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Abstract and Applied Analysis
VolumeΒ 2010Β (2010), Article IDΒ 680572, 12 pages
http://dx.doi.org/10.1155/2010/680572
Research Article

On theBlow-UpSet for Non-Newtonian Equation with a Nonlinear Boundary Condition

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 9 June 2010; Accepted 15 September 2010

Academic Editor: NicholasΒ Alikakos

Copyright Β© 2010 Zhilei Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We identify the blow-up set of solutions to the problem , , , , , and , , where . We obtain that the blow up set satisfies . The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.

1. Introduction

Consider a one-dimensional process of diffusion in a medium that occupies the half space ; that is, where and is an appropriately smooth function with some compatibility conditions. Problem (1.1) describes the non-Newtonian fluid with a power dependence of the tangential stress on the velocity of the displacement under nonlinear condition. It has many applications and has been intensively studied; see [1–3] and the references cited therein. For the local in time existence, we refer to [4]. Also it is known that (1.1) has no classical solution in general due to the possible degeneration at So we usually understand the weak solution defined in the following sense.

Definition 1.1. A nonnegative function with is said to be a weak solution of (1.1), if the integral identity is fulfilled for all .

An interesting phenomenon is that, due to the boundary effect, the solution of (1.1) may exist for and becomes unbounded as for some . Namely, the solutionoccurs blow-upphenomenon. In this connection, Galaktionov and Levine proved in [5] that the solutions are global in time when but occur blow-up for the range while for blow-up happens or not depending on the size of the initial data. The main concern in this work is on the set of points at which solutions becomes unbounded, that is, the blow-up set, which is defined as A problem which has attracted a lot of attention in the literature is the identification of possible blow-up sets. Numerical analysis hints that the blow-up set should be a single point (single point blow-up) when , a proper subset of the spatial domain (regional blow-up) when and the whole half line (global blow-up) when . In fact, based on the Gilding and Herrero's work in [6], QuirΓ³s and Rossi considered the porous medium type equation They proved in [1]: the blow-up set if but if however, the blow-up set is regional in case of , namely, . Afterwards, CortΓ‘zar et al. given a detailed description on the regional blow-up set. They proved in [7] that if then the blow-up set satisfies

In the light of previous works, we discuss the blow-up set of solutions for the -Laplacian equation (1.1). In the current paper, we identify the set of blow-up points in case of . So, in the following we consider where We take to be a , nonincreasing and compactly supported function with some compatibility conditions. The below theorem is our main result.

Theorem 1.2. Assume that is a nonnegative, nonincreasing and compactly supported function, then all the nontrivial solutions of problem (1.5) occur blow-up; moreover, the blow-up set satisfies

Remark 1.3. The nonincreasing assumption on makes the proof much simpler (see also [7]).

Remark 1.4. One could expect that if the solutions of (1.1) occur blow-up, then the blow-up set if , but if

2. Proof of Theorem 1.2

In order to study the solution of (1.5) near the blow-up time as in [8] we introduce the rescaled function where Hence, we obtain the below equations in terms of by substituting (2.1) into (1.5),

If is independent, then is called a self-similar solution. Let , we have A direct integration shows that the non-negative solution of (2.3) is or where

Theorem 1.2 is a direct consequence of the following two propositions.

Proposition 2.1. Let be a solution of (2.2), then there exists a function such that the limit holds uniformly on Moreover, has the explicit form (2.4) and solves problem (2.3).

Proposition 2.1 and the formula (2.1) generate

Proposition 2.2. Suppose that the constants satisfy then there exists a large time point such that for all where constant depends on and

Proposition 2.2, along with (2.1), shows that is uniformly bounded for all if . This claims

Hence, Theorem 1.2 immediately follows from (2.6) and (2.8). So the remaining task in this paper is to prove the validity of Propositions 2.1 and 2.2. This will be discussed in Section 3.

3. Proof of Propositions 2.1 and 2.2

In this section we prove Propositions 2.1 and 2.2. The argument contains several lemmas.

Lemma 3.1. All the nontrivial solutions of (1.5) occur blow-up.

The proof is available in [5], (see also [3]).

Lemma 3.2. If the initial function is nonincreasing, then so does the solution for all existent times, that is,

Proof. Consider the regularized problem where is nonincreasing and converges to uniformly as . By uniqueness, the solution of (1.5) should be the limit function of that of (3.1). Setting and then differentiating the equation in to obtain Clearly, (3.2) is a linear parabolic equation with respect to . On account of the boundary conditions and we deduce via comparison theorem. The proof is completed.

Remark 3.3. Using the maximum principle for problem (3.1), we obtain By sending it follows that

Lemma 3.4. There exists a constant such that

Proof. Integrating (1.5)1, the first equation of (1.5), over yields Now multiplying (1.5)1 by and integrating the resulting expression once more, we have Thanks to Remark 3.3, we estimate the last sign comes from the boundary condition. Thus, (3.7) becomes Multiplying (3.6) by a constant and then subtracting (3.9) produces If we choose so large such that then Inequalities (3.10) and (3.11) lead to Putting and integrating above inequality over brings Because is bounded when time is away from the blow-up time (3.13) and thus (3.5) is valid for all

The next lemma proves that is localized. Namely, the support of is uniformly bounded for all .

Lemma 3.5. There exists a constant depending on and such that if then

Proof. Let us introduce the function where satisfies where and is an arbitrary positive constant. It is easy to check that satisfies (1.5)1 By differentiating (3.5), we have This, together with the monotonicity of , deduces Because the initial support is bounded, we choose a suitably large point to guarantee . Setting it has By virtue of (3.16), there exists some large constant such that for all , it holds So comparison theorem concludes Hence, for every constant satisfying , (3.19) ensures Thus Denoted by , it follows that for all This completes the proof.

Corollary 3.6. Lemma 3.5 and (2.1) claim that is localized as well. Moreover, (2.1) and (3.16) guarantee the boundedness of for all and By a similar argument as that in Lemma 3.2, one can prove that also is bounded.

In what follows, we analyze the large time behavior of the solution of problem (2.2).

Multiplying (2.2) by and then integrating by parts, we deduce that where is the Liapunov functional and Based on the estimates the -limit set consists of solutions of the problem (2.3). Indeed, using the estimates (3.23) in passing to the limit in (2.2), we have that, given a monotone sequence , where depending on solves (2.3) in a weak sense, and It follows from (3.23) that uniformly in This means that the limit function does not rely on and is a weak solution for the (2.3). Finally, the independence of the choice of the sequence follows from the nonincreasing of in time. Furthermore, the convergence is uniform in due to the boundness of In conclusion we arrive to the following lemma.

Lemma 3.7. The limit holds as where satisfies the problem (2.3).

To finish Proposition 2.1, it remains to confirm that is not null and thus to be of the form (2.4). The proof will be given at the end of the paper. Now let us turn to Proposition 2.2.

Consider the initial-boundary-value problem where are positive constants and are suitably small. We have the following.

Lemma 3.8. Let suitably small with that satisfies Then there exists a constant such that for all ,

Proof. It is easy to check that the function of the form solves Moreover, it has since is a super solution to problem (3.30). Choosing and subtracting (3.30) from (3.27), we obtain Integrating (3.31) over gives rise to Noting that is symmetric with respect to and nonincreasing on , thus we have for all where sign is the sign function. Then, Similarly, Therefore, (3.32) becomes Applying the Gronwall inequality yields On the other hand, since is symmetric and nondecreasing on with -variable, for all it has where we have used the fact That is, This completes Lemma 3.8.

We are now in a position to prove Proposition 2.2. From Lemma 3.7 we observe that tends to the limit function which is none for all points Taking and a small constant to satisfy . Then, for a given , the inequality for all holds provided , where is a large point which depends on . Using comparison theorem, one has The proof ends up with (3.40) and Lemma 3.8.

Finally, We prove that the function appeared in Lemma 3.7 takes the form (2.4). For this purpose, it is enough to illuminate that is impossible. We argue by contradiction. Assuming that for a moment. Then blows up only at the boundary point By this we can find a sequence as such that Moreover, By the maximin principle (see Remark 3.3), one has as long as is chosen large enough. Recalling that is nonincreasing with respect to -variable, from (3.41) we conclude Integrating (3.42) gives rise to This shows that occurs blow-up at least in the interval which contradicts with the assumption that blow-up happens only at

Now the proof of Theorem 1.2 is completed.

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