Blow-up rate for a porous medium equation with convection.

Abstract

This paper deals with the blow-up rate for a porous medium equation with convection and a nonlinear boundary flux. By using the scaling method, we obtain the blow-up rate which is determined by the interaction between the diffusion and the boundary flux. Comparing with the previous results for the porous medium model without convection, we observer that the gradient term included here does not affect the blow-up rates for solutions.

AMS (2000) Subject Classification: 35K50, 35K55, 35K65, 35B33. Keywords: Parabolic equation, Convection, Nonlinear boundary flux, Blow-up.

1. Introduction

In this paper, we consider the following porous medium equation with convection

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.1)

where the parameters q > m > 1; [u.sub.0](x) [greater than or equal to] 1 is continuous and satisfies the compatibility condition.

Remark 1.1. Since we study the blow-up rate, the conditions q > m > 1 and [u.sub.0](x) [greater than or equal to] 1 are reasonable (if q < m the solution is global existence for this problem (see [18])).

The porous medium equations without convection were considered extensively in recently years (see articles [4, 15] or reviews [3, 5, 12] or books [14, 16, 20, 21]), for example in [4], Galaktionov and Levine studied the following single-equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.2)

It was shown if 0 < q [less than or equal to] [q.sub.0] = (m + 1)/2, then all nonnegative solutions of (1.2) are global in time, while for q > [q.sub.0] there are solutions with finite time blow-up. Moreover, they show if [q.sub.0] < q [less than or equal to] [q.sub.c] = m+ 1, the all nontrivial nonnegative solutions blow up in a finite time, while global nontrivial nonnegative solutions exist if q > [q.sub.c]. However, they do not consider the blow-up rate in this paper. The blow-up rates were considered by F. Quiros and J. D. Rossi in a recent paper [15]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.3)

with notationa

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

They proved that the solutions of (1.3) are global if pq < (m + 1)(n + 1)/4, and may blow up in finite time if pq > (m+1)(n+1)/4. In the case of pq > (m+1)(n+1)/4, if [[alpha].sub.1] + [[beta].sub.1] [less than or equal to] 0, or [[alpha].sub.2] + [[beta].sub.2] [less than or equal to] 0, then every nonnegative, nontrivial solution of (1.3) blows up in finite time; if [[alpha].sub.1] + [[beta].sub.1] > 0, and [[alpha].sub.2] + [[beta].sub.2] > 0, then there exist blow-up solutions for large initial data and global solutions for small initial data. They get the blow-up rate of the positive solution is O[((T - t).sup.-[[alpha].sub.1])for component u and O[((T - t).sup.[[alpha].sub.2]) for component v as t [right arrow] T.

Comparing with the porous medium equation, the porous medium equations with convection are difficult to study. There are also many open problems (see [19]). The elegant work for this kind of problem was done by F. Andreu et al. [1], where they study the following problem with convection

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.4)

where [OMEGA] is a bounded smooth domain in [[??].sup.N] (N [greater than or equal to] 1), m [greater than or equal to] 1, [alpha] > 0; p [greater than or equal to] 1 and q [greater than or equal to] 1. However, They only get the conditions for the global existence of weak solution.

The blow-up conditions for the kind of porous medium equation with convention was studied in [22]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.5)

where [OMEGA] [subset] [[??].sup.N] is a bounded domain with smooth boundary [partial derivative][OMEGA], [p.sub.2] > 0, [q.sub.1] > 0, [p.sub.1] [greater than or equal to] 0, [q.sub.2] [greater than or equal to] 0, [[epsilon].sub.0] > 0, [m.sub.i] > 0, [a.sub.i] > 0, [b.sub.i] [greater than or equal to] 0(i = 1, 2), smooth functions [u.sub.0](x), [v.sub.0](x) [greater than or equal to] [[eta].sub.0] > 0 on [[bar.OMEGA]], compatible on [partial derivative][OMEGA]. They get the blow-up conditions for (1.5), but they dont study the blow-up rate.

In this paper, we study the blow-up rate of (1.1), the main result of this work is the following blow-up rate estimate theorem.

Theorem 1.1 Let u be the solution of problem (1.1) with (m - 1)/m[([u.sup.m.sub.0]).sub.xx] - (m - 1)[u.sup.m-2.sub.0] [|[u.sub.0x]|.sup.2] [greater than or equal to] 0 on [0, 1] and blows up at a finite time T, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for t [member of] 2 (0, T) and positive constants c, C.

Remark 1.2. This result for the solution u of problem (1.1) agrees with those without convection obtained in [10,11], namely, the gradient term in (1.1) makes no contribution to the blow-up rate. In other words, the convection here is insufficient to affect the result of the interaction between the boundary flux and the diffusion term.

2. Blow-up rate

In this section, we will establish the blow-up rate for (1.1) to prove Theorem 1.1. The main tool used here is a scaling argument [2, 7, 8, 9].

If we make the transformation w = log u, the problem (1.1) takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

Then the result in [18] ensures that every solution of (2.1) blows up in finite for q > m > 1.

Proof of Theorem 1.1. Let w be a solution of (2.1) with blow-up time T. Since [([e.sup.m-1][w.sub.0]).sub.xx] [greater than or equal to] 0 (the condition in Theorem 1.1 ensures) on [0, 1], the maximum principle yields [w.sub.t] [greater than or equal to] 0 and hence [([e.sup.(m-1)w]).sub.xx] [greater than or equal to] 0, namely, [(e.sup.(m-1)w]).sub.x] is nondecreasing with respect to x. Taking this together with the boundary condition [w.sub.x](0, t) = 0, we know that [w.sub.x](x, t) [greater than or equal to] 0 for (x, t) [member of] [0, 1] x [0, T). For any t > 0, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.3)

where a, b are parameters depending on m, q, M. Clearly, [e.sup.-(m-1)M([t.sup.*])] [less than or equal to] [[phi].sub.a,b] [less than or equal to] 1, [[phi].sub.a,b](0, 0) = 1, [[partial derivative].sub.[[phi].sub.a,b]/[partial derivative]s [greater than or equal to] 0. Choosing a = [e.sup.(m-q)M], b = [e.sup.(1+m-2q)M] and a direct computation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.4)

We remark that the positive parameters a and b go to zero as [t.sup.*] [right arrow] T due to q > m > 1.

Next, we claim that there exist positive constants [C.sub.1] and [C.sub.2] such that

[C.sub.1] [less than or equal to] [[partial derivative].sub.[[phi].sub.a,b]/[partial derivative]s (0, 0) [less than or equal to] [C.sub.2] holds for a, b small. (2.5)

The proof of (2.5) relies on the uniform boundedness of {[[phi].sub.a,b]} and {[([[phi].sub.a,b]).sub.y]}. Indeed, it is easy to see by (2.4) with 0 [less than or equal to] [[phi].sub.a,b] [less than or equal to] 1 that {[([[phi].sub.a,b]).sub.y]} is also uniformly bounded. From the results for bounded solutions of porous medium type equations (see [6, 23]), {[[phi].sub.a,b]} is equicontinuous on compact subsets of their common domain. Let [a.sub.j] = a([t.sup.*.sub.j]), [b.sub.j] = b([t.sup.*.sub.j]) with [t.sup.*.sub.j] [right arrow] T as j [right arrow] +1. Passing to a subsequence if necessary, we have that [[phi].sub.[a.sub.j],[b.sub.j]] [right arrow] [phi] uniformly on compact subsets of A = {y [less than or equal to] 0, s [less than or equal to] 0}. The limit function [phi] is continuous with [phi](0, 0) = 1. Hence, for any [[epsilon].sub.0] [member of] (0, 1), there exists a neighborhood of (0, 0), denote by U [subset] A, such that [phi] > [[epsilon].sub.0] in U, and [[epsilon].sub.0]/2 [less than or equal to] [[phi].sub.[a.sub.j],[b.sub.j]] [less than or equal to] 1 on [bar.U] for j large enough. By Schauder estimates [13], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.6)

The second inequality in (2.5) follows immediately.

Now we come to the proof of the first inequality of (2.5). If the first inequality in (2.5) fails, then there exists a sequence ([a.sub.j], [b.sub.j]) [right arrow] (0, 0) such that [[partial derivative].sub.[[phi][a.sub.j],[b.sub.j]]/[partial derivative]s(0, 0) [right arrow] 0. We proceed as before to obtain that [[phi].sub.[a.sub.j],[b.sub.j]] [right arrow] [phi], and the estimate (2.6) holds on compact subset of {(y, s) : [phi] > 0}. Thus, we have [[phi].sub.[a.sub.j],[b.sub.j]] [right arrow] [phi] in [C.sup.2+[beta],1+[beta]/2] for some [beta] < [alpha] satisfying 0 [less than or equal to] [phi] [less than or equal to] 1, [phi](0, 0) = 1, [phi]=[partial derivative]s [greater than or equal to] 0, and [phi] is a weak solution of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.7)

in {y < 0} x (-1, 0]. Define z = [[phi].sub.s] and we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.8)

in the positivity set of [phi], it follows by Hopf lemma [14, 17] that z [equivalent to] 0, namely, [phi] is independent of s. Therefore, [phi] = [phi](y) satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.9)

However, the problem (2.9) does not have bounded weak solutions. In terms of w, it follows from (2.5) that

[C.sub.1] [less than or equal to] [(m - 1).sup.[e.sup.(1+m-2q).sup.M] [M.sub.t]([t.sup.*]) [less than or equal to] [C.sub.2]. (2.10)

Integrating (2.10) from t to T, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

with c = ([C.sub.2](2q - m - 1)/[(m - 1)).sup.-1/(2q-m-1)]

and C = ([C.sub.1](2q - m - 1)/[(m - 1)).sup.-1/(2q-m-1)]. The proof of Theorem 1.1 is complete.

Acknowledgements. This work is supported in part by NNSF of China (10571126) and in part by Program for New Century Excellent Talents in University.

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Jun Zhou (a) (1) and Chunlai Mu (b)

(a) School of mathematics and statistics, Southwest University,Chongqing, 400715, P. R. China

(b) School of mathematics and physics, Chongqing University, Chongqing, 400044, P. R. China

(1) Corresponding author: zhoujun [e-mail].