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Academic research paper on topic "Profiles of blow-up solution of a weighted diffusion system"

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Profiles of blow-up solution of a weighted diffusion system

Weili Zeng1,2*, Chunxue Liu,,Xiaobo Lu1,2 and Shumin Fei1,2

"Correspondence: zwlseu@163.com 1 Schoolof Automation, Southeast University, Nanjing, 210096, China 2Key Laboratory of Measurement and Controlof CSE, Ministry of Education, Southeast University, Nanjing, 210096, China

Abstract

In this paper, we study the blow-up profiles for a coupled diffusion system with a weighted source term involved in a product with local term. We prove that the solutions have a global blow-up and the profile of the blow-up is precisely determined in all compact subsets of the domain.

Keywords: diffusion system; weighted localized source; blow-up profile

1 Introduction

In this paper, we consider the following coupled diffusion system with a weighted nonlinear localized sources:

ut - Au = a(x)up(x, t)va(0, t), x e B, 0 < t < T*, vt - Av = b(x)u3(0, t)vq(x, t), x e B,0< t < T*, u(x, t) = v(x, t) = 0, x e dB, t >0, u(x, 0) = u0(x), v(x, 0) = v0(x), x e B,

where B is an open ball of RN, N > 2 with radius R; a, ¡3, p, q are nonnegative constants and satisfy a + p >0 and ¡3 + q >0.

System (1.2) is usually used as a model to describe heat propagation in a two-component combustible mixture [1]. In this case u and v represent the temperatures of the interacting components, thermal conductivity is supposed constant and equal for both substances, a volume energy release given by some powers of u and v is assumed.

The problem with a nonlinear reaction in a dynamical system taking place only at a single site, of the form

ut - Au = up(0, t)va (0, t), x e a, 0 < t < T*, vt - Av = uß(0, t)vq(0, t), x e a, 0 < t < T*, u(x, t) = v(x, t) = 0, x e 9a, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), x e a,

ft Spri

ringer

was studied by Pao and Zheng [2] and they obtained the blow-up rates and boundary layer profiles of the solutions.

As for problem (1.2), it is well known that problem (1.2) has a classical, maximal in time solution and that the comparison principle is true (using the methods of [3]). A number of

© 2014 Zeng et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly credited.

papers have studied problem (1.2) from the point of view of blow-up and global existence (see [4, 5]). In [6], Chen studied the following problem:

ut - Au = upva, x e 0< t < T*, vt - Av = ufivq, x e ^,0< t < T*, u(x, t) = v(x, t) = 0, x e 9^, t > 0, u(x,0) = u0(x), v(x,0) = v0(x), x e

assumingp > 1, or q > 1, or afi >(1 -p)(1 - q), he proved that the solution blows up in finite time if the initial data u0(x) and v0(x) are large enough.

In the case of a(x) = b(x) = 1, Li and Wang [7] discussed the blow-up properties for this system, and they proved that:

(i) If m, q < 1, this system possesses uniform blow-up profiles.

(ii) If m, q > 1, this system presents single point blow-up patterns.

Recently, Zhang and Yang [8] studied the problem of (1.1), but they only obtained the estimation of the blow-up rate, which is not precisely determined. In [9], the authors proved there are initial data such that simultaneous and non-simultaneous blow-up occur for a diffusion system with weighted localized sources, but they did not study the profile of the blow-up solution. There are many known results concerning blow-up properties for parabolic system equations, of which the reaction terms are of a nonlinear localized type. For more details as regards a parabolic system with localized sources, see [10-14].

Our present work is partially motivated by [15-18]. The purpose of this paper is to determine the blow-up rate of solutions for a nonlinear parabolic equation system with a weighted localized source. That is, we prove that the solutions u and v blow up simultaneously and that the blow-up rate is uniform in all compact subsets of the domain. Moreover, the blow-up profiles of the solutions are precisely determined.

In the following section, we will build the profile of the blow-up solution of (1.1).

2 Blow-up profile

Throughout this paper, we assume that the functions a(x), b(x), u0(x) and v0(x) satisfythe following three conditions: (Al) a(x), b(x), u0(x), v0(x) e C2(B); a(x), b(x), u0(x), v0(x) > 0 inB and

a(x) = b(x) = u0(x) = v0(x) = 0 on dB. (A2) a(x), b(x), u0(x) and v0(x) are radially symmetric; a(r), b(r), u0(r) and v0(r) are

non-increasing for r e (0,R] (r = |x|). (A3) u0(x) and v0(x) satisfy Au0(x) + a(x)up0(x)va(0, t) > 0 and Av0(x) + b(x)ufi (x)vq(0, t) > 0 in B, respectively.

Theorem 2.1 Assume (A1), (A2), and (A3) hold. Let (u, v) be the blow-up solution of (1.1), non-decreasing in time, and let the following limits hold uniformly in all compact subsets ofB:

(i) If p <1, q <1 and afi >(1 -p)(1 - q), then

lim u(x, t)(T* - t)6 = a(x)ll(1-p)C20B(a 19)filafi-(1-p)(1-q),

lim v(x, t)(T* - t)a = b(x)1l(1-p)C1aa(9la)alafi-(1-p)(1-q), t^r* '

9 = (a + 1 - q)l(afi - (1 -p)(1 - q)), a = (fi + 1 -p)l(afi - (1 -p)(1 - q)),

fi fi9 a a9

C1 = (a(0)b(0)) (1-p>(1~q>~afi (b(0))1-q, C2 = (a(0)b(0)) (1-p)(1-q)-afi (a(0))1-q.

(ii) If p < 1 and q = 1, then

Hm u(x, t)(T* - t)1lfi = a(x)1l(1-p)(a(0))1lp-1( ^(1lfi)1lfi,

, (1-p)b(x)

(1+p-fi)b(x) -a(x) (1+fi-p)b(x) / 1 + fi - afib(0)

lim v(x, t)(T* - t) afib<°> = (a(0)) ab(°) (1lfi) afib(0) —fi_£ .

t^r* y ' y ' \ ab(0) /

(iii) If p = 1 and q = 1, then

ate /1 \ fia(0)

lim u(x, t)(T* - t)^ = -r-- , m* y 'K ' \ab(0)J

MxL / 1 lim v(x, t)(T* - t) ab(0) = —— . m* y ' 'K ' \fib(0))

(iv) If p = 1 and q <1, then

(1-q)a(x)

(1+a-q)b(x) -a(x) (1+a-q)b(x) / 1 + a - q \ afia(0)

l a^^ / n\\ fi77riT /1 / „A I ^ \

lim* u(x, t)(T* - t) afia(0) = (b(0)) (1la) afia(0)

lim v(x, t)(T* - t)1la = b(x)1l(1-q)(b(0))1lq-Y-fia(0^Vla(1la)1lfi.

t^T* y ' v ' \1 + a - q/

Throughout this section, we denote

g1(t) = ufi(0,t), Gi(t) = i g1(s)ds, g2(t) = va(0,t), G2(t) = i g2(s)ds.

Lemma 2.1 Assume that (u, v) is the positive solution of (1.1), which blow up in finite time T*. Letp < 1 and q < 1, then

lim g1(t) = lim G1(t) = (, lim g2(t) = lim G2(t) = (.

t^T* t^T * t^T* t^T*

Proof First we claim that limt^T* G2(t) = (. Since u(0, t) = max^ u(x, t), we have

ut(0, t) < a(0)up(0, t)g2(t). By integrating the above inequality over (0, t), we get

u1-p(0, t) < (1-p)a(0) i g2(s) ds + uQ-p(0), ifp < 1,

ln u(0, t) < a(0)G2(t) + ln uq(0), ifp = 1.

From limt^T* u(0, t) = to, it follows that limt^T* G2(t) = to. Applying similar arguments as above to the equation of v in system (1.2), it is reasonable that limt^T*g\(t) = limt ^t* Gi(t) = to. □

The following lemma will play a key role in proving Theorem 2.1, which will give the relationships among u, v, G1(t), and G2(t).

Lemma 2.2 Under the conditions of Theorem 2.1, the following statements hold uniformly in any compact subsets ofB: (i) p <1 and q <1, then

lim u - (x, t) =(1 _p)a(x), lim V — ^ =(1 - q)b(x).

t^T* G2(t) (ii) p = 1 and q <1, then ln u(x, t)

lim „ , t^T* G2(t)

= a(x),

t^T* G1(t)

lim v _(x',t) =(1 - q)b(x).

(iii) p = 1 and q = 1, then ln u(x, t)

lim „ , t^T* G2(t)

= a(x),

t^T* G1(t)

ln v(x, t)

lim -—— = b(x).

t^T* G1(t) ( )

(iv) p <1 and q = 1, then

u1-p(x> i) In v(x, t)

lim = (1 -p)a(x), lim = b(x).

t^r* G2(t) t^r* G1(t)

Proof (i) When p <1 and q < 1. A simple computation shows that

dv1-p dt

= Au1-p + p(1 -p)u-1-p|Vu|2 + (1 -p)a(x)g2(t), x e Q, 0 < t < T*, = Av1-q + q(1 - q)v-1-q|Vu|2 + (1 - q)b(x)g1(t), x e Q, 0 < t < T*,

(2.1) (2.2)

and the initial and boundary conditions are given by

| u1-p (x, t) = v1-q(x, t) = 0, x e dB, t > 0, { u1-p(x, 0) = u0-p(x), v1-q(x, 0) = v0-q(x), x e B.

Denote X2, the first eigenvalue of -A in Hj(B) and by y(x) > 0 and 0(x) > 0 the corresponding eigenfunction, normalized by fB a(x)y(x) dx = 1 and fB b(x)0(x) dx = 1.

Multiplying both sides of (2.1) and (2.2) by y and 0, respectively, and integrating over B x (0, t), we have, for 0 < t < T*

f u1-py dx - f u0-Py dx = -.2 f f u1-py dxds Jb Jb j0 Jb

np(1 - p)u-p-1|V u|V dxds + (1 -p)G2(t),

I vlp\$ dx - I Vqp\$ dx = —.2 / I v1 dxds Jb Jb Jo Jb

np(1 -p)v-p- 1|V v|2^ dxds + (1 - p)Gi(t).

We claim that lim^T* u1 -p(0, t)/g2(t) = 0 and lim^T* v1-q(0, t)/gi(t) = 0. In fact, we have ut(0, t) < up(0, t)V (0, t), for 0 < t < T* that is,

u1-p(o t)

limsup —< (1-p)a(0). (2.3)

t!T* G2(t)

Since g2(t) is non-decreasing, it follows that for all e >0,

T* - e

0 < GO) < f0T -e g2 (s) ds + e < g2 (t) < g2(t) + e,

and using limt!T*g2(t) = to, we deduce that limt!T* G2(t)/g2(t) = 0, so that (2.3) implies limt!T* u1-p(0, t)/g2(t) = 0. By a process analogous to above, we arrive at limt!T* v1-p(0, t)/ g1(t) = 0.

Analogous to the proof of Theorem 2.2 in Ref. [18], it can be inferred that

L u1-pp dx / L v1-qV dx / / N

!* ^^ !* = (1-q). (24)

From (2.1) and (2.2), we know (u1-p, v1-q) is a sub-solution of the following problem:

'di = Aw +(1 -p)a(x)g2(t), x e B, 0 < t < T*, d = Az +(1 - q)b(x)g1(t), x e B,0< t < T*, w(x, t) = z(x, t) = 0, x e dB, t >0, w(x, 0) = u0p(x), z(x, 0) = v0~q(x), x e B,

Equation (2.5) and Lemma 2.1 assert that

w(x, t) z(x, t)

!* gw=(1 -p)a(x), !* =(1"q)b(x), (2.6)

uniformly in all compact subsets of B.

The rest of the proof of case (i) is similar to Lemma 2.2(i). Cases (ii), (iii), and (iv) can be treated similarly. Now we prove Theorem 2.1 by using Lemma 2.2. □

Proof of Theorem 2.1 (i) If p <1 and q < 1. By Lemma 2.2(i), we know that for choosing positive constants 5 <1<t, there exists t0 < T* such that

(5(1 -p)a(0)G2(t))M1-p) < G1 (t) < (t(1 -p)a(0)G2(t))M1-p), t e [to, T*), (S(1 - q)b(0)G1(t))a/(1-q) < G2(t) < (t(1 - q)b(0)G1(t))a/(1-q), t e [to, T*).

Therefore,

(5(1-p)a(0)G2(t)f/(1-p) ^ dG1(t) ^ (t(1-p)a(0)G2(t))p/(1-p] (t(1- q)b(0)G1(t))a/(1-q) < dG2(t) < (5(1- q)b(0)G1(t))a/(1-q).

From the right-hand side of (2.7),

(5(1 - q)b(0)G1(t))al(1-q)dG1(t) < (t(1 -p)a(0)G2(t))fil(1-p)dG2(t), t e [to, T*). Integrating the above inequality over [0, t) yields

(1 - q)(S(l -

-i-G1 (t)

1 + a - q

(1 -p)(T(1 -p)b(O))ß/(1-rtMi+ß-p)/(i-p)^ t

<-r+ß-p-G (t) to

< (1-p)(t(1-P)b(O))ß/(1-p) G(1+ß-p)/d-p)(t). (2.8)

1 + ß - p

Since limt^T* G1(t) = to and q <1, for any constant O < e <1, there exists iO : tO < iO < T* such that G11+a-q)/(1-q)(to) < (1 - e)G^+a-qV(1-q)(t) for t e fo, T*). Hence, from (2.8) it can be deduced that for t e [iO, T*),

e(5b(O))a/(1-q)(1 + ß -p)((1 - q)G1(t))(1+a-q)/(1-q)

< (tß(O))ß/(1-p)(1 + 9 - q)((1 -p)G2(t))(1+ß-p)/(1-p). (2.9)

By an argument similar to above, there exists iO < T* such that iO < t < T*,

e(5a(O))ß/(1-p)(1 + 9 - q)((1 -p)G2(t))(1+a-q)/(1-q)

< (tb(O))a/(1-q)(1 + ß -p)((1 - q)G1 (t))(1+a-q)/(1-q). (2.1O)

Set t* = maxjiO, ¿o), then (2.9) and (2.1O) hold simultaneously for all t e [t*, T*). Next we choose №)°?l, (ei),?!, {ri)i=1, satisfying O < 5;, ei <1 and Ti >1 with Si, ei, Ti ^ 1 as i ^to. Let t* < T* such that (2.9) and (2.1O) hold for t* < t < T*. From Lemma 2.2(i), it follows that for such sequences (5i)°=1, and (ri)?=1, there exists (ti)°=1 : ti < T* with ti ^ T*, as i ^ to such that

(Si(1 -p)a(O)G2(t))ß/(1-p) < Gl(t) < (n(1 -p)a(O)G2(t))ß/(1-p), t e [ti, T*). (2.11) Taking Ti = max(t*, ti), in terms of (2.9), (2.1O), and (2.11), we deduce that for T < t < T* Gl(t) > (Si(1-p)a(O)G2(t))ß/(1-p)

0 / Sib(O) \ (i-p)(i+ß-p) ß

> (8ib(O)) ^^ TOO)) (eia/0) 1+ß-p ((1-q)Gi(t))(2.12)

ß0 / T b(O)\ (i-p)(i+ß-p) ß ß0

Gl(t) < (xib(O))^SOO)) (a/ei0)1+Fp ((1-q)Gi(t^(2.13)

where C = (a(O)/b(O))ß/(1-p).

Since 1 - //0/(a (1 - q)) = -1/(a (1 - q)) < 0 and limt^T* G^t) = ro, integrating (2.12) and (2.13) over (t, T*) we have, for T < t < T*,

£>-1a(a/0)-1+b < (T* - t)((1-q)G1(t))1/a(1-q) < d-1a(a/0)-1+h, (2.14)

/a(0) \b/(1-p), / &b(0) \ (1-p)(1+b-p) b

/(1-p) /r.hinW (1-pm

a(0)\b/(1-p), f/ x,^/tib(0) \(i-p)(i+b -p) x ' v b(0v v ' n Via(0)J ( ')

Clearly,

a(0) \b/(1-p), , xx / b(0) \ (i-p)(i+b-p)

d— lblf >0)) ■• a. — 1.

By plugging i ^ ro into (2.14) we get

((1 - q)G1(t))1/(1-q) - Qaa(0/a)3/a/-(1-P)(1-q) (T* - t)-a, (2.15)

/ BL + /

where C1 = (a(0)) (1-p)(1-q)-a/ (b(0))1-q + (1-p)(1-q)-a/ .

Applying a similar proof to the one above, we can conclude that

((1 - q)G2(t))1/(1-p) - C200(a/0)B/a/-(1"P)(1-q)(T* - t)-°, (2.16)

_a__a0 _a_

where C2 = (b(0)) (1-p)(1-q)-a/ (a(0))1-q + a-p)a-q)-a/ .

According to Lemma 2.2(i), (2.15), and (2.16), it follows that uniformly in all compact subsets of B

lim u(x, t)(T* - t) = a(x)1/(1-p)C20°(a/0)//a/-(1-p)(1-q),

t^T* v '

lim v(x, t)(T* - t)a = b(x)1/(1-p)C1 aa(0/a)a/a/-(1-p)(1-q).

t^T* v '

The arguments of cases (ii), (iii), and (iv) are very similar to the above, we omit the details. Therefore, we have completed the proof of Theorem 2.1. □

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All the authors typed, read, and approved the final manuscript. Acknowledgements

This work was supported by the National Natural Science Foundation of China underGrant 61374194, the China Postdoctoral Science Foundation Founded Project underGrant 2013M540405, the National Key Technologies R&D Program of China underGrant 2009BAG13A06, the National High-tech R&D Program of China (863 Program) underGrant 2008AA040202, and the Natural Science Foundation of Jiangsu Province under grant BK20140638.

Received: 8 May 2014 Accepted: 27 May 2014 Published online: 11 September 2014

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doi:10.1186/s13661-014-0143-1

Cite this article as: Zeng et al.: Profiles of blow-up solution of a weighted diffusion system. Boundary Value Problems 2014 2014:143.

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