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:
(1.1)∂∂tu(x,t)-Δu(x,t)=F(u(x,t)),u(x,0)=u0(x),
where x∈ℝn, 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
(1.2)∂∂tu-Δu=|u|pe|u|q,
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
(1.3)‖u(t)‖L∞≤Ct-1/(p-1)
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 ℝn. 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 ℝn 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 ℝn. 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 limt↑T*∥u(t)∥X=∞ in an appropriate function space X. We choose an open ball Bδ of radius δ that contains the support of u0. We proceed by choosing an orthonormal basis {wj}j=1∞ for L2(Bδ), where wj∈H01(Bδ) is an eigenfunction corresponding to each eigenvalue λj of -Δ:
(2.1)-Δwj=λjwjinBδ,wj=0on∂Bδ,
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
(2.2)λ1=r02δ2,
where r0>0 is the first positive zero of the Bessel function Jn/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
(2.3)w1>0inBδ,∫Bδw1(x)dx=1.
A smooth solution u in H01(Bδ) can be expressed by a linear combination of {wj}j=1∞: u(x,t)=∑j=1∞aj(t)wj(x) (0≤t<T*,x∈Bδ), where aj(t)=∫Bδu(x,t)wj(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 tpetq>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 Tη* of the first eigen-coefficient η(t) is less than or equal to the positive number
(2.4)δ2(qm1+p-1)r02ln(δ2η0qm1+p-1δ2η0qm1+p-1-m1!r02),
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
(2.5)m1!(qm1+p-1)η0qm1+p-1.

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 Tη* 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
(3.1)ddtη(t)=∫Bδutw1dx=∫Bδ(Δu+|u|pe|u|q)w1dx=-λ1η(t)+∑k=0∞1k!∫Bδ|u|qk+pw1dx.
Hölder’s inequality and (2.3), on the other hand, yield that for each k(3.2)|η(t)|≤∫Bδ|u|w1dx≤(∫Bδ|u|qk+pw1dx)1/(qk+p)(∫Bδw1dx)(qk+p-1)/(qk+p)=(∫Bδ|u|qk+pw1dx)1/(qk+p).
Therefore we have|η(t)|qk+p≤∫Bδ|u|qk+pw1dx. Apply this inequality on (3.1) to find that for 0≤t<T*,
(3.3)ddtη(t)≥-λ1η(t)+∑k=0∞1k!|η(t)|qk+p=-λ1η(t)+|η(t)|pe|η(t)|q.

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:
(3.4)ddtϕ(t)=-λ1ϕ(t)+|ϕ(t)|pe|ϕ(t)|q
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 ∫0t((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+∑m=0k(1/m!)|x|qm+p. Then it is clear that g(x)≥h(x)for all 0≤x<∞. Let ρk be a solution for a Bernoulli-type equation:
(3.5)ddtρk(t)=h(ρk(t))
with the initial condition:
(3.6)ρk(0)=(1-1k+2)ϕ(0).

Lemma 3.1.

For each k, ϕ(t)≥ρk(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(ρk(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 ρk(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(ρk(t))-H(ρk(0)). Hence we arrive at G(ϕ(t))=H(ρk(t)). From this together with the fact that the function G is dominated by H, we can realize that ρk should be dominated by ϕ, that is, ϕ(t)≥ρk(t)for all0≤t<T*.

We assert that the sequence {ρk(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 {ρk(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
(3.7)ρk(t)=ρk(0)-λ1∫0tρk(τ)dτ+∫0t∑m=0k1m!|ρk(τ)|qm+pdτ.
Lebesgue dominated convergence theorem together with Lemma 3.1 leads to the (pointwise) limit of (3.7): ξ(t)=ϕ(0)-λ1∫0tξ(τ)dτ+∫0t|ξ(t)|pe|ξ(t)|qdτ, 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 ρk by observing that ρk=∑m=0kϱm, where ϱm are solutions for classical Bernoulli's equations: (d/dt)ϱm=-λ1ϱm+(1/m!)ϱmqm+p with initial values:
(3.8)ϱm(0)=(1m+1-1m+2)ϕ(0).
By solving each Bernoulli's equation and summing up the solutions, we obtain
(3.9)ρk(t)=∑m=0k(λ1m!λ1m!-ϱm(0)qm+p-1(1-e-(qm+p-1)λ1t))1/(qm+p-1)e-λ1tϱm(0),
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 ϱm0(0)e((1/m0!)-λ1)t. Therefore we obtain
(3.10)ϕ(t)=∑m=0∞(λ1m!λ1m!-ϱm(0)qm+p-1(1-e-(qm+p-1)λ1t))1/(qm+p-1)e-λ1tϱm(0).
The first blow-up time at the right hand side of (3.10) is
(3.11)T1≡-1(qm1+p-1)λ1ln(1-{(m1+1)(m1+2)}qm1+p-1m1!λ1η(0)qm1+p-1),
(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 {ak}k=1∞ satisfying 0=a1<a2<⋯<limk→∞ak=1. Then by replacing the initial conditions in (3.6) and (3.8) with ρk(0)=ak+2ϕ(0) and ϱm(0)=(am+2-am+1)ϕ(0), respectively, we have
(3.12)Tη*≤-1(qm1+p-1)λ1ln(1-m1!λ1{am1+2-am1+1}qm1+p-1η(0)qm1+p-1)
instead of (3.11). The estimate (3.12) holds for any sequence {am}m=1∞ with 0<am1+1<am1+2<1. Therefore letting the number am1+2-am1+1 go to 1, we finally get a better upper bound
(3.13)1(qm1+p-1)λ1ln(η(0)qm1+p-1η(0)qm1+p-1-m1!λ1)
of T*. This completes the proof.

Acknowledgment

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

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