Chunxue Liu

F Li

H Chen

H Li

J Bebernes

KB Bimpong

L Kong

L Pao

L Wang

M Escobedo

M Pedersen

MX Wang

P Souplet

QL Liu

R Zhang

Shumin Fei

W Deng

Weili Zeng

WL Zeng

Xiaobo Lu

Z Ling

Boundary Value Problems

English

Crossref

Zeng et al. Boundary Value Problems 2014, 2014:143http://www.boundaryvalueproblems.com/content/2014/1/143RESEARCH Open AccessProﬁles of blow-up solution of a weighteddiffusion systemWeili Zeng1,2*, Chunxue Liu1, Xiaobo Lu1,2 and Shumin Fei1,2*Correspondence: zwlseu@163.com1School of Automation, SoutheastUniversity, Nanjing, 210096, China2Key Laboratory of Measurementand Control of CSE, Ministry ofEducation, Southeast University,Nanjing, 210096, ChinaAbstractIn this paper, we study the blow-up proﬁles for a coupled diﬀusion system with aweighted source term involved in a product with local term. We prove that thesolutions have a global blow-up and the proﬁle of the blow-up is preciselydetermined in all compact subsets of the domain.Keywords: diﬀusion system; weighted localized source; blow-up proﬁle1 IntroductionIn this paper, we consider the following coupled diﬀusion system with a weighted nonlin-ear localized sources:⎧⎪⎪⎪⎨⎪⎪⎪⎩ut –u = a(x)up(x, t)vα(, t), x ∈ B,  < t < T∗,vt –v = b(x)uβ(, t)vq(x, t), x ∈ B,  < t < T∗,u(x, t) = v(x, t) = , x ∈ ∂B, t > ,u(x, ) = u(x), v(x, ) = v(x), x ∈ B,(.)where B is an open ball of RN , N ≥  with radius R; α, β , p, q are nonnegative constantsand satisfy α + p >  and β + q > .System (.) is usually used as amodel to describe heat propagation in a two-componentcombustible mixture []. In this case u and v represent the temperatures of the interactingcomponents, 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 asingle site, of the form⎧⎪⎪⎪⎨⎪⎪⎪⎩ut –u = up(, t)vα(, t), x ∈ ,  < t < T∗,vt –v = uβ (, t)vq(, t), x ∈ ,  < t < T∗,u(x, t) = v(x, t) = , x ∈ ∂, t > ,u(x, ) = u(x), v(x, ) = v(x), x ∈ ,(.)was studied by Pao and Zheng [] and they obtained the blow-up rates and boundary layerproﬁles of the solutions.As for problem (.), it is well known that problem (.) has a classical, maximal in timesolution and that the comparison principle is true (using themethods of []). A number of© 2014 Zeng et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly credited.Zeng et al. Boundary Value Problems 2014, 2014:143 Page 2 of 8http://www.boundaryvalueproblems.com/content/2014/1/143papers have studied problem (.) from the point of view of blow-up and global existence(see [, ]).In [], Chen studied the following problem:⎧⎪⎪⎪⎨⎪⎪⎪⎩ut –u = upvα , x ∈ ,  < t < T∗,vt –v = uβvq, x ∈ ,  < t < T∗,u(x, t) = v(x, t) = , x ∈ ∂, t > ,u(x, ) = u(x), v(x, ) = v(x), x ∈ ,(.)assuming p > , or q > , or αβ > (–p)(–q), he proved that the solution blows up in ﬁnitetime if the initial data u(x) and v(x) are large enough.In the case of a(x) = b(x) = , Li and Wang [] discussed the blow-up properties for thissystem, and they proved that:(i) If m,q≤ , this system possesses uniform blow-up proﬁles.(ii) If m,q > , this system presents single point blow-up patterns.Recently, Zhang and Yang [] studied the problem of (.), but they only obtained the es-timation of the blow-up rate, which is not precisely determined. In [], the authors provedthere are initial data such that simultaneous and non-simultaneous blow-up occur for adiﬀusion system with weighted localized sources, but they did not study the proﬁle ofthe blow-up solution. There are many known results concerning blow-up properties forparabolic 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 [–].Our present work is partially motivated by [–]. The purpose of this paper is to de-termine the blow-up rate of solutions for a nonlinear parabolic equation system with aweighted localized source. That is, we prove that the solutions u and v blow up simultane-ously 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 (.).2 Blow-up proﬁleThroughout this paper, we assume that the functions a(x), b(x), u(x) and v(x) satisfy thefollowing three conditions:(A) a(x),b(x),u(x), v(x) ∈ C(B); a(x),b(x),u(x), v(x) >  in B anda(x) = b(x) = u(x) = v(x) =  on ∂B.(A) a(x), b(x), u(x) and v(x) are radially symmetric; a(r), b(r), u(r) and v(r) arenon-increasing for r ∈ (,R] (r = |x|).(A) u(x) and v(x) satisfy u(x) + a(x)up(x)vα(, t)≥  andv(x) + b(x)uβ (x)vq(, t)≥  in B, respectively.Theorem . Assume (A), (A), and (A) hold. Let (u, v) be the blow-up solution of (.),non-decreasing in time, and let the following limits hold uniformly in all compact subsetsof B:(i) If p < , q <  and αβ > ( – p)( – q), thenlimt→T∗u(x, t)(T∗ – t)θ = a(x)/(–p)Cθθ (σ /θ )β/αβ–(–p)(–q),limt→T∗v(x, t)(T∗ – t)σ = b(x)/(–p)Cσσ (θ/σ )α/αβ–(–p)(–q),Zeng et al. Boundary Value Problems 2014, 2014:143 Page 3 of 8http://www.boundaryvalueproblems.com/content/2014/1/143whereθ = (α +  – q)/(αβ – ( – p)( – q)), σ = (β +  – p)/(αβ – ( – p)( – q)),C =(a()b()) β(–p)(–q)–αβ(b()) βθ–q , C =(a()b()) α(–p)(–q)–αβ(a()) αθ–q .(ii) If p <  and q = , thenlimt→T∗u(x, t)(T∗ – t)/β = a(x)/(–p)(a())/p–(αb() + β – p)/β(/β)/β ,limt→T∗v(x, t)(T∗ – t) (+p–β)b(x)αβb() =(a()) –a(x)αb() (/β)(+β–p)b(x)αβb()( + β – pαb()) (–p)b(x)αβb().(iii) If p =  and q = , thenlimt→T∗u(x, t)(T∗ – t) a(x)βa() =( αb()) a(x)βa(),limt→T∗v(x, t)(T∗ – t) b(x)αb() =( βb()) b(x)αb().(iv) If p =  and q < , thenlimt→T∗u(x, t)(T∗ – t) (+α–q)b(x)αβa() =(b()) –a(x)βa() (/α)(+α–q)b(x)αβa()( + α – qβa()) (–q)a(x)αβa(),limt→T∗v(x, t)(T∗ – t)/α = b(x)/(–q)(b())/q–(βa() + α – q)/α(/α)/β .Throughout this section, we denoteg(t) = uβ (, t), G(t) =∫ tg(s)ds, g(t) = vα(, t), G(t) =∫ tg(s)ds.Lemma . Assume that (u, v) is the positive solution of (.), which blow up in ﬁnitetime T∗. Let p≤  and q ≤ , thenlimt→T∗g(t) = limt→T∗ G(t) =∞, limt→T∗ g(t) = limt→T∗ G(t) =∞.Proof First we claim that limt→T∗ G(t) =∞. Since u(, t) = max u(x, t), we haveut(, t)≤ a()up(, t)g(t).By integrating the above inequality over (, t), we getu–p(, t)≤ ( – p)a()∫ tg(s)ds + u–p (), if p < ,lnu(, t)≤ a()G(t) + lnu(), if p = .Zeng et al. Boundary Value Problems 2014, 2014:143 Page 4 of 8http://www.boundaryvalueproblems.com/content/2014/1/143From limt→T∗ u(, t) = ∞, it follows that limt→T∗ G(t) = ∞. Applying similar argu-ments as above to the equation of v in system (.), it is reasonable that limt→T∗ g(t) =limt→T∗ G(t) =∞. The following lemma will play a key role in proving Theorem ., which will give therelationships among u, v, G(t), and G(t).Lemma . Under the conditions of Theorem ., the following statements hold uniformlyin any compact subsets of B:(i) p <  and q < , thenlimt→T∗u–p(x, t)G(t)= ( – p)a(x), limt→T∗v–q(x, t)G(t)= ( – q)b(x).(ii) p =  and q < , thenlimt→T∗lnu(x, t)G(t)= a(x), limt→T∗v–q(x, t)G(t)= ( – q)b(x).(iii) p =  and q = , thenlimt→T∗lnu(x, t)G(t)= a(x), limt→T∗ln v(x, t)G(t)= b(x).(iv) p <  and q = , thenlimt→T∗u–p(x, t)G(t)= ( – p)a(x), limt→T∗ln v(x, t)G(t)= b(x).Proof (i) When p <  and q < . A simple computation shows thatdu–pdt =u–p + p( – p)u––p|∇u| + ( – p)a(x)g(t), x ∈ ,  < t < T∗, (.)dv–pdt =v–q + q( – q)v––q|∇u| + ( – q)b(x)g(t), x ∈ ,  < t < T∗, (.)and the initial and boundary conditions are given by{u–p(x, t) = v–q(x, t) = , x ∈ ∂B, t > ,u–p(x, ) = u–p (x), v–q(x, ) = v–q (x), x ∈ B.Denote λ, the ﬁrst eigenvalue of – in H(B) and by ϕ(x) >  and φ(x) >  the corre-sponding eigenfunction, normalized by∫B a(x)ϕ(x)dx =  and∫B b(x)φ(x)dx = .Multiplying both sides of (.) and (.) by ϕ and φ, respectively, and integrating overB× (, t), we have, for  < t < T∗∫Bu–pϕ dx –∫Bu–p ϕ dx = –λ∫ t∫Bu–pϕ dxds+∫ t∫Bp( – p)u–p–|∇u|ϕ dxds + ( – p)G(t),Zeng et al. Boundary Value Problems 2014, 2014:143 Page 5 of 8http://www.boundaryvalueproblems.com/content/2014/1/143∫Bv–pφ dx –∫Bv–p φ dx = –λ∫ t∫Bv–pφ dxds+∫ t∫Bp( – p)v–p–|∇v|φ dxds + ( – p)G(t).We claim that limt→T∗ u–p(, t)/g(t) =  and limt→T∗ v–q(, t)/g(t) = . In fact, we haveut(, t)≤ up(, t)vα(, t), for  < t < T∗ that is,limt→T∗supu–p(, t)G(t)≤ ( – p)a(). (.)Since g(t) is non-decreasing, it follows that for all ε > ,≤ G(t)g(t) ≤∫ T∗–ε g(s)dsg(t)+ ε,and using limt→T∗ g(t) = ∞, we deduce that limt→T∗ G(t)/g(t) = , so that (.) implieslimt→T∗ u–p(, t)/g(t) = . By a process analogous to above, we arrive at limt→T∗ v–p(, t)/g(t) = .Analogous to the proof of Theorem . in Ref. [], it can be inferred thatlimt→T∗∫u–pϕ dxG(t)= ( – p), limt→T∗∫v–qϕ dxG(t)= ( – q). (.)From (.) and (.), we know (u–p, v–q) is a sub-solution of the following problem:⎧⎪⎪⎪⎨⎪⎪⎪⎩dwdt =w + ( – p)a(x)g(t), x ∈ B,  < t < T∗,dzdt =z + ( – q)b(x)g(t), x ∈ B,  < t < T∗,w(x, t) = z(x, t) = , x ∈ ∂B, t > ,w(x, ) = u–p (x), z(x, ) = v–q (x), x ∈ B,(.)Equation (.) and Lemma . assert thatlimt→T∗w(x, t)G(t)= ( – p)a(x), limt→T∗z(x, t)G(t)= ( – q)b(x), (.)uniformly in all compact subsets of B.The rest of the proof of case (i) is similar to Lemma .(i). Cases (ii), (iii), and (iv) can betreated similarly. Now we prove Theorem . by using Lemma .. Proof of Theorem . (i) If p <  and q < . By Lemma .(i), we know that for choosingpositive constants δ <  < τ , there exists t < T∗ such that(δ( – p)a()G(t))β/(–p) ≤G′(t)≤(τ ( – p)a()G(t))β/(–p), t ∈ [t,T∗),(δ( – q)b()G(t))α/(–q) ≤G′(t)≤(τ ( – q)b()G(t))α/(–q), t ∈ [t,T∗).Therefore,(δ( – p)a()G(t))β/(–p)(τ ( – q)b()G(t))α/(–q)≤ dG(t)dG(t) ≤(τ ( – p)a()G(t))β/(–p)(δ( – q)b()G(t))α/(–q). (.)Zeng et al. Boundary Value Problems 2014, 2014:143 Page 6 of 8http://www.boundaryvalueproblems.com/content/2014/1/143From the right-hand side of (.),(δ( – q)b()G(t))α/(–q)dG(t)≤(τ ( – p)a()G(t))β/(–p)dG(t), t ∈[t,T∗).Integrating the above inequality over [, t) yields( – q)(δ( – q)b())α/(–q) + α – q G(+α–q)/(–q) (t)∣∣∣tt≤ ( – p)(τ ( – p)b())β/(–p) + β – p G(+β–p)/(–p) (t)∣∣∣tt≤ ( – p)(τ ( – p)b())β/(–p) + β – p G(+β–p)/(–p) (t). (.)Since limt→T∗ G(t) = ∞ and q < , for any constant  < ε < , there exists t¯ : t ≤ t¯ ≤ T∗such that G(+α–q)/(–q) (t) ≤ ( – ε)G(+α–q)/(–q) (t) for t ∈ [t¯,T∗). Hence, from (.) it canbe deduced that for t ∈ [t¯,T∗),ε(δb())α/(–q)( + β – p)(( – q)G(t))(+α–q)/(–q)≤ (τa())β/(–p)( + ∂ – q)(( – p)G(t))(+β–p)/(–p). (.)By an argument similar to above, there exists t˜ < T∗ such that t˜ < t < T∗,ε(δa())β/(–p)( + ∂ – q)(( – p)G(t))(+α–q)/(–q)≤ (τb())α/(–q)( + β – p)(( – q)G(t))(+α–q)/(–q). (.)Set t∗ = max{t¯, t˜}, then (.) and (.) hold simultaneously for all t ∈ [t∗,T∗). Next wechoose {δi}∞i=, {εi}∞i=, {τi}∞i=, satisfying  < δi, εi <  and τi >  with δi, εi, τi →  as i → ∞.Let t∗ < T∗ such that (.) and (.) hold for t∗i ≤ t < T∗. From Lemma .(i), it followsthat for such sequences {δi}∞i=, and {τi}∞i=, there exists {ti}∞i= : ti < T∗ with ti → T∗, asi→ ∞ such that(δi( – p)a()G(t))β/(–p) ≤G′(t)≤(τi( – p)a()G(t))β/(–p), t ∈ [ti,T∗). (.)Taking Ti = max{t∗i , ti}, in terms of (.), (.), and (.), we deduce that for T ≤ t < T∗G′(t)≥(δi( – p)a()G(t))β/(–p)≥ (δib()) βθσ (–q)(δib()τia()) β(–p)(+β–p)(εiσ /θ )β+β–p(( – q)G(t)) βθσ (–q) , (.)G′(t)≤(τib()) βθσ (–q)(τib()δia()) β(–p)(+β–p)(σ /εiθ )β+β–p(( – q)G(t)) βθσ (–q) , (.)where C = (a()/b())β/(–p) .Zeng et al. Boundary Value Problems 2014, 2014:143 Page 7 of 8http://www.boundaryvalueproblems.com/content/2014/1/143Since  – βθ/(σ ( – q)) = –/(σ ( – q)) <  and limt→T∗ G(t) =∞, integrating (.) and(.) over (t,T∗) we have, for T ≤ t < T∗,D–i σ (σ /θ )– β+β–p ≤ (T∗ – t)(( – q)G(t))/σ (–q) ≤ d–i σ (σ /θ )–β+β–p , (.)wheredi =(a()b())β/(–p)(δib()) βθσ (–q)(δib()τia()) β(–p)(+β–p)(εi)β+β–p ,Di =(a()b())β/(–p)(τib()) βθσ (–q)(τib()δia()) β(–p)(+β–p)(εi)β+β–p .Clearly,di,Di →(a()b())β/(–p)(b()) βθσ (–q)(b()a()) β(–p)(+β–p), as εi,∈?, τi → .By plugging i→ ∞ into (.) we get(( – q)G(t))/(–q) ∼ Cσσ (θ/σ )β/αβ–(–p)(–q)(T∗ – t)–σ , (.)where C = (a())β(–p)(–q)–αβ (b())βθ–q +β(–p)(–q)–αβ .Applying a similar proof to the one above, we can conclude that(( – q)G(t))/(–p) ∼ Cθθ (σ /θ )β/αβ–(–p)(–q)(T∗ – t)–θ , (.)where C = (b())α(–p)(–q)–αβ (a())αθ–q +α(–p)(–q)–αβ .According to Lemma .(i), (.), and (.), it follows that uniformly in all compactsubsets of Blimt→T∗u(x, t)(T∗ – t)θ = a(x)/(–p)Cθθ (σ /θ )β/αβ–(–p)(–q),limt→T∗v(x, t)(T∗ – t)σ = b(x)/(–p)Cσσ (θ/σ )α/αβ–(–p)(–q).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 .. Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll the authors typed, read, and approved the ﬁnal manuscript.AcknowledgementsThis work was supported by the National Natural Science Foundation of China under Grant 61374194, the ChinaPostdoctoral Science Foundation Founded Project under Grant 2013M540405, the National Key Technologies R&DProgram of China under Grant 2009BAG13A06, the National High-tech R&D Program of China (863 Program) under Grant2008AA040202, and the Natural Science Foundation of Jiangsu Province under grant BK20140638.Received: 8 May 2014 Accepted: 27 May 2014Zeng et al. Boundary Value Problems 2014, 2014:143 Page 8 of 8http://www.boundaryvalueproblems.com/content/2014/1/143References1. Bebernes, J, Eberly, D: Mathematical Problems from Combustion Theorem. Springer, New York (1989)2. Pao, L, Zheng, S: Critical exponents and asymptotic estimates of solutions to parabolic systems with localizednonlinear sources. J. Math. Anal. Appl. 292, 621-635 (2004)3. Deng, W: Global existence and ﬁnite time blow up for a degenerate reaction-diﬀusion system. Nonlinear Anal. 60,977-991 (2005)4. Wang, L, Chen, Q: The asymptotic behavior of blow-up solution of localized nonlinear equation. J. Math. Anal. Appl.200, 315-321 (1996)5. Wang, MX: Global existence and ﬁnite time blow up for a reaction-diﬀusion system. Z. Angew. Math. Phys. 51,160-167 (2000)6. Chen, H: Global existence and blow-up for a nonlinear reaction-diﬀusion system. J. Math. Anal. Appl. 212, 481-492(1997)7. Li, H, Wang, M: Properties of blow-up solution to a parabolic system with nonlinear localized terms. Discrete Contin.Dyn. Syst. 13, 683-700 (2005)8. Zhang, R, Yang, Z: Uniform blow-up rates and asymptotic estimates of solutions for diﬀusion systems with weightedlocalized sources. J. Appl. Math. Comput. 32, 429-441 (2010)9. Ling, Z, Wang, Z: Simultaneous and non-simultaneous blow-up criteria of solutions for a diﬀusion system withweighted localized sources. J. Appl. Math. Comput. 40, 183-194 (2012)10. Bimpong, KB, Ross, PO: Far-from-equilibrium phenomena at local sites of reaction. J. Chem. Phys. 80, 3124-3133(1974)11. Pedersen, M, Lin, ZG: Coupled diﬀusion systems with localized nonlinear reactions. Comput. Math. Appl. 42, 807-816(2001)12. Souplet, P: Blow up in nonlocal reaction-diﬀusion equations. SIAM J. Math. Anal. 29(6), 1301-1334 (1998)13. Escobedo, M, Herrero, M: A semi-linear parabolic system in a bounded domain. Ann. Mat. Pura Appl. (4) CLXV,315-336 (1993)14. Li, F, Liu, B: Non-simultaneous blow-up in parabolic equations coupled via localized sources. Appl. Math. Lett. 23,871-874 (2010)15. Liu, QL, Li, YX, Gao, HJ: Uniform blow-up rate for diﬀusion equations with nonlocal nonlinear source. Nonlinear Anal.67, 1947-1957 (2007)16. Kong, L, Wang, J, Zheng, S: Asymptotic analysis to a parabolic equation with a weighted localized source. Appl. Math.Comput. 197, 819-827 (2008)17. Liu, QL, Li, YX, Gao, HJ: Uniform blow-up rate for diﬀusion equations with localized nonlinear source. J. Math. Anal.Appl. 320, 771-778 (2006)18. Zeng, WL, Lu, XB, Tan, XH: Uniform blow-up rate for a porous medium equation with a weighted localized source.Bound. Value Probl. 2011, 57 (2011)doi:10.1186/s13661-014-0143-1Cite this article as: Zeng et al.: Proﬁles of blow-up solution of a weighted diffusion system. Boundary Value Problems2014 2014:143.

Profiles of blow-up solution of a weighted diffusion system

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Profiles of blow-up solution of a weighted diffusion system

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