SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM
|
|
- Darrell Lloyd
- 6 years ago
- Views:
Transcription
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
MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationBLOW-UP OF SOLUTIONS FOR A NONLINEAR WAVE EQUATION WITH NONNEGATIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 213 (213, No. 115, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu BLOW-UP OF SOLUTIONS
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationUNIQUENESS OF SELF-SIMILAR VERY SINGULAR SOLUTION FOR NON-NEWTONIAN POLYTROPIC FILTRATION EQUATIONS WITH GRADIENT ABSORPTION
Electronic Journal of Differential Equations, Vol. 2015 2015), No. 83, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIQUENESS OF SELF-SIMILAR
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationOSGOOD TYPE REGULARITY CRITERION FOR THE 3D NEWTON-BOUSSINESQ EQUATION
Electronic Journal of Differential Equations, Vol. 013 (013), No. 3, pp. 1 6. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu OSGOOD TYPE REGULARITY
More informationMULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER THREE-POINT p-laplacian BOUNDARY-VALUE PROBLEMS
Electronic Journal of Differential Equations, Vol. 27(27, No. 23, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp MULTIPLE POSITIVE
More informationPERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian
Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationBLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST
More informationExistence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1
Journal of Mathematical Analysis and Applications 257, 321 331 (2001) doi:10.1006/jmaa.2000.7347, available online at http://www.idealibrary.com on Existence and Multiplicity of Solutions for a Class of
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationHARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.
Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp. 1 8. ISSN: 107-6691. UL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationUNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS
Electronic Journal of Differential Equations, Vol. 16 16, No. 7, pp. 1 11. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS
More informationPositive Solutions of a Third-Order Three-Point Boundary-Value Problem
Kennesaw State University DigitalCommons@Kennesaw State University Faculty Publications 7-28 Positive Solutions of a Third-Order Three-Point Boundary-Value Problem Bo Yang Kennesaw State University, byang@kennesaw.edu
More informationResearch Article On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 21, Article ID 68572, 12 pages doi:1.1155/21/68572 Research Article On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary
More informationSUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON R N
Electronic Journal of Differential Equations, Vol. 015 015), No. 17, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SUPER-QUADRATIC CONDITIONS
More informationCritical exponents for a nonlinear reaction-diffusion system with fractional derivatives
Global Journal of Pure Applied Mathematics. ISSN 0973-768 Volume Number 6 (06 pp. 5343 535 Research India Publications http://www.ripublication.com/gjpam.htm Critical exponents f a nonlinear reaction-diffusion
More informationBlow-up for a Nonlocal Nonlinear Diffusion Equation with Source
Revista Colombiana de Matemáticas Volumen 46(2121, páginas 1-13 Blow-up for a Nonlocal Nonlinear Diffusion Equation with Source Explosión para una ecuación no lineal de difusión no local con fuente Mauricio
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationMULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS
More informationULAM-HYERS-RASSIAS STABILITY OF SEMILINEAR DIFFERENTIAL EQUATIONS WITH IMPULSES
Electronic Journal of Differential Equations, Vol. 213 (213), No. 172, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ULAM-HYERS-RASSIAS
More informationLORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR ELLIPTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 27 27), No. 2, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationVarious behaviors of solutions for a semilinear heat equation after blowup
Journal of Functional Analysis (5 4 7 www.elsevier.com/locate/jfa Various behaviors of solutions for a semilinear heat equation after blowup Noriko Mizoguchi Department of Mathematics, Tokyo Gakugei University,
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationBLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF TYPE WITH ARBITRARY POSITIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 6 6, No. 33, pp. 8. ISSN: 7-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR THE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES
Electronic Journal of Differential Equations, Vol. 29(29), No. 129, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationTRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT BOUNDARY-VALUE PROBLEMS
Electronic Journal of Differential Equations, Vol. 24(24), No. 6, pp. 8. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) TRIPLE POSITIVE
More informationSimultaneous vs. non simultaneous blow-up
Simultaneous vs. non simultaneous blow-up Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F.C.E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More informationOn critical Fujita exponents for the porous medium equation with a nonlinear boundary condition
J. Math. Anal. Appl. 286 (2003) 369 377 www.elsevier.com/locate/jmaa On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition Wenmei Huang, a Jingxue Yin, b andyifuwang
More informationMULTIPLE SOLUTIONS FOR THE p-laplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 37, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MULTIPLE
More informationTHE CAUCHY PROBLEM AND STEADY STATE SOLUTIONS FOR A NONLOCAL CAHN-HILLIARD EQUATION
Electronic Journal of Differential Equations, Vol. 24(24), No. 113, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) THE CAUCHY
More informationSOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS
Electronic Journal of Differential Equations, Vol. 214 (214), No. 225, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTION OF AN INITIAL-VALUE
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS
More informationSYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT
More informationPOTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem
Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL
More informationExplosive Solution of the Nonlinear Equation of a Parabolic Type
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 5, 233-239 Explosive Solution of the Nonlinear Equation of a Parabolic Type T. S. Hajiev Institute of Mathematics and Mechanics, Acad. of Sciences Baku,
More informationNonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains
Electronic Journal of Differential Equations, Vol. 22(22, No. 97, pp. 1 19. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp Nonexistence of solutions
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationCONVERGENCE BEHAVIOUR OF SOLUTIONS TO DELAY CELLULAR NEURAL NETWORKS WITH NON-PERIODIC COEFFICIENTS
Electronic Journal of Differential Equations, Vol. 2007(2007), No. 46, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) CONVERGENCE
More informationGENERIC SOLVABILITY FOR THE 3-D NAVIER-STOKES EQUATIONS WITH NONREGULAR FORCE
Electronic Journal of Differential Equations, Vol. 2(2), No. 78, pp. 1 8. ISSN: 172-6691. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) GENERIC SOVABIITY
More informationEXISTENCE AND APPROXIMATION OF SOLUTIONS OF SECOND ORDER NONLINEAR NEUMANN PROBLEMS
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 03, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationEXISTENCE OF POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
Electronic Journal of Differential Equations, Vol. 013 013, No. 180, pp. 1 8. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE
More informationA Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s
A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationNonexistence of solutions for quasilinear elliptic equations with p-growth in the gradient
Electronic Journal of Differential Equations, Vol. 2002(2002), No. 54, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonexistence
More informationALEKSANDROV-TYPE ESTIMATES FOR A PARABOLIC MONGE-AMPÈRE EQUATION
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 11, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ALEKSANDROV-TYPE
More informationCRITICAL EXPONENTS FOR A SEMILINEAR PARABOLIC EQUATION WITH VARIABLE REACTION.
CRITICAL EXPONENTS FOR A SEMILINEAR PARAOLIC EQUATION WITH VARIALE REACTION. R. FERREIRA, A. DE PALO, M. PÉREZ-LLANOS AND J. D. ROSSI Abstract. In this paper we study the blow-up phenomenon for nonnegative
More informationEXISTENCE, UNIQUENESS AND QUENCHING OF THE SOLUTION FOR A NONLOCAL DEGENERATE SEMILINEAR PARABOLIC PROBLEM
Dynamic Systems and Applications 6 (7) 55-559 EXISTENCE, UNIQUENESS AND QUENCHING OF THE SOLUTION FOR A NONLOCAL DEGENERATE SEMILINEAR PARABOLIC PROBLEM C. Y. CHAN AND H. T. LIU Department of Mathematics,
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM
Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationSOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS ON BOUNDED DOMAINS
Electronic Journal of Differential Equations, Vol. 018 (018), No. 11, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOURTH-ORDER BOUNDARY-VALUE PROBLEMS IN BANACH SPACES
Electronic Journal of Differential Equations, Vol. 9(9), No. 33, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationRELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 147 161. RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL
More informationHOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS
Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO
More informationEXISTENCE OF SOLUTIONS TO P-LAPLACE EQUATIONS WITH LOGARITHMIC NONLINEARITY
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 87, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationDYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS. Wei Feng
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 7 pp. 36 37 DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS Wei Feng Mathematics and Statistics Department
More informationCONSEQUENCES OF TALENTI S INEQUALITY BECOMING EQUALITY. 1. Introduction
Electronic Journal of ifferential Equations, Vol. 2011 (2011), No. 165, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu CONSEQUENCES OF
More informationELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC
More informationBLOW-UP ON THE BOUNDARY: A SURVEY
SINGULARITIES AND DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 33 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1996 BLOW-UP ON THE BOUNDARY: A SURVEY MAREK FILA Department
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationExistence Of Solution For Third-Order m-point Boundary Value Problem
Applied Mathematics E-Notes, 1(21), 268-274 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Existence Of Solution For Third-Order m-point Boundary Value Problem Jian-Ping
More informationREGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY
Electronic Journal of Differential Equations, Vol. 00(00), No. 05, pp. 5. ISSN: 07-669. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REGUARITY OF GENERAIZED NAVEIR-STOKES
More informationViscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationPOSITIVE GROUND STATE SOLUTIONS FOR QUASICRITICAL KLEIN-GORDON-MAXWELL TYPE SYSTEMS WITH POTENTIAL VANISHING AT INFINITY
Electronic Journal of Differential Equations, Vol. 017 (017), No. 154, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu POSITIVE GROUND STATE SOLUTIONS FOR QUASICRITICAL
More informationHIGHER ORDER CRITERION FOR THE NONEXISTENCE OF FORMAL FIRST INTEGRAL FOR NONLINEAR SYSTEMS. 1. Introduction
Electronic Journal of Differential Equations, Vol. 217 (217), No. 274, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu HIGHER ORDER CRITERION FOR THE NONEXISTENCE
More informationSYNCHRONIZATION OF NONAUTONOMOUS DYNAMICAL SYSTEMS
Electronic Journal of Differential Equations, Vol. 003003, No. 39, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp SYNCHRONIZATION OF
More informationA New Regularity Criterion for the 3D Navier-Stokes Equations via Two Entries of the Velocity Gradient
Acta Appl Math (014) 19:175 181 DOI 10.1007/s10440-013-9834-3 A New Regularity Criterion for the 3D Navier-Stokes Euations via Two Entries of the Velocity Gradient Tensor Zujin Zhang Dingxing Zhong Lin
More informationEXISTENCE OF POSITIVE SOLUTIONS OF A NONLINEAR SECOND-ORDER BOUNDARY-VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 236, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE
More informationAN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationBOUNDARY-VALUE PROBLEMS FOR NONLINEAR THIRD-ORDER q-difference EQUATIONS
Electronic Journal of Differential Equations, Vol. 211 (211), No. 94, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu BOUNDARY-VALUE PROBLEMS
More informationStability properties of positive solutions to partial differential equations with delay
Electronic Journal of Differential Equations, Vol. 2001(2001), No. 64, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Stability properties
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationThe Journal of Nonlinear Science and Applications
J Nonlinear Sci Appl 3 21, no 4, 245 255 The Journal of Nonlinear Science and Applications http://wwwtjnsacom GLOBAL EXISTENCE AND L ESTIMATES OF SOLUTIONS FOR A QUASILINEAR PARABOLIC SYSTEM JUN ZHOU Abstract
More informationResearch Article On Behavior of Solution of Degenerated Hyperbolic Equation
International Scholarly Research Network ISRN Applied Mathematics Volume 2012, Article ID 124936, 10 pages doi:10.5402/2012/124936 Research Article On Behavior of Solution of Degenerated Hyperbolic Equation
More informationPERMANENCE IN LOGISTIC AND LOTKA-VOLTERRA SYSTEMS WITH DISPERSAL AND TIME DELAYS
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 60, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) PERMANENCE
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationA PATTERN FORMATION PROBLEM ON THE SPHERE. 1. Introduction
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 007), pp. 97 06. ISSN: 07-669. URL: http://ejde.math.txstate.edu
More informationPOSITIVE GROUND STATE SOLUTIONS FOR SOME NON-AUTONOMOUS KIRCHHOFF TYPE PROBLEMS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 1, 2017 POSITIVE GROUND STATE SOLUTIONS FOR SOME NON-AUTONOMOUS KIRCHHOFF TYPE PROBLEMS QILIN XIE AND SHIWANG MA ABSTRACT. In this paper, we study
More informationSYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP
lectronic ournal of Differential quations, Vol. 2013 2013, No. 108, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMTRY AND RGULARITY
More informationHYERS-ULAM STABILITY FOR SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 011 (011), No. 80, pp. 1 5. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HYERS-ULAM STABILITY
More informationWEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 94, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu WEAK ASYMPTOTIC
More informationON THE OSCILLATION OF THE SOLUTIONS TO LINEAR DIFFERENCE EQUATIONS WITH VARIABLE DELAY
Electronic Journal of Differential Equations, Vol. 008(008, No. 50, pp. 1 15. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp ON THE
More informationUniqueness of ground states for quasilinear elliptic equations in the exponential case
Uniqueness of ground states for quasilinear elliptic equations in the exponential case Patrizia Pucci & James Serrin We consider ground states of the quasilinear equation (.) div(a( Du )Du) + f(u) = 0
More informationDECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME
Electronic Journal of Differential Equations, Vol. 218 218), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION
More informationONE-DIMENSIONAL PARABOLIC p LAPLACIAN EQUATION. Youngsang Ko. 1. Introduction. We consider the Cauchy problem of the form (1.1) u t = ( u x p 2 u x
Kangweon-Kyungki Math. Jour. 7 (999), No. 2, pp. 39 50 ONE-DIMENSIONAL PARABOLIC p LAPLACIAN EQUATION Youngsang Ko Abstract. In this paper we establish some bounds for solutions of parabolic one dimensional
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
More informationThe Singular Diffusion Equation with Boundary Degeneracy
The Singular Diffusion Equation with Boundary Degeneracy QINGMEI XIE Jimei University School of Sciences Xiamen, 36 China xieqingmei@63com HUASHUI ZHAN Jimei University School of Sciences Xiamen, 36 China
More informationFinite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation. CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO
Finite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation JOSÉ ALFREDO LÓPEZ-MIMBELA CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO jalfredo@cimat.mx Introduction and backgrownd
More informationBulletin of the. Iranian Mathematical Society
ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 42 (2016), No. 1, pp. 129 141. Title: On nonlocal elliptic system of p-kirchhoff-type in Author(s): L.
More informationGROWTH OF SOLUTIONS TO HIGHER ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS IN ANGULAR DOMAINS
Electronic Journal of Differential Equations, Vol 200(200), No 64, pp 7 ISSN: 072-669 URL: http://ejdemathtxstateedu or http://ejdemathuntedu ftp ejdemathtxstateedu GROWTH OF SOLUTIONS TO HIGHER ORDER
More informationMemoirs on Differential Equations and Mathematical Physics
Memoirs on Differential Equations and Mathematical Physics Volume 51, 010, 93 108 Said Kouachi and Belgacem Rebiai INVARIANT REGIONS AND THE GLOBAL EXISTENCE FOR REACTION-DIFFUSION SYSTEMS WITH A TRIDIAGONAL
More information