Research Article | Open Access
Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity
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.
We are concerned with the initial value problem of nonstationary nonlinear heat equations: where , is a given nonlinear function and 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 for some positive real numbers , . 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 , 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 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 , there exists a uniform constant such that under certain constraints before the blow-up, see [19, 22]. It also has been noticed after the blow-up that for such subcritical cases 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 . The first main contribution in this direction seems to be the work of Baras and Cohen  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 ). 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 (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  and also in . 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 , are of obvious interest.
2. The Main Theorem
Let be a function with compact support in and let be a (smooth) solution of (1.2) inside of with a homogeneous Dirichlet's boundary condition and the initial condition . It is clear that for all if we employ the trivial extension of to the whole space . By virtue of maximum principle, if the initial source is nonnegative, so is . It is also well known that a positive solution of (1.2) with sufficiently large initial datum blows up within a finite time; that is, there exists a positive constant (the maximal existence time) so that in an appropriate function space . We choose an open ball of radius that contains the support of . We proceed by choosing an orthonormal basis for , where is an eigenfunction corresponding to each eigenvalue of : for . In particular, we are interested in the eigenfunctions corresponding to the principal eigenvalue .
We recall a relationship between the volume of the domain and the principal eigenvalue of the Laplacian, which says that where is the first positive zero of the Bessel function of order which can be expressed by elementary functions (for , see page 45 in ). Also, we may choose an eigenfunction satisfying A smooth solution in can be expressed by a linear combination of : (), where . In particular, we denote the eigen-coefficient of with respect to the eigenfunction by .
We introduce two specific real numbers and as follows: is the smallest positive integer among satisfying , and is the smallest nonnegative number such that holds for all .
Theorem 2.1. Let the spatial dimension be greater than 1. With the notations above, assume that the given initial source is large enough that the initial eigen-coefficient is greater than both and . Then the (first) blow-up time of the first eigen-coefficient is less than or equal to the positive number where .
Remark 2.2. We notice that as the diameter of the support of gets bigger, (2.4) converges to
Remark 2.3. By virtue of Hölder's inequality on , it is noted that the blow-up time of cannot exceed the (first) blow-up time of . Here the space can be one of any function spaces that obey Hölder's inequality together with the dual space . Classical Lebesgue spaces, , Besov spaces, Triebel-Lizorkin spaces, and Orlicz spaces are some of the examples.
3. The Arguments
We are now going to find a lower bound function for . To do it, take to be a solution of the ordinary differential equation: with . We also define a real-valued function by . A closer look at (3.3) and a chain of considerations on the choice of deliver that , which in turn implies that . Integrate both sides with respect to , and we have . Consider an indefinite integral of to get . Similarly, we can obtain . Hence these facts together with yield that . Note that is monotone increasing on , and so we can deduce that for .
We will find the first blow-up time for . First, for a fixed , we consider two real-valued functions defined by and . Then it is clear that for all . Let be a solution for a Bernoulli-type equation: with the initial condition:
Lemma 3.1. For each , for all .
Proof. We choose indefinite integrals and of and , respectively, with the conditions that and . We have for all , which follows from the facts that for all and . On the other hand, the argument used above leads to get , and similarly . Hence we arrive at . From this together with the fact that the function is dominated by , we can realize that should be dominated by , that is, for all.
We assert that the sequence is monotone increasing and converges to for . In fact, by the same argument used in Lemma 3.1, it can be noticed that is monotone increasing and bounded above by , and so it converges to some . The integral representation of (3.5) can be written as Lebesgue dominated convergence theorem together with Lemma 3.1 leads to the (pointwise) limit of (3.7): , which implies that is the solution of (3.4). The uniqueness of the solution for (3.4) yields that .
We can explicitly compute the solutions by observing that , where are solutions for classical Bernoulli's equations: with initial values: By solving each Bernoulli's equation and summing up the solutions, we obtain provided that the denominator is not zero. In case is a positive integer, to say , then the -th term in the summation above should be replaced by . Therefore we obtain The first blow-up time at the right hand side of (3.10) is ( is defined at page 3) which implies that , and so the solution blows up before the finite time .
We now present a better upper bound than of the blow-up time . In fact, the number “” 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 satisfying . Then by replacing the initial conditions in (3.6) and (3.8) with and , respectively, we have instead of (3.11). The estimate (3.12) holds for any sequence with . Therefore letting the number go to , we finally get a better upper bound of . This completes the proof.
The author was supported by the research fund of Dankook University in 2010.
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