Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity

Abstract

A blow-up time for nonlinear heat equations with transcendental nonlinearity is investigated. An upper bound of the first blow-up time is presented. It is pointed out that the upper bound of the first blow-up time depends on the support of the initial datum.

1. Introduction

We are concerned with the initial value problem of nonstationary nonlinear heat equations: 𝜕𝑢()()(𝑢(,𝑢()𝜕𝑡𝑥,𝑡−Δ𝑢𝑥,𝑡=𝐹𝑥,𝑡))𝑥,0=𝑢₀(𝑥), (1.1) where 𝑥∈ℝ^𝑛, 𝐹 is a given nonlinear function and 𝑢 is unknown. Due to the mathematical and physical importance, existence and uniqueness theories of solutions of nonlinear heat equations have been extensively studied by many mathematicians and physicists, for example, [1–10] and references therein. Unlike other studies, we focus on the nonlinear heat equations with transcendental nonlinearities such as 𝜕|𝑢|𝜕𝑡𝑢−Δ𝑢=^𝑝𝑒^|𝑢|𝑞, (1.2) for some positive real numbers 𝑝, 𝑞. The nonlinearity in the above problem grows so fast that the solutions may blow up very fast. We are interested in how fast! Even though we present only one problem with the specific nonlinear function 𝐹(𝑢)≡|𝑢|^𝑝𝑒^|𝑢|𝑞, this nonlinearity exemplifies (analytic) nonlinearities with rapid growth.

The study of the blow-up problem has attracted a considerable attention in recent years. The latest developments for the case of power type nonlinear terms 𝐹(𝑢)≡|𝑢|^𝑝−1𝑢 are mainly devoted to the subjects of blow-up rate, set, profiles, and the possible continuation after blow-up. The continuity with respect to the initial data also has been studied.

The studies on finite time blow-up rates were conducted in [11–21]. For example, it has been proved that for 1<𝑝<(𝑛+2)/(𝑛−2), there exists a uniform constant 𝐶 such that ‖𝑢(𝑡)‖_𝐿∞≤𝐶𝑡^{−1/(𝑝−1)} (1.3) under certain constraints before the blow-up, see [19, 22]. It also has been noticed after the blow-up that for such subcritical cases 1<𝑝<(𝑛+2)/(𝑛−2) the blow-up is complete, that is to say, a proper continuation of the solution beyond the blow-up point identically equals +∞ in the whole space ℝ^𝑛. The first main contribution in this direction seems to be the work of Baras and Cohen [23] who looked into the complete blow-up of semi-linear heat equations with subcritical power type nonlinear terms, and thus established the validity of a conjecture of H. Brezis (page 143 in [23]). Further results were obtained in [18, 24, 25]; see also the references therein.

It seems to be very natural and important to find the explicit blow-up time in study of the blow-up problem. To the author's knowledge, explicit blow-up time has not been uncovered yet—even for the case of power type nonlinearity. One only began to understand that the blow-up time is continuous with respect to the initial data 𝑢₀ (for a certain topological sense) for details, see [8, 23, 24, 26–28].

This paper is mainly concerned with the blow-up time. For the power type nonlinearity, when the blow-up phenomena are established, a partial representation for an upper bound of the (first) blow-up time can be found in Section 9 in [29] and also in [30]. One preliminary observation of this research is that an upper bound of the blow-up time for the case of the power type nonlinear term is related with the explicit solution of the classical Bernoulli’s equations (see (3.5) below). For the case of transcendental nonlinearities, we prove a series of ordinary differential inequalities and equations to disclose an effective upper bound of the blow-up time for positive solutions with a large initial datum. We have found that the blow-up time (of the positive solutions) may depend not only on the norm of given initial datum but also on the area of the support of the initial datum.

The upper bound of the blow-up time we present here is universal in the sense that it is an upper bound for many popular function spaces as explained at Remark 2.3. A better upper bound and a lower bound in a special space, for example the Lebesgue space 𝐿^∞, are of obvious interest.

2. The Main Theorem

Let 𝑢₀ be a function with compact support in ℝ^𝑛 and let 𝑢 be a (smooth) solution of (1.2) inside of supp𝑢₀ with a homogeneous Dirichlet's boundary condition and the initial condition 𝑢(𝑥,0)=𝑢₀(𝑥). It is clear that supp𝑢(𝑡)⊂supp𝑢₀ for all 𝑡≥0 if we employ the trivial extension of 𝑢 to the whole space ℝ^𝑛. By virtue of maximum principle, if the initial source 𝑢₀ is nonnegative, so is 𝑢. It is also well known that a positive solution 𝑢 of (1.2) with sufficiently large initial datum blows up within a finite time; that is, there exists a positive constant 𝑇^∗(the maximal existence time) so that lim_{𝑡↑𝑇∗}‖𝑢(𝑡)‖_𝑋=∞ in an appropriate function space 𝑋. We choose an open ball 𝐵_𝛿 of radius 𝛿 that contains the support of 𝑢₀. We proceed by choosing an orthonormal basis {𝑤_𝑗}^∞_𝑗=1 for 𝐿²(𝐵_𝛿), where 𝑤_𝑗∈𝐻¹₀(𝐵_𝛿) is an eigenfunction corresponding to each eigenvalue 𝜆_𝑗 of −Δ: −Δ𝑤_𝑗=𝜆_𝑗𝑤_𝑗in𝐵_𝛿,𝑤_𝑗=0on𝜕𝐵_𝛿, (2.1) for 𝑗=1,2,…. In particular, we are interested in the eigenfunctions corresponding to the principal eigenvalue 𝜆₁>0.

We recall a relationship between the volume of the domain and the principal eigenvalue of the Laplacian, which says that 𝜆₁=𝑟²₀𝛿², (2.2) where 𝑟₀>0 is the first positive zero of the Bessel function 𝐽_𝑛/2−1 of order (𝑛/2)−1 which can be expressed by elementary functions (for 𝑛≥2, see page 45 in [31]). Also, we may choose an eigenfunction 𝑤₁ satisfying 𝑤₁>0in𝐵_𝛿,_𝐵𝛿𝑤₁(𝑥)𝑑𝑥=1. (2.3) A smooth solution 𝑢 in 𝐻¹₀(𝐵_𝛿) can be expressed by a linear combination of {𝑤_𝑗}^∞_𝑗=1: 𝑢(𝑥,𝑡)=∑^∞_𝑗=1𝑎_𝑗(𝑡)𝑤_𝑗(𝑥) (0≤𝑡<𝑇^∗,𝑥∈𝐵_𝛿), where 𝑎_𝑗(𝑡)=∫_𝐵𝛿𝑢(𝑥,𝑡)𝑤_𝑗(𝑥)𝑑𝑥. In particular, we denote the eigen-coefficient of 𝑢 with respect to the eigenfunction 𝑤₁ by 𝜂(𝑡)≡𝑎₁(𝑡).

We introduce two specific real numbers 𝑚₁ and 𝑐₀ as follows: 𝑚₁ is the smallest positive integer among 𝑚 satisfying 𝑞𝑚+𝑝>1, and 𝑐₀ is the smallest nonnegative number such that 𝑡^𝑝𝑒^𝑡𝑞>𝜆₁𝑡 holds for all 𝑡>𝑐₀.

Theorem 2.1. Let the spatial dimension 𝑛 be greater than 1. With the notations above, assume that the given initial source 𝑢₀ is large enough that the initial eigen-coefficient 𝜂₀≡∫_𝐵𝛿𝑢₀(𝑥)𝑤₁(𝑥)𝑑𝑥 is greater than both (𝑚₁!𝜆₁)^{1/(𝑞𝑚1+𝑝−1)} and 𝑐₀. Then the (first) blow-up time 𝑇^∗_𝜂 of the first eigen-coefficient 𝜂(𝑡) is less than or equal to the positive number 𝛿²𝑞𝑚₁+𝑝−1𝑟²₀𝛿ln²𝜂^𝑞𝑚1₀^+𝑝−1𝛿²𝜂^𝑞𝑚1₀^+𝑝−1−𝑚₁!𝑟²₀, (2.4) where 𝛿=(1/2)max{|𝑥−𝑦|∶𝑥,𝑦∈supp𝑢₀}.

Remark 2.2. We notice that as the diameter 𝛿 of the support of 𝑢₀ gets bigger, (2.4) converges to 𝑚₁!𝑞𝑚₁+𝑝−1𝜂^𝑞𝑚1₀^+𝑝−1. (2.5)

Remark 2.3. By virtue of Hölder's inequality on 𝜂(𝑡)=∫_𝐵𝛿𝑢(𝑥,𝑡)𝑤₁(𝑥)𝑑𝑥, it is noted that the blow-up time 𝑇^∗ of ‖𝑢‖_𝑋 cannot exceed the (first) blow-up time 𝑇^∗_𝜂 of 𝜂(𝑡). Here the space 𝑋 can be one of any function spaces that obey Hölder's inequality together with the dual space 𝑋′. Classical Lebesgue spaces, 𝐵𝑀𝑂, Besov spaces, Triebel-Lizorkin spaces, and Orlicz spaces are some of the examples.

3. The Arguments

The monotone convergence theorem implies that 𝑑𝜂(𝑡)=𝑑𝑡_𝐵𝛿𝑢_𝑡𝑤₁𝑑𝑥=_𝐵𝛿Δ𝑢+|𝑢|^𝑝𝑒^|𝑢|𝑞𝑤₁𝑑𝑥=−𝜆₁𝜂(𝑡)+^∞_𝑘=01𝑘!_𝐵𝛿|𝑢|^{𝑞𝑘+𝑝}𝑤₁𝑑𝑥. (3.1) Hölder’s inequality and (2.3), on the other hand, yield that for each 𝑘𝜂(𝑡)≤_𝐵𝛿|𝑢|𝑤₁𝑑𝑥≤_𝐵𝛿|𝑢|^{𝑞𝑘+𝑝}𝑤₁𝑑𝑥^{1/(𝑞𝑘+𝑝)}_𝐵𝛿𝑤₁𝑑𝑥^{(𝑞𝑘+𝑝−1)/(𝑞𝑘+𝑝)}=_𝐵𝛿|𝑢|^{𝑞𝑘+𝑝}𝑤₁𝑑𝑥^{1/(𝑞𝑘+𝑝)}. (3.2) Therefore we have|𝜂(𝑡)|^{𝑞𝑘+𝑝}≤∫_𝐵𝛿|𝑢|^{𝑞𝑘+𝑝}𝑤₁𝑑𝑥. Apply this inequality on (3.1) to find that for 0≤𝑡<𝑇^∗, 𝑑𝜂(𝑡)𝑑𝑡≥−𝜆₁𝜂(𝑡)+^∞_𝑘=01𝜂(𝑡𝑘!)^{𝑞𝑘+𝑝}=−𝜆₁𝜂(𝑡)+𝜂(𝑡)^𝑝𝑒^{|𝜂(𝑡)|𝑞}. (3.3)

We are now going to find a lower bound function for 𝜂(𝑡). To do it, take 𝜙 to be a solution of the ordinary differential equation: 𝑑𝜙(𝑡)𝑑𝑡=−𝜆₁𝜙(𝑡)+𝜙(𝑡)^𝑝𝑒^{|𝜙(𝑡)|𝑞} (3.4) with 𝜂(0)=𝜙(0). We also define a real-valued function 𝑓 by 𝑓(𝑡)≡−𝜆₁𝑡+|𝑡|^𝑝𝑒^|𝑡|𝑞. A closer look at (3.3) and a chain of considerations on the choice of 𝑐₀ deliver that 𝜂(𝑡)≥𝜂(0)=𝜂₀>𝑐₀, which in turn implies that (𝑑/𝑑𝑡)𝜂(𝑡)/𝑓(𝜂(𝑡))≥1. Integrate both sides with respect to 𝑡, and we have ∫^𝑡₀((𝑑/𝑑𝑠)𝜂(𝑠)/𝑓(𝜂(𝑠)))𝑑𝑠≥𝑡. Consider an indefinite integral 𝐹 of 1/𝑓(𝑥) to get 𝐹(𝜂(𝑡))−𝐹(𝜂(0))≥𝑡. Similarly, we can obtain 𝐹(𝜙(𝑡))−𝐹(𝜙(0))=𝑡. Hence these facts together with 𝜂(0)=𝜙(0) yield that 𝐹(𝜂(𝑡))≥𝐹(𝜂(0))+𝑡=𝐹(𝜙(0))+𝑡=𝐹(𝜙(𝑡)). Note that 𝐹 is monotone increasing on (𝑐₀,∞), and so we can deduce that 𝜂(𝑡)≥𝜙(𝑡) for 0≤𝑡<𝑇^∗.

We will find the first blow-up time for 𝜙(𝑡). First, for a fixed 𝑘∈ℕ, we consider two real-valued functions 𝑔,ℎ∶[0,∞)→ℝ defined by 𝑔(𝑥)≡−𝜆₁𝑥+|𝑥|^𝑝𝑒^|𝑥|𝑞 and ℎ(𝑥)≡−𝜆₁𝑥+∑^𝑘_𝑚=0(1/𝑚!)|𝑥|^{𝑞𝑚+𝑝}. Then it is clear that 𝑔(𝑥)≥ℎ(𝑥)for all 0≤𝑥<∞. Let 𝜌_𝑘 be a solution for a Bernoulli-type equation: 𝑑𝜌𝑑𝑡^𝑘(𝑡)=ℎ𝜌_𝑘(𝑡) (3.5) with the initial condition: 𝜌_𝑘(0)=11−𝜙(0).𝑘+2 (3.6)

Lemma 3.1. For each 𝑘, 𝜙(𝑡)≥𝜌_𝑘(𝑡) for all 𝑡∈[0,𝑇^∗).

Proof. We choose indefinite integrals 𝐺 and 𝐻 of 1/𝑔 and 1/ℎ, respectively, with the conditions that 𝐺(0)=0 and 𝐻(𝜌_𝑘(0))=𝐺(𝜙(0)). We have 𝐺(𝑥)≤𝐻(𝑥) for all 𝑥, which follows from the facts that 𝑔(𝑥)≥ℎ(𝑥) for all 𝑥 and 𝜌_𝑘(0)<𝜙(0). On the other hand, the argument used above leads to get 𝐺(𝜙(𝑡))−𝐺(𝜙(0))=𝑡, and similarly 𝑡=𝐻(𝜌_𝑘(𝑡))−𝐻(𝜌_𝑘(0)). Hence we arrive at 𝐺(𝜙(𝑡))=𝐻(𝜌_𝑘(𝑡)). From this together with the fact that the function 𝐺 is dominated by 𝐻, we can realize that 𝜌_𝑘 should be dominated by 𝜙, that is, 𝜙(𝑡)≥𝜌_𝑘(𝑡)for all0≤𝑡<𝑇^∗.

We assert that the sequence {𝜌_𝑘(𝑡)}^∞_𝑘=1 is monotone increasing and converges to 𝜙(𝑡) for 𝑡∈[0,𝑇^∗). In fact, by the same argument used in Lemma 3.1, it can be noticed that {𝜌_𝑘(𝑡)}^∞_𝑘=1 is monotone increasing and bounded above by 𝜙(𝑡), and so it converges to some 𝜉(𝑡). The integral representation of (3.5) can be written as 𝜌_𝑘(𝑡)=𝜌_𝑘(0)−𝜆₁^𝑡₀𝜌_𝑘(𝜏)𝑑𝜏+^𝑡₀^𝑘_𝑚=01𝜌𝑚!^𝑘(𝜏)^{𝑞𝑚+𝑝}𝑑𝜏. (3.7) Lebesgue dominated convergence theorem together with Lemma 3.1 leads to the (pointwise) limit of (3.7): 𝜉(𝑡)=𝜙(0)−𝜆₁∫^𝑡₀𝜉(𝜏)𝑑𝜏+∫^𝑡₀|𝜉(𝑡)|^𝑝𝑒^{|𝜉(𝑡)|𝑞}𝑑𝜏, which implies that 𝜉 is the solution of (3.4). The uniqueness of the solution for (3.4) yields that 𝜉=𝜙.

We can explicitly compute the solutions 𝜌_𝑘 by observing that 𝜌_𝑘=∑^𝑘_𝑚=0𝜚_𝑚, where 𝜚_𝑚 are solutions for classical Bernoulli's equations: (𝑑/𝑑𝑡)𝜚_𝑚=−𝜆₁𝜚_𝑚+(1/𝑚!)𝜚_𝑚^{𝑞𝑚+𝑝} with initial values: 𝜚_𝑚(0)=1−1𝑚+1𝜙(0).𝑚+2 (3.8) By solving each Bernoulli's equation and summing up the solutions, we obtain 𝜌_𝑘(𝑡)=^𝑘_𝑚=0𝜆₁𝑚!𝜆₁𝑚!−𝜚_𝑚(0)^{𝑞𝑚+𝑝−1}1−𝑒^{−(𝑞𝑚+𝑝−1)𝜆1𝑡1/(𝑞𝑚+𝑝−1)}𝑒^−𝜆1𝑡𝜚_𝑚(0), (3.9) provided that the denominator is not zero. In case (1−𝑝)/𝑞 is a positive integer, to say 𝑚₀, then the 𝑚₀-th term in the summation above should be replaced by 𝜚_𝑚0(0)𝑒^{((1/𝑚0!)−𝜆1)𝑡}. Therefore we obtain 𝜙(𝑡)=^∞_𝑚=0𝜆₁𝑚!𝜆₁𝑚!−𝜚_𝑚(0)^{𝑞𝑚+𝑝−1}1−𝑒^{−(𝑞𝑚+𝑝−1)𝜆1𝑡1/(𝑞𝑚+𝑝−1)}𝑒^−𝜆1𝑡𝜚_𝑚(0). (3.10) The first blow-up time at the right hand side of (3.10) is 𝑇₁1≡−𝑞𝑚₁+𝑝−1𝜆₁𝑚ln1−¹+1𝑚₁+2^{𝑞𝑚1+𝑝−1}𝑚₁!𝜆₁𝜂(0)^{𝑞𝑚1+𝑝−1}, (3.11) (𝑚₁ is defined at page 3) which implies that 𝑇^∗≤𝑇₁, and so the solution blows up before the finite time 𝑇₁.

We now present a better upper bound than 𝑇₁ of the blow-up time 𝑇^∗. In fact, the number “{(𝑚₁+1)(𝑚₁+2)}^{q𝑚1+𝑝−1}” in (3.11) can be improved by taking another initial data in (3.6) and (3.8). We choose a strictly increasing sequence of real numbers {𝑎_𝑘}^∞_𝑘=1 satisfying 0=𝑎₁<𝑎₂<⋯<lim_𝑘→∞𝑎_𝑘=1. Then by replacing the initial conditions in (3.6) and (3.8) with 𝜌_𝑘(0)=𝑎_𝑘+2𝜙(0) and 𝜚_𝑚(0)=(𝑎_𝑚+2−𝑎_𝑚+1)𝜙(0), respectively, we have 𝑇^∗_𝜂1≤−𝑞𝑚₁+𝑝−1𝜆₁𝑚ln1−¹!𝜆₁𝑎_𝑚1+2−𝑎_𝑚1+1^{𝑞𝑚1+𝑝−1}𝜂(0)^{𝑞𝑚1+𝑝−1} (3.12) instead of (3.11). The estimate (3.12) holds for any sequence {𝑎_𝑚}^∞_𝑚=1 with 0<𝑎_𝑚1+1<𝑎_𝑚1+2<1. Therefore letting the number 𝑎_𝑚1+2−𝑎_𝑚1+1 go to 1, we finally get a better upper bound 1𝑞𝑚₁+𝑝−1𝜆₁𝜂(0)ln^{𝑞𝑚1+𝑝−1}𝜂(0)^{𝑞𝑚1+𝑝−1}−𝑚₁!𝜆₁ (3.13) of 𝑇^∗. This completes the proof.

Acknowledgment

The author was supported by the research fund of Dankook University in 2010.