and 𝑝>1. Let 𝜙 be a bounded continuous function on 𝐑^𝑛.

Theorem 1.3 (see [3]). Suppose that 𝑓 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)∫^𝜀₀₊𝑓(𝜆)/𝜆^2+(2+𝑛)𝑑𝜆 for some 𝜀>0,(c)there exists a positive constant 𝑐≤1 such that𝑓𝜆𝜇≥𝜇^1+(2+𝑛)𝑓(𝜆),𝑓0<𝜆≤𝜇,𝜆<𝑐𝜆𝜇≥𝜇^2+(2+𝑛)𝑓(𝜆)0<𝜇≤𝜆<𝑐. (1.6) 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: 𝑢(𝑥,𝑡)=𝐺_𝑡∗𝜙(𝑥)+^𝑡₀𝐺_𝑡−𝑠∗𝑢^𝑝(𝑠)(𝑥)𝑑𝑠. (1.7)
(ii) The critical nonlinearity of power type 𝑓(𝑢)=𝑢^1+2/𝑛 satisfies the assumptions (a), (b), and (c) in [3].

Weissler proved the nonexistence of global solution in 𝐿^𝑝-framework in [4]. The proof is quite short and elegant.

Theorem 1.5 (see [4]). Suppose 𝑝=𝑝_𝐹 and that 𝜙≥0 in 𝐿^𝑞(𝐑^𝑛) (𝑞≥1) is not identically zero. Then there is no nonnegative global solution 𝑢∶[0,∞)→𝐿^𝑞 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 solution 𝑢(𝜀,𝑥)≥𝑘𝐺_𝛼(𝑥)(𝑘,𝛼>0) for some 𝜀>0, we can obtain that ‖𝑢(𝑡)‖_{𝐿1(𝐑𝑛)}≥𝐶(𝑘,𝑝,𝑛)∫^𝑡₀(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)Let 𝑝∈(1,𝑝_𝐹]. Then every nontrivial solution of (1.1) blows up in finite time.(ii)Let 𝑝∈(𝑝_𝐹,∞). 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:𝜕𝑢𝜕𝑡=Δ𝑢^𝑚+𝑡^𝑠|𝑥|^𝜎𝑢^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (1.8) where 𝑚>max{(𝑛−2)/𝑛,0}, 𝜎>max{−𝑛,−2} and 𝑝>max{𝑚,1}. They showed that the critical exponent is 𝑝_𝑐1∶=𝑚+(𝑚−1)𝑠+(2+2𝑠+𝜎)/𝑛.

Theorem 1.6 (see [6]). (i) If 1<𝑝≤𝑝_𝑐1, then every nontrivial solution of (1.8) blows up in finite time.
(ii) If 𝑝>𝑝_𝑐1, then (1.8) has global classical solutions for small initial data.

Another extension is the following quasilinear parabolic equation:𝜕𝑢||𝜕𝑡=𝑑𝑖𝑣∇𝑢^𝑚−1∇𝑢+𝑡^𝑠|𝑥|^𝜎𝑢^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (1.9) 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 𝑝_𝑐2∶=𝑚+(1+𝑚+2𝑠+𝜎)/𝑛.

Theorem 1.7 (see [7]). (i) If 1<𝑝≤𝑝_𝑐2, then every nontrivial solution of the (1.9) blows up in finite time.
(ii) If 𝑝>𝑝_𝑐2, 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: 𝜕𝑢|𝑥|𝜕𝑡=Δ𝑢+^𝜎1𝑢^𝑝11𝑣^𝑝12,(𝑥,𝑡)∈𝐑^𝑛×(0,∞),𝜕𝑣|𝑥|𝜕𝑡=Δ𝑣+^𝜎1𝑢^𝑝21𝑣^𝑝22,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛,𝑣(𝑥,0)=𝜓(𝑥)≥0,𝑥∈𝐑^𝑛, (1.10) where 𝜎_𝑗>max{−2,−𝑛} and 𝑝_𝑗𝑘≥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 𝜇_𝑅 be the first Dirichlet eigenvalue of −Δ in the ball 𝐵_𝑅.

Theorem 1.9 (see [16]). The solution of (1.1) blows up in finite time if liminf_𝑥→∞|𝑥|^2/(𝑝−1)𝜙(𝑥)>𝜇₁^1/(𝑝−1). (1.11)

We put Ω={(𝑟,𝜔)∈(0,∞)×𝐒^𝑛−1;𝑟>𝑅,𝑑(𝜔,𝜔₀)<𝑐𝑟^−𝜇} for some 𝑅>0, 𝑐>0, 𝜔₀∈𝐒^𝑛−1, and 0≤𝜇<1, where 𝑑(⋅,⋅) denotes the usual distance on the unit sphere 𝐒^𝑛−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:𝜕𝑢|𝑢|𝜕𝑡=Δ𝑢+^𝑝−1𝑢,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙,𝑥∈𝐑^𝑛, (1.12) 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 data 𝜙 may change sign. Suppose that 𝜙∈𝑊^1,∞(𝐑^𝑛) satisfies the following: 𝜙≥𝐾₁𝑟^−𝛼𝑖𝑛Ω𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼>0,𝐾₁>0,∇𝜙≥𝐾₂𝑟^{−𝛼−1+𝜇}𝑖𝑛Ω𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼>0,𝐾₂>0, (1.13) 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 data 𝜙 is nonnegative. Suppose that 𝜙∈𝐿^∞(𝐑^𝑛) satisfies that 𝜙≥𝐾₁𝑟^−𝛼𝑖𝑛Ω𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼>0𝐾₁>0, (1.14) 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 Sections 2 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:𝜕𝑢𝜕𝑡=Δ𝑢+𝑢^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜆𝜓≥0,𝑥∈𝐑^𝑛,𝜕𝑢|𝑢|𝜕𝑡=Δ𝑢+^𝑝−1𝑢,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜆𝜓,𝑥∈𝐑^𝑛, (2.1)(2.2) where 𝑛∈𝐍, 𝑝>1. Let 𝜓 be a bounded continuous function on 𝐑^𝑛 and 𝜆 be a positive parameter.

In this paper, we define the lifespan (or blow-up time) 𝑇_max as𝑇_max∶=sup𝑇>0∣.Theproblempossessesauniqueclassicalsolutionin𝐑^𝑛×[0,𝑇). (2.3)

We first introduce the result about asymptotic behavior of the life span 𝑇_max(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 that 𝜓 is nonnegative.(i)There exist constants 𝐶₁>0 and 𝐶₂>0 such that 𝐶₁𝜆^1−𝑝≤𝑇_max(𝜆)≤𝐶₂𝜆^1−𝑝 for large 𝜆.(ii)If liminf_|𝑥|→∞𝜓(𝑥)>0, then there exists constants 𝐶₁>0 and 𝐶₂>0 such that 𝐶₁𝜆^1−𝑝≤𝑇_max(𝜆)≤𝐶₂𝜆^1−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 that 𝜓 is nonnegative.(i)We have lim_𝜆→∞𝑇_max(𝜆)⋅𝜆^𝑝−1=1𝜓𝑝−1^{𝐿1−𝑝∞(𝐑𝑛)}. (2.4)(ii)If   lim_|𝑥|→∞𝜓(𝑥)=𝜓_∞>0, then lim_𝜆→0𝑇_max(𝜆)⋅𝜆^𝑝−1=1𝜓𝑝−1^∞1−𝑝. (2.5)

It is noteworthy that the limiting values as 𝜆→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:𝜕𝑢(𝑥)𝑢𝜕𝑡=Δ𝑢+𝑎^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜆𝜓≥0,𝑥∈𝐑^𝑛, (2.6) where 𝑎(𝑥)≥0 is Hölder continuous, 0≤𝜓∈𝐵𝐶(𝐑^𝑛), 𝜓≢0, 𝑝>1 and 𝜆>0 is a parameter. The author treated the initial data in the following two classes:ClassS0≤𝜓(𝑥)≤𝛿exp−𝛾|𝑥|²,where𝛿,𝛾>0.ClassL𝑐₁≤𝜓(𝑥)≤𝑐₂,where𝑐₁,𝑐₂>0. (2.7)

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 that 𝜓 belongs to Class S.(i)Let 𝑝∈(1,1+(2+𝑚)/𝑛). Then 𝑇_max(𝜆)∼𝜆^{2(1−𝑝)/(2+𝑚−𝑛(1−𝑝))},as𝜆⟶0. (2.8)(ii)Let 𝑝=1+(2+𝑚)/𝑛. Then there exists a constant 𝑐₁>0 and for every 𝜀>0, a constant 𝑐₂>0 such that 𝑐₁𝜆^{−(2+𝑚)/𝑛}≤log𝑇_max(𝜆)≤𝑐₂𝜆^{−𝜀−(2+𝑚)/𝑛},forsmall𝜆. (2.9)

Theorem 2.4 (see [34] Class L for small 𝜆). Assume that 𝜓 belongs to Class L. Then 𝑇_max(𝜆)∼𝜆^{2(1−𝑝)/(2+𝑚)},as𝜆⟶0. (2.10)

Theorem 2.5 (see [34], for large 𝜆). Assume that 𝑎 is bounded and let 𝜓 be arbitrary initial data.(i)If there exists an 𝑥₀∈𝐑^𝑛 such that 𝑎(𝑥₀),𝜓(𝑥₀)>0, then 𝑇_max(𝜆)∼𝜆^1−𝑝,as𝜆⟶∞. (2.11)(ii)If dist(supp(𝑎),supp(𝜓))>0, then 𝑇_max(𝜆)∼log𝜆⁻¹,as𝜆⟶∞. (2.12)

Kobayashi extended the results to the following system of equations:𝜕𝑢(𝑥)𝑣𝜕𝑡=Δ𝑢+𝑎^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(0,∞),𝜕𝑣(𝑥)𝑢𝜕𝑡=Δ𝑣+𝑏^𝑞,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()0,∞𝑥,0=𝜆^𝜇𝜙(𝑥),𝑥∈𝐑^𝑛,𝑣(𝑥,0)=𝜆^𝜈𝜓(𝑥),𝑥∈𝐑^𝑛, (2.13) 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]:𝜕𝑢𝜕𝑡=Δ𝑢^𝑚+𝑢^𝑝,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜆𝜓,𝑥∈𝐑^𝑛, (2.14) where 1<𝑚<𝑝, and 𝜓 is a nonnegative and bounded function.

Theorem 2.6 (see [31]). (i) Let 𝑝>𝑚. Suppose that ‖𝜓‖_{𝐿∞(𝐑𝑛)}=𝜓(0)>0. Then lim_𝜆→∞𝜆^𝑝−1𝑇_max(𝜆)=1𝜓(0)𝑝−1^1−𝑝. (2.15)
(ii) Let 𝑝>𝑚. Suppose that ‖𝜓‖_{𝐿∞(𝐑𝑛)}=lim_|𝑥|→∞𝜓(𝑥)>0. Then lim_𝜆→0𝜆^𝑝−1𝑇_max(𝜆)=1𝜓𝑝−1^{𝐿1−𝑝∞(𝐑𝑛)}. (2.16)
(iii) Assume that 𝑎≥0. Let 𝑚<𝑝<𝑚+2/𝑛 or 𝑎<2/(𝑝−𝑚). Suppose that 𝜓(𝑥)∼|𝑥|^−𝑎 for large |𝑥|. Then 𝑇_max(𝜆)∼𝜆^{(1−𝑝)/(1−min{𝑎,𝑛}(𝑝−𝑚)/2)}as𝜆⟶0. (2.17)

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 𝑇_max(𝜆) for (2.2) as 𝜆→∞. The authors introduced the following function space:𝐷=𝜓∈𝐵𝐶𝐑^𝑛∶exp−𝑐|𝑥|²∇𝜓∈𝐿²𝐑^𝑛forsome𝑐>0. (2.18)

Theorem 2.7 (see [39]). Suppose that 𝜓∈𝐷 satisfies the following assumptions; (A1)|𝜓| attains its maximum at some point 𝑥=𝑎, and 𝜓 satisfies that 𝜓(𝑥)−𝜓(𝑎)=|𝑥−𝑎|^𝑘𝜓(𝑥)+𝑜|𝑥−𝑎|^𝑘+2, (2.19) at 𝑥=𝑎 with some 𝑘>−2, where 𝜓 is twice continuously differentiable at 𝑥=𝑎 and satisfies 𝜓(𝑎)=0, ∇𝜓(𝑎)=0 and Δ𝜓(𝑎)≥0.(A2)There exist 𝑅>0 and 𝛿>0 such that 𝜓(𝑥)<𝜓_{𝐿∞(𝐑𝑛)}−𝛿∀|𝑥|>𝑅. (2.20)
If |𝜓| attains its maximum at only one point 𝑥=𝑎, then 𝑇_max(𝜆)=1𝜓𝑝−1^{𝐿1−𝑝∞(𝐑𝑛)}𝜆^1−𝑝+𝐾𝜓_𝐿^{(2+𝑘/2)(1−𝑝)−1}_{∞(𝐑𝑛)}Δ𝜓(𝑎)𝜆^{(2+𝑘/2)(1−𝑝)}+𝑜𝜆^{(2+𝑘/2)(1−𝑝)}, (2.21) as 𝜆→∞ with 2𝐾=^𝑘+1Γ((𝑘+𝑛+2)/2)𝑝−1^1+𝑘/2Γ)).((𝑛+2/2 (2.22) In particular, if 𝜓 is smooth at 𝑥=𝑎, then 𝑇_max(𝜆)=1𝜓𝑝−1^{𝐿1−𝑝∞(𝐑𝑛)}𝜆^1−𝑝+2𝜓𝑝−1^{𝐿2(1−𝑝)−1∞(𝐑𝑛)}Δ𝜓(𝑎)𝜆^2(1−𝑝)+𝑜𝜆^2(1−𝑝), (2.23) as 𝜆→∞.

2.2. Life Span for the Equation with Large Diffusion

We shall consider the following Cauchy problem:𝜕𝑢|𝑢|𝜕𝑡=𝐷Δ𝑢+^𝑝−1𝑢,(𝑥,𝑡)∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜆+𝜙,𝑥∈𝐑^𝑛, (2.24) where 𝐷>0, 𝑝>1, 𝑛≥3, 𝜆>0, and 𝜙∈𝐿^∞(𝐑^𝑛)∩𝐿¹(𝐑^𝑛,(1+|𝑥|)²𝑑𝑥).

In [24, 25] Fujishima and Ishige obtained the asymptotics of the life span 𝑇_max(𝐷) of the solution of (2.24) as 𝐷→∞. The situation is divided into three cases:_𝐑𝑛𝜙(𝑥)𝑑𝑥>0,_𝐑𝑛𝜙(𝑥)𝑑𝑥=0,and_𝐑𝑛𝜙(𝑥)𝑑𝑥<0. (2.25) We prepare some notation. Put the following:𝑀𝜙∶=_𝐑𝑛𝜙(𝑥)𝑑𝑥,Ξ𝜙∶=_𝐑𝑛𝑥𝜙(𝑥)𝑑𝑥,𝑆_𝜆𝜆∶=^1−𝑝.𝑝−1 (2.26)

Theorem 2.8 (see [24, 25]). (i) Assume that 𝑀(𝜙)>0. Then 𝑇_max(𝐷)≤𝑆_𝜆 for any 𝐷>0, and 𝑆_𝜆−𝑇_max(𝐷)=4𝜋𝑆_𝜆^−𝑛/2𝜆^−𝑝𝐷^−𝑛/2𝑀𝜙+𝑂𝐷⁻¹ (2.27) as 𝐷→∞.
(ii) Assume that 𝑀(𝜙)=0. Then 𝑇_max(𝐷)≤𝑆_𝜆 for any 𝐷>0, and 𝑆_𝜆−𝑇_max(𝐷)=4𝜋𝑆_𝜆^−𝑛/2𝜆^𝑝2𝑒𝑆_𝜆𝐷^{(−𝑛−1)/2}Ξ𝜙+𝑂𝐷^−1/2, (2.28) as 𝐷→∞.
(iii) Assume that 𝑀(𝜙)<0. Then 𝑇_max(𝐷)≤𝑆_𝜆 for any 𝐷>0, and 𝑆_𝜆−𝑇_max(𝐷)=𝑂𝐷^{−𝑛/2−1}, (2.29) 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]):𝜕𝑢(𝑢),()𝜕𝑡=Δ𝑢+𝑓𝑥,𝑡∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (3.1) where 𝜙 is a bounded continuous function on 𝐑^𝑛. Suppose that[),𝑓(𝜉)(),𝑓𝑖𝑠𝑙𝑜𝑐𝑎𝑙𝑙𝑦𝐿𝑖𝑝𝑠𝑐ℎ𝑖𝑡𝑧𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑖𝑛0,∞>0𝜉>0_∞1𝑑𝜉𝑓(𝜉)<∞. (3.2) Applying the comparison principle to (3.1), we always have𝑇_max≥^∞_{𝜙𝐿∞𝑛)(𝐑}𝑑𝜉𝑓(𝜉). (3.3) When 𝑓(𝑢)=𝑢^𝑝, we always have𝑇_max≥1𝜙𝑝−1^{𝐿1−𝑝∞(𝐑𝑛)}. (3.4) A solution 𝑢 to (3.1) with initial data 𝜙 is said to blow up at minimal blow-up time provided that𝑇_max=^∞_{𝜙𝐿∞𝑛)(𝐑}𝑑𝜉𝑓(𝜉). (3.5) We put 𝜌(𝑥)∶=𝑒^−|𝑥|/(∫_𝐑𝑛𝑒^−|𝑦|𝑑𝑦) and 𝐴_𝜌(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)/𝜉^𝑝 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 Thereexistsasequence𝑥_𝑛⊂𝐑^𝑛𝑥suchthat_𝑛⟶∞and𝜙𝑥+𝑥_𝑛⟶𝜙_{𝐿∞(𝐑𝑛)}a.e.in𝐑^𝑛as𝑛⟶∞; (3.6)sup_{𝑥∈𝐑𝑛}𝐴_𝜌𝑥;𝜙=𝜙_{𝐿∞(𝐑𝑛)}. (3.7)

In [36], Seki et al. consider the following quasilinear equations:𝜕𝑢(𝑢)(𝑢),()𝜕𝑡=ΔΦ+𝑓𝑥,𝑡∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (3.8) where 𝜙 is a bounded continuous function on 𝐑^𝑛. Suppose thatΦ(𝜉),𝑓(𝜉)∈𝐶¹[0,∞)∩𝐶^∞(),Φ(𝜉)0,∞>0,Φ^′(𝜉)>0,Φ^′′(𝜉)(),𝑓(𝜉)(),Φ(0)≥0𝜉>0>0𝜉>0=0. (3.9) The authors proved that if there exist a function Ψ(𝜂) and constants 𝑐>0 and 𝜂₁>0 such thatΨ𝜂>0,Ψ^′𝜂≥0,Ψ^′′𝜂≥0𝜂>𝜂1_∞𝜂1+1,𝑑𝜂Ψ𝜂𝑓Φ<∞,₋₁𝜂^′Ψ𝜂−𝑓Φ⁻¹𝜂Ψ^′𝜂≥𝑐Ψ𝜂Ψ^′𝜂𝜂>𝜂1, (3.10) 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 𝑢_𝑡=Δ𝑢^𝑚+𝑢^𝑝   (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:𝜕𝑢(𝑢),()𝜕𝑡=Δ𝑢+𝑓𝑥,𝑡∈𝐑^𝑛×(),𝑢()(𝑥)0,∞𝑥,0=𝜙≥0,𝑥∈𝐑^𝑛, (4.1) where 𝑛∈𝐍, and 𝜙 is a bounded continuous function on 𝐑^𝑛. We assume that 𝐹(𝑢) satisfies𝑓(𝑢)≥𝑢^𝑝for𝑢≥0, (4.2) with 𝑝>1.

In order to state the results, we prepare several notations. For 𝜉′∈𝐒^𝑛−1, and 𝛿∈(0,√2), we set neighborhood 𝑆_𝜉′(𝛿):𝑆_𝜉′(𝛿)∶=𝜂′∈𝐒^𝑛−1;𝜂′−𝜉′<𝛿. (4.3) Define𝑀_∞∶=sup_{𝜉′∈𝐒𝑛−1,𝛿>0}ess.inf_{𝑥′∈𝑆𝜉′(𝛿)}liminf_𝑟→+∞𝜙𝑟𝑥^′. (4.4)

Theorem 4.1 (cf. [33, 37, 38]). Let 𝑛≥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: 𝑇_max≤1𝑀𝑝−1^∞1−𝑝. (4.5)

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 values 𝑀_∞ and the initial data 𝜙 in space dimensions 𝑛=2. For simplicity, we employ polar coordinates.

(1)  𝜙(𝑟,𝜃)=1−exp(−𝑟²).

  Since liminf_𝑟→+∞𝜙(𝑟𝑥′)=1, we have 𝑀_∞=1.

(2)  𝜙(𝑟,𝜃)={1−exp(−𝑟²)}(1+cos𝜃).

  Since liminf_𝑟→+∞𝜙(𝑟𝑥′)=1+cos𝜃, we have 𝑀_∞=2.

(3)  𝜙(𝑟,𝜃)={1−exp(−𝑟²)}(1+cos𝜃)(2−cos𝑟).

  Since liminf_𝑟→+∞𝜙(𝑟𝑥′)=1+cos𝜃, we have 𝑀_∞=2.

Once we admit the theorem, we can prove the following corollary immediately.

Corollary 4.2. Let 𝑛≥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 𝑇_max.

Outline of the Proof of the Theorem
The proof is based on a slight modification of Kaplan’s method. We first prepare the sequence {𝑤_𝑗(𝑡)}. For 𝜉′∈𝐒^𝑛−1 and 𝛿>0, we first determine the sequences {𝑎_𝑗}⊂𝐑^𝑛 and {𝑅_𝑗}⊂(0,∞). Let {𝑎_𝑗}⊂𝐑^𝑛 be a sequence satisfying that |𝑎_𝑗|→∞ as 𝑗→∞, and that 𝑎_𝑗/|𝑎_𝑗|=𝜉′ for any 𝑗∈𝐍. Put 𝑅_𝑗=(𝛿4−𝛿²/2)|𝑎_𝑗|. For 𝑅_𝑗>0, let 𝜌_𝑅𝑗 be the first eigenfunction of −Δ on 𝐵_𝑅𝑗(0)={𝑥∈𝐑^𝑛;|𝑥|<𝑅_𝑗} with zero Dirichlet boundary condition under the normalization ∫_{𝐵𝑅𝑗(0)}𝜌_𝑅𝑗(𝑥)𝑑𝑥=1. Moreover, let 𝜇_𝑅𝑗 be the corresponding first eigenvalue. For the solutions for (1.1), define 𝑤_𝑗(𝑡)∶=_{𝐵𝑅𝑗(0)}𝑢𝑥+𝑎_𝑗,𝑡𝜌_𝑅𝑗(𝑥)𝑑𝑥. (4.6) Now we introduce the following two lemmas.Lemma 4.4 (see [13]). The blow-up time of 𝑤_𝑗 is estimated from above as follows: 𝑇^∗_𝑤𝑗≤log1−𝜇_𝑅𝑗𝑤_𝑗^1−𝑝(0)−𝑝−1𝜇_𝑅𝑗 (4.7) for large 𝑗.Lemma 4.5 (cf. [13]). (i) We have liminf_𝑗→+∞𝑤_𝑗(0)≥ess.inf_{𝑥′∈𝑆𝜉′(𝛿)}𝜙_∞𝑥^′. (4.8)
(ii) We have lim_𝑗→+∞log1−𝜇_𝑅𝑗𝑤_𝑗^1−𝑝(0)−𝜇_𝑅𝑗𝑤_𝑗^1−𝑝(0)=1. (4.9)From the definition of 𝑤_𝑗(𝑡), 𝑇_max≤𝑇^∗_𝑤𝑗 holds for large 𝑗. Using the lemmas, we obtain that 𝑇_max≤limsup_𝑗+→∞𝑇^∗_𝑤𝑗≤limsup_𝑗+→∞log1−𝜇_𝑅𝑗𝑤_𝑗^1−𝑝(0)−𝑝−1𝜇_𝑅𝑗=1𝑝−1lim_𝑗→+∞log1−𝜇_𝑅𝑗𝑤_𝑗^1−𝑝(0)−𝜇_𝑅𝑗𝑤_𝑗^1−𝑝(0)⋅liminf_𝑗→+∞𝑤_𝑗(0)^1−𝑝≤1𝑝−1ess.inf_{𝑥′∈𝑆𝜉′(𝛿)}𝜙_∞𝑥^′1−𝑝≤1𝑀𝑝−1^∞1−𝑝. (4.10) This completes the proof.

Remark 4.6. For the problem (1.1), it is well known that 𝑢(𝑥,𝑡)=𝑒^𝑡Δ𝜙(𝑥)1−𝑝−1∫^𝑡₀𝑒^𝑠Δ𝜙_𝐿^𝑝−1_{∞(𝐑𝑛)}𝑑𝑠^{−1/(𝑝−1)},𝑢(𝑥,𝑡)=𝑒^𝑡Δ𝜙(𝑥)^1−𝑝−𝑝−1𝑡^{−1/(𝑝−1)} (4.11) are a supersolution and a subsolution, respectively, where 𝑒^𝑡Δ𝜙=∫_𝐑𝑛𝐺_𝑡(𝑥−𝑦)𝜙(𝑦)𝑑𝑥. Hence, we note that if there exist constants 𝑇₁,𝑇₂>0 satisfying 1=𝑝−1^𝑇1₀𝑒^𝑠Δ𝜙_𝐿^𝑝−1_{∞(𝐑𝑛)}𝑒𝑑𝑠,_𝑇2Δ𝜙_𝐿^1−𝑝_{∞(𝐑𝑛)}=𝑝−1𝑇₂, (4.12) then we have 𝑇₁≤𝑇_max≤𝑇₂.

At last, we introduce the result in 𝑛=1. The proof does work in the same way as in 𝑛≥2.

Theorem 4.7 (see [13]). Let 𝑛=1. Assume that maxliminf_{𝑥→−∞}𝜙(𝑥),liminf_𝑥→+∞𝜙(𝑥)>0. (4.13) Then the classical solution for (1.1) blows up in finite time, and the blow-up time is estimated as follows: 𝑇_max≤1𝑝−1maxliminf_{𝑥→−∞}𝜙(𝑥),liminf_𝑥→+∞𝜙(𝑥)^1−𝑝. (4.14)

Acknowledgment

The author would like to thank Professor Tohru Ozawa for his helpful advice and suggestions.