SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM

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1 Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: URL: or ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM ZHENGQIU LING, ZEJIA WANG Abstract. This article concerns the blow-up solutions of a reaction-diffusion system with nonlocal sources, subject to the homogeneous Dirichlet boundary conditions. The criteria used to identify simultaneous and non-simultaneous blow-up of solutions by using the parameters p and in the model are proposed. Also, the uniform blow-up profiles in the interior domain are established.. Introduction and description of results In this article, we investigate the following reaction-diffusion system with nonlocal sources u t = u + uv p α, (x, t) Ω (, T ), (.) v t = v + uv β, (x, t) Ω (, T ) (.2) u(x, ) = u (x), v(x, ) = v (x), x Ω, (.3) u(x, t) =, v(x, t) =, (x, t) Ω (, T ), (.4) where Ω = B R = { x < R} R N (N ), α, β, p, >, and the continuous functions u (x), v (x) are nonnegative, nontrivial, radially symmetric, decreasing with x, and vanish on B R, where α α = Ω α dx. Nonlinear parabolic systems (.)-(.4) can be used to describe some reaction diffusion phenomena, Such as heat propagations in a two-component combustible mixture [3], chemical reactions [6], interaction of two biological groups without self-iting [], etc., where u and v represent the temperatures of two different materials during a propagation, the thicknesses of two kinds of chemical reactants, the densities of two biological groups during a migration, etc. Using the methods of [7, 2, 4] we know that (.)-(.4) has a local nonnegative classical solution. Moreover, if p,, then the uniueness holds. In recent years, many results on blow-up solutions have been obtained for the nonlinear parabolic system. We will recall several results in the following. As for the other related works on the global existence and blow-up of solutions of the nonlinear parabolic system, they can be found in [5,, 5, 4] and references therein. 2 Mathematics Subject Classification. 35B33, 35B4, 35K55, 35K57. Key words and phrases. Simultaneous and non-simultaneous blow-up; uniform blow-up profile; reaction-diffusion system; nonlocal sources. c 22 Texas State University - San Marcos. Submitted August 6, 22. Published November 24, 22.

2 2 Z. LING, Z. WANG EJDE-22/26 Li, Huang and Xie in [8] and Deng, Li and Xie in [2] considered the following two systems, respectively, u t = u + u m (x, t)v n (x, t) dx, v t = v + u p (x, t)v (x, t) dx, Ω with x Ω, t > ; and u t = u m + a v p α, v t = v n + b u β, (x, t) Ω (, T ). The authors showed some results on the global solutions, the blow-up solutions and the blow-up profiles. In 22, Zheng, Zhao and Chen in [8] studied the problem u t = u + f (u, v), v t = v + f 2 (u, v), (x, t) Ω (, T ) (.5) with homogeneous Dirichlet boundary conditions, where f (u, v) = e mu(x,t)+pv(x,t), f 2 (u, v) = e u(x,t)+v(x,t). The simultaneous blow-up rates are obtained for radially symmetric blow-up solutions in the exponent region { m <, n < p}. Later, Zhao and Zheng in [7], Li and Wang in [9] studied the localized problem (.5) with the more general Ω R N and f (u, v) = e mu(x,t)+pv(x,t), f 2 (u, v) = e u(x,t)+nv(x,t), x Ω. The critical blow-up exponents were discussed. Uniform blow-up profiles for simultaneous blow-up solutions were proved in the exponent region { m, n p}. Our present work is motivated by the above mentioned papers, the main purpose of this paper is to identify the simultaneous and non-simultaneous blow-up of the solutions and establish the uniform blow-up profiles for the system (.) (.4). For convenience, we introduce a pair of parameters σ and θ, the solution of ( p p ) ( ) σ = θ ( Ω ), (.6) namely, p ( ) (p ) σ =, θ = p + p +. (.7) This paper is organized as follows. In the next Section, we investigate the simultaneous and non-simultaneous blow-up of the solutions for the system (.) (.4), and give the blow-up criteria. In Section 3, we deal with the blow-up rates of the solutions. 2. Simultaneous and non-simultaneous blow-up In this section, we discuss the simultaneous and non-simultaneous blow-up phenomena for the system (.) (.4), and propose a complete and optimal classification to identify the simultaneous and non-simultaneous blow-up solutions. For problem (.)-(.4), because of the nonlinear sources, there exist solution (u, v) that blow up in finite time, T, if and only if the exponents p, verify any of conditions, p >, > or p > ( )(p ). In particular, the component u(or v) can blow up for the large initial data if p > (or > p ), see [9, 2]. So there may be non-simultaneous blow-up, that is to say that one component blows

3 EJDE-22/26 SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP 3 up while the other remains bounded. On the other hand, the simultaneous blow-up means that sup u(, t) = sup v(, t) = +. t T t T Assume the initial data u (x), v (x) satisfy u (x) + u v p α εϕ(x)u p ()vp (), x B R, (2.) v (x) + u v β εϕ(x)u ()v (), x B R (2.2) for some a constant ε (, ), where ϕ(x) is the first eigenfunction of ϕ = λϕ, x B R ; ϕ =, x B R, normalized by ϕ =, ϕ > in B R. In addition, by using the methods in [6], it is easy to check that u t, v t for (x, t) B R (, T ) by the comparison principle. Our results about the simultaneous and non-simultaneous blow-up criteria are as follows. Theorem 2.. If p + >, then there exists initial data such that the nonsimultaneous blow-up occurs in (.) (.4) if and only if σ < (or θ < ) ( for v(or u) blowing up alone, respectively). Theorem 2.2. If p + >, then any blow-up in (.) (.4) is non-simultaneous if and only if σ with θ < ( for u blowing up alone ), or θ with σ < ( for v blowing up alone). Corollary 2.3. If p + >, then any blow-up in (.) (.4) is simultaneous if and only if σ and θ. Similar to the study in[8], it is seen that Corollary 2.4. All solutions are global in (.) (.4) if and only if σ < and θ < (i.e., p + < ). In summary, the complete and optimal classification for simultaneous and nonsimultaneous blow-up solutions of the problem (.)-(.4) can be described by Figure v blows up p = = p alone simultaneous blow-up u blows up alone p + = (non-simultaneous blow-up) global (, ) p Figure. Regions of simultaneous and non-simultaneous blow-up The key clues for the classification of simultaneous and non-simultaneous blowup solutions are the signs of p ( ), (p ) and p +. The conditions p > and p + > imply that u may blow up by itself but cannot provide

4 4 Z. LING, Z. WANG EJDE-22/26 sufficient help to the blow-up of v (with small v ), while < p ensures that v can provide effective help to the blow-up of u, but v remains bounded. Before we give the proof of Theorem 2., we first introduce the following lemma. Let φ(x, t) satisfy with φ t = φ, (x, t) B R (, T ); φ =, (x, t) B R (, T ) φ(x, ) = ϕ(x), x B R. Lemma 2.5. Under conditions (2.) and (2.2), the solution (u, v) of (.) (.4) satisfies u t (x, t) εφ(x, t)u p (, t)v p (, t), (x, t) B R [, T ), (2.3) v t (x, t) εφ(x, t)u (, t)v (, t), (x, t) B R [, T ). (2.4) Proof. Since that the proofs of the ineualities (2.3) and (2.4) are similar, we prove only (2.3). Let J(x, t) = u t (x, t) εφ(x, t)u p (, t)v p (, t). It is easy to check that for ε small enough since u t, v t, we obtain J t J = ( uv p ) α t εφ( u p (, t)v p (, t) ) t, (x, t) B R (, T ), J(x, t) =, (x, t) B R (, T ), J(x, ) = u (x) + u v p α εϕ(x)u p ()vp (), x B R. Conseuently, (2.3) is true by the comparison principle. Proof of Theorem 2.. Without loss of generality, we only prove that there exist suitable initial data such that u blows up while v remains bounded if and only if θ <. Assume θ <, namely, p > and p > by Figure and (.7). From (2.3), we obtain that u t (, t) εφ(, T )u p (, t)v p (), t [, T ). (2.5) Integrating the above ineuality (2.5) from t to T, we have the estimate for u as follows u(, t) ( ε(p )φ(, T )v p () ) /(p )(T t) /(p ), t [, T ). (2.6) At the same time, since the initial data (u, v ) is radially symmetric and nonincreasing, therefore the (u, v) is also radial symmetrical and non-increasing; i.e., u r (r, t), v r (r, t) for r [, R). Thus, u(x, t) and v(x, t) always reach their maxima at x =, which means that u(, t), v(, t). Hence, from (.) and (.2), we know that there exist constants C, C 2 > such that u t (, t) uv p α C u p (, t)v p (, t), t [, T ) v t (, t) uv β C 2u (, t)v (2.7) (, t), t [, T ). Let Γ(x, y, t, s) = [4π(t s)] exp { x y 2 } N/2 4(t s) be the fundamental solution of the heat euation. Suppose that (ũ, ṽ ) is a pair of initial data such that the solution of (.) (.4) blows up. Fix radially symmetrical

5 EJDE-22/26 SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP 5 v ( ṽ ) in B R and take constant M > v (x). By the proof of [, Theorem.], we know that if u is large with v fixed then T becomes small. Therefore, let u ( ũ ) be large such that T becomes small and satisfies M v () + p ( ε(p )φ(, T )v p p ()) p T p p M β, where M β = ( Ω M β dx)/β. Consider the following auxiliary problem v t = v + ( ε(p )φ(, T )v p ()) p (T t) p M β, (x, t) B R (, T ), v(x, t) =, (x, t) B R (, T ), v(x, ) = v (x), x B R. Since p >, we obtain by Green s identity that v v () + p ) (ε(p )φ(, T )v p p () p T p p M β M, and hence v satisfies v t v + ( ε(p )φ(, T )v p ()) p (T t) p v(x, t) On the other hand, v satisfies v t v + ( ε(p )φ(, T )v p ()) p (T t) p v(x, t) Therefore, by the comparison principle, we conclude v v M. Now assume that u blows up while v remains bounded. By (2.7) we have u t (, t) Cu p (, t), for t [, T ). This implies p > and the estimate for u that Therefore, by using (2.4), we have u(, t) ( C(p ) ) /(p ) (T t) /(p ). v t (, t) εφ(, T ) ( C(p ) ) By integrating, we obtain that p v ()(T t) p. v(, t) v () + εφ(, T ) ( C(p ) ) t p v () (T s) p ds. (2.8) The boundedness of v reuires p > from (2.8), that is θ <. Thus, the proof is complete. Proof of Theorem 2.2. We only treat the case of u blowing up and v remains bounded. Assume σ with θ < ; that is p, < p and p > by Figure and (.7). From (2.3) and (2.7), we have v p (, t)v t (, t) C 2 εφ(, T ) u p (, t)u t (, t), t [, T ). (2.9) By Theorem 2., it is impossible for v blowing up alone under σ with θ <. Then we show that v is bounded. In fact, by integrating the ineuality (2.9) from to t, we have v p + (, t) C Cu (p ) (, t) for some a C >. Therefore, we can get the boundedness of v(, t). β. β.

6 6 Z. LING, Z. WANG EJDE-22/26 Now, assume that any blow-up must be the case for u blowing up alone. This reuires θ < by Theorem 2.. Again by Theorem 2., if in addition σ <, there exists the initial data such that v blows up alone. Therefore, it has to be satisfied that σ. Then, the proof is complete. 3. Uniform Blow-up Profiles In this section, we study the uniform blow-up profiles for system (.) (.4). At first, the following result of Souplet for a single diffusion euation with nonlocal nonlinear sources [3, Theorem 4.] will play a basic role in our discussion. Lemma 3.. Let u C 2, ( Ω (, T )) be a solution of the problem u t = u + g(t), (x, t) Ω (, T ), u(x, t) =, (x, t) Ω (, T ), u(x, ) = u (x), x Ω, where g(t) is nonnegative and may depend on the solution u. Then u(, t) t T = + (3.) if and only if t g(s) ds = +. Furthermore, if (3.) is fulfilled, then u(x, t) u(, t) = = t T G(t) t T G(t) uniformly on compact subsets of Ω, where G(t) = t g(s) ds. For convenience, we denote t f(t) = uv p α, g(t) = uv β, F (t) = f(s) ds, G(t) = According to the Lemma 3., we have the following result. t g(s) ds. Lemma 3.2. Assume u, v C 2, ( Ω [, T )) are the solutions of (.) (.4). If u and v blow up simultaneously in the finite time T, then we have uniformly on compact subsets of Ω, and u(x, t) v(x, t) =, t T F (t) t T G(t) = t T t T F (t) = G(t) =. We remark that if we assume that only u (or v) blows up in finite time T, then the above conclusions about u ( or v) and F (or G) are also valid. Throughout this section the notation f(t) g(t) is used to describe such functions f(t) and g(t) satisfying f(t)/g(t) as t T. When u and v blow up simultaneously, we have the following results about the uniform blow-up profiles for u and v. Theorem 3.3. Let (u, v) be a solution of (.) (.4) with simultaneous blow-up time T. Then the following its hold uniformly on any compact subset of Ω: () If σ > and θ >, then u(x, t)(t t) σ = t T p/α ( Ω β p σ α σ θ )p/(p+ )) σ, (3.2)

7 EJDE-22/26 SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP 7 (2) If σ =, then v(x, t)(t t) θ = t T /β ( Ω p α θ β θ σ )/(+ p)) θ. (3.3) t T u2 (x, t) ln(t t) = 2 p Ω p α β, (3.4) t T vp (x, t) ( ln v(x, t) ) 2 (T t) = p Ω /β( 2 Ω p α ) β /2. (3.5) (3) If θ =, then we have t T u (x, t) ( ln u(x, t) ) p 2 (T t) = Ω p/α( 2 Ω β ) p α p/2, (3.6) t T v2 (x, t) ln(t t) = 2 Ω β α. (3.7) Proof. From Lemma 3.2, we know that u(x, t) F (t) and v(x, t) G(t), then u α (x, t) v α (x, t) t T F α = (t) t T G α (t) =, u β (x, t) v β (x, t) t T F β = (t) t T G β (t) =. By the Lebesgue dominated convergence theorem, we find that Hence, F (t) = f(t) = uv p α Ω p/α F p (t)g p (t), (3.8) G (t) = g(t) = uv β Ω /β F (t)g (t). (3.9) F p df Ω p α β G p dg. (3.) () Note that the conditions σ > and θ > imply that p + >, + > p since p + >. Integrating (3.) from to t, we obtain F + p (t) Ω p α + p β p + Gp+ (t) = Ω p α θ β σ Gp+ (t). (3.) Combining (3.9) and (3.), we can obtain Since G (t) Ω /β p α β θ σ ) + p G 2 + p = p + + p = θ < 2 p + p (t). (3.2) and t T G(t) =, by integrating (3.2), we obtain /β ( p G(t) Ω α θ ) ) θ(t β + p t) θ. (3.3) θ σ From (3.3) and Lemma 3.2, we have v(x, t)(t t) θ /β ( p = Ω α θ ) ) θ, β + p t t θ σ which holds uniformly on the compact subsets of Ω.

8 8 Z. LING, Z. WANG EJDE-22/26 Combining (3.8) and (3.), and applying the similar proofs of F and u, we obtain that u(x, t)(t t) σ p/α ( = Ω β p σ ) p ) σ α p+ t T σ θ holds uniformly on the compact subsets of Ω. (2) When σ =, or p + =, noticing (3.9) and (3.), we see that G (t) Ω /β( 2 Ω p α β ) /2G (t) ( ln G(t) ) /2. (3.4) Note that t T G(t) =, integrating (3.4) from t(> ) to T asserts Furthermore, t T That is to say that p G(t) G(t) s (ln s) ds Ω /β( 2 Ω p /2 α ) β /2(T t). (3.5) G(t) s (ln s) /2 ds G (t) ( ln G(t) ) /2 = G G s (ln s) /2 ds G ( ln G ) /2 = = p. s (ln s) /2 ds G (t)(ln G(t)) /2 = G p (t)(ln G(t)) /2. (3.6) By (3.5) and (3.6), it indicates G p (t)(ln G(t)) /2 p Ω /β( 2 Ω p α β ) /2(T t). (3.7) Since t T v(x, t) = uniformly on the compact subset of Ω and t T G(t) =, we may claim that the following euivalent is valid uniformly on the compact subset of Ω, v(x, t) G(t) ln v(x, t) ln G(t). And thus by (3.7), we reach the conclusion v p (x, t)(ln v(x, t)) /2 p Ω /β( 2 Ω p α β ) /2(T t). Then uniformly on the compact subsets of Ω, it yields Since t T vp (x, t)(ln v(x, t)) /2 (T t) = p Ω /β( 2 Ω p α it follows from (3.8) and (3.7) that In view of (3.8), we have ln G(t) 2 Ω β p α F 2 (t), ) β /2. F (t)f p (t) Ω p/α G p (t) F (t) p(t t) Ω p α β. (3.8) Therefore, by Lemma 3.2, we obtain 2 F 2 (t) p Ω p α β ln(t t). u 2 (x, t) 2 p Ω p α β ln(t t) ;

9 EJDE-22/26 SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP 9 that is to say t T u2 (x, t) ln(t t) = 2 p Ω p α holds uniformly on the compact subsets of Ω. (3) When θ =, the proof is similar to that of the case (2). Then, the proof is completed. Acknowledgements. This work was supported by the NNSF of China. The authors want to thank the anonymous referees for their helpful suggestions. References [] H. W. Chen; Global existence and blow-up for a nonlinear reaction-diffusion system, J. Math. Anal. Appl., 22 (997), [2] W. B. Deng, Y. X. Li, C. H. Xie; Blow-up and global existence for a nonlocal degenerate parabolic system, J. Math. Anal. Appl., 277 (23), [3] E. Escobedo, M. A. Herrero; Boundedness and blow-up for a semilinear reaction-diffusion system, J. Diff. Euations, 89() (99), [4] M. Escobedo, M.A. Herrero; A semilinear parabolic system in a bounded domain, Ann. Mate. Pura Appl., CLXV(IV) (993), [5] M. Escobedo, H. A. Levine; Critical blow-up and global existence numbers for a weakly coupled system of reaction-diffusion euations, Arch. Rational Mech. Anal., 29 (995), 47-. [6] P. Glansdorff, I. Prigogine; Thermodynamic Theory of Structure, Stability and Fluctuation, Wiley-interscience, London, 97. [7] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva; Linear and Quasi-linear Euations of Parabolic Type, Amer. Math. Soc., Rhode Island, 968. [8] F. C. Li, S. X. Huang, C. H. Xie; Global existence and blow-up of solutions to a nonlocal reaction-diffusion system, Discrete Contin Dyn. Syst., 9(6) (23), [9] H. L. Li, M. X. Wang; Blow-up properties for parabolic systems with localized nonlinear sources, Appl. Math. Lett., 7 (24), [] H. Meinhardt; Models of Biological Pattern Formation, Academic Press, London, 982. [] F. Quirós, J. D. Rossi; Non-simultaneous blow-up in a semilinear parabolic system, Z. angew. Math. Phys., 52 (2), [2] P. Souplet; Blow up in nonlocal reaction-diffusion euations, SIMA J. Math. Anal., 29(6) (998), [3] P. Souplet; Uniform Blow-up profiles and boundary behavior for diffusion euations with nonlocal nonlinear source, J. Diff. Euations, 53 (999), [4] L. W. Wang; The blow-up for weakly coupled reaction-diffusion system, Proc. Amer. Math. Soc., 29 (2), [5] M. X. Wang; Global existence and finite time blow-up for a reaction-diffusion system, Z. Angew. Math. Phys., 5 (2), [6] M. X. Wang; Blow-up rate estimates for semilinear parabolic systems, J. Diff. Euations, 7(2) [7] L. Z. Zhao, S. N. Zheng; Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources, J. Math. Anal. Appl., 292 (24), [8] S. N. Zheng, L. Z. Zhao, F. Chen; Blow-up rates in a parabolic system of ignition model, Nonlinear Anal. 5 (22), Zhengiu Ling Institute of Mathematics and Information Science, Yulin Normal University, Yulin 537, China address: lingz@tom.com Zejia Wang College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 3322, China address: wangzj979@gmail.com β

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