ON THE EXISTENCE AND NONEXISTENCE OF GLOBAL SIGN CHANGING SOLUTIONS ON RIEMANNIAN MANIFOLDS

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1 Nonlinear Functional Analysis and Applications Vol. 2, No. 2 (25), pp Copyright c 25 Kyungnam University Press KUPress ON THE EXISTENCE AND NONEXISTENCE OF GLOBAL SIGN CHANGING SOLUTIONS ON IEMANNIAN MANIFOLDS Smail Chemikh, Makkia Dammak 2 and Ibrahim Dridi 3 Department of Mathematics University of Sciences and Technology of Houari Boumedien, Alger, Algeria sm.chemikh@gmail.com 2 Department of Mathematics, ISTMT University of Tunis El Manar, Tunis, Tunisia makkia.dammak@gmail.com 3 Department of Mathematics, College of Preparatory Year Umm Alqura University, Makkah, Saudi Arabia Kingdom dridibr@gmail.com Abstract. In this paper, we study the global existence and nonexistence of solutions to the following nonlinear reaction-diffusion problem ut u = u p in M n (, ), u(x, ) = u (x) in M n, where M n, n 3, is a non-compact complete iemannian manifold, is the Laplace- Beltrami operator and p >.We assume that u (x) is smooth, bounded function satisfying M n u (x)dx >. We prove that there is an exponent p which is critical in the following sense: when p (, p ), the above problem has no global solution and when p (p, ), the problem has a global solution for some u (x).. Introduction and statement of the results In this paper, we study the global existence and nonexistence of solutions to the following semilinear parabolic equation eceived November 22, 24. Accepted February 9, Mathematics Subject Classification: 35K55, 35J65, 58G. Keywords: Critical exponent of Fujita type, iemannian manifold, sign changing solution, global solution.

2 29 S. Chemikh, M. Dammak and I. Dridi ut u = u p in M n (, ), u(x, ) = u (x) in M n, (.) where (M n, g), n 3, is a non-compact complete iemannian manifold, is the Laplace-Beltrami operator, u are smooth functions and p >. The study of existence or nonexistence of global positive solutions to homogeneous semilinear parabolic equations and systems have been studied widely since the result of Fujita [3] in 966 who proved the following Theorem. Theorem.. Consider the Cauchy problem ut u = u p in n (, ), u(x, ) = u (x) in n, where p >, n and the standard Laplace operator. conclusions hold. (.2) The following (a) When < p < + 2 n, and u, problem (.2) possesses no global solutions. (b) When p > + 2 n and u smaller then a small Gaussian, then (.2) has global positive solutions. We call p = + 2 the critical exponent of Fujita type for the semilinear n heat equation (.2). These equations are model cases of many equations applied to wide scope and we can find an account of the historic development in the survey papers [2, 6]. After that, Zhang [2] has undertaken the research to prove that Fujita s results on critical exponents can be extended to solutions that may change sign. He considered the problem ut u = u p in n (, ), u(x, ) = u (x) in n (.3). He proved the following result. Theorem.2. Let p >, n and the standard Laplacien. The following results hold (a) If < p < + 2 n, and n u (x) dx >, then (.3) has no global solutions. (b) If p = + 2 n and n u dx >, then (.3) has no global solutions provided that u is compactly supported.

3 On the existence and nonexistence of global sign changing solutions 29 (c) If p > + 2 n, then (.3) has global solutions for some u. Also, Zhang [] has undertaken the research on semilinear parabolic operators on iemannian manifold and obtains a lot of important results in the study of the positive global existence and nonexistence (see [] and []). The method he uses is rather technical, and the main tools are fixed point theorems and many estimates. As an expansion of [9,, ], we study our problem (.) and obtain several meaningful results. Throughout the paper, for a fixed x M n, we make the following assumptions (see [, ]). (i) There are positive constants α > 2, k and c such that: icci k and for all x M n, c r α B r (x) cr α, when r is large. (ii) G(x, y, t) is the fundamental solution of the linear operator t, and satisfies c d(x,y) 2 B(x, t 2 ) e b t G(x, y, t), in M n (, ) and when t s d(x, y) 2, b is a positive constant, the fundamental solution G(x, y, t s) satisfies } c G(x, y, t s) min B(x, (t s) 2 ), c. B(y, (t s) 2 ) (iii) log g 2 r c r, when r = d(x, x ) is smooth, here g 2 is the volume of density of the manifold. Since the above assumptions are satisfied, the following lemmas hold. Lemma.3. ([]) There exists a constant c >, depending only on n, α and δ >, such that sup x M n d(x, y) α 2 [ + d(y, x ) 2+δ ] dy c. Lemma.4. ([]) There exists a constant c 2 >, depending only on n, α and δ >, such that M n d(x, y) α 2 [ + d(y, x ) α+δ ] dy c 2 + d(x, x ) α 2.

4 292 S. Chemikh, M. Dammak and I. Dridi Lemma.5. ([]) Let Γ(x, y) the Green s function for the Laplacien, then there exists a constant c 3 > such that Γ(x, y) M n + d(x, y) α+δ dy c 3 + d(x, x ) α 2 Lemma.6. ([]) Given δ >, there exists a constant c 4 > such that G(x, y, t) h(x, t) = M n + d(y, x ) α+δ dy c 4 + d(x, x ) α Since we are dealing with sign changing solutions on a iemannian manifold, we take the following definition. Definition.7. u(x, t) L loc (M n (, ), ) is called a solution of (.) if t u(x, t) = G(x, y, t)u (y)dy + G(x, y, t s) u(y, s) p dyds. M n M n Definition.8. On a complete iemannian manifold, we define Green s function Γ(x, y) = G(x, y, s)ds, if the integral on the right hand side converges. We have t G(x, y, t s)ds = t Our results are as follows. Γ(x, y) >, Γ = δ x y, G(x, y, w)dw G(x, y, w)dw = Γ(x, y). (.4) Theorem.9. If < p < + 2 α and M n u (y)dy >, then (.) has no global solutions. Theorem.. If p > + 2 α M and u (y)dy >, then (.) has global ɛ n solutions, whenever u (x) < + d(x, x ) α for some δ > and some + δ sufficiently small ɛ >.

5 On the existence and nonexistence of global sign changing solutions 293 emark.. By Theorems (.9) (.), it is easy to see that p = + 2 α is the critical exponent of Fujita type for the semilinear parabolic equation (.). 2. Nonexistence of global solutions In this section, we prove the Theorem.9 and c is always a constant that may be change for line to line. Proof of Theorem.9. Step. Let ω(x, t) = G(x, y, t)u (y)dy, (2.) M n where G is the fundamental solution to the heat equation u u t, we will prove in this step that ω(x, t) c (2.2) when (x, t) = B () [4 2, 8 2 ] for a sufficiently large. Here c can be chosen independently of. Choosing > to be determined later, we have ω(x, t) = G(x, y, t)u (y)dy + G(x, y, t)u (y)dy and hence ω(x, t) B () B () B C () t α 2 G(x, y, t)u + (y)dy G(x, y, t)u (y)dy. B C () B () G(x, y, t)u (y)dy Suppose (x, t), by assumption (i) and (ii) we obtain for >, ω(x, t) c u + t α (y)dy c d(x,y)2 b e 2 t α t u (y)dy 2 Since b >, we have c t α 2 B () ω(x, t) c t α 2 B C () e B () d(x,y)2 b t B () u (y)dy. u (y)dy c t α 2 B C () u (y)dy.

6 294 S. Chemikh, M. Dammak and I. Dridi By our assumption on u, there exists a δ > such that u (y)dy δ, B () when is sufficiently large. For >, we have ω(x, t) cδ c u t α 2 t α (y)dy. (2.3) 2 B C () Since u L (M n ) and the constants in the inequality (2.3) are independent of, we can take sufficiently large so that for (x, t), ω(x, t) c δ (2.4) t α 2 Step 2. We will use the methode of contradiction. Suppose that u is a global solution to (.). By Definition.7 and the inequality (2.4), we have u(x, t) ω(x, t) > for (x, t). (2.5) Let φ, η C be tow functions satisfying φ(r) [, ], if r [, + ), φ(r) =, if r [, 2], φ(r) =, if r [, + ), η(t) [, ], if t [, + ), η(t) =, if t [, 4], η(t) =, if t [, + ), C φ (r) ; φ (r) C; η (t) C. For >, we define a cut-off function Ψ (x, t) = φ (x)η (t), where φ (x) = φ( x ) and η (t) = η( t 42 ). Clearly φ 4 2 is radial function and we have C φ r, φ 2 r 2 C 2, η (t) C 2. (2.6) For >, we set I = u p (x, t)ψ q (x, t)dxdt, (2.7) where p + q =. Since u is a solution of (.), we have I = [u t (x, t) u(x, t)] Ψ q (x, t)dxdt, (2.8)

7 On the existence and nonexistence of global sign changing solutions 295 which implies, via integration by parts (Stokes Formula) I u(x,.)ψ q (x,.) 82 4 dx u(x, t)φ q 2 (x)qηq (t)η (t)dxdt B () u(x, t) φq (x) 4 2 B () n ηq (t) ds xdt 8 2 (2.9) Ψ q (x, t) u 4 2 B () n (x, t)ds xdt u(x, t) φ q (x)ηq (t)dxdt. Here and later n denotes the outward normal derivative with respect to the relevant bounded domain. Using the boundary conditions on the cut-off function and (2.9), and since we obtain Because φ q φ q n = qφq φ ( r n ) on B (), I q u(x, t)φ q (x)ηq (t)η (t)dxdt u(x, t) φ q (x)ηq (t)dxdt. (x) = qφq (x) φ (x) + q(q )φ q 2 (x) φ (x) 2, (2.) the inequality (2.) yields I q u(x, t)φ q (x)ηq (t)η (t)dxdt q u(x, t)φ q (x) φ (x)η q (t)dxdt. (2.) ecalling of the supports of φ (x) and η (t) that is η (t) =, η (t) =, if t [42, 7 2 ], (2.2) φ (x) =, φ (x) =, if r = x [, 2 ],

8 296 S. Chemikh, M. Dammak and I. Dridi we can reduce (2.) to 8 2 I q q B () u(x, t)φ q B ()\B 2 () Since φ is radial, we have φ = φ + (x, t)ηq (t)η (t)dtdx u(x, t)φ q (x, t) φ (x)η q (t)dxdt. (2.3) [ ] n + log g 2 φ r r. Taking sufficiently large, by assumption (iii), that is log g 2 r C, we obtain, when x B () \ B () 2 Merging (2.3), (2.4) and (2.6), we know I c c B () φ (x) c 2. (2.4) u(x, t)φ q B ()\B 2 () Therefore, as φ and η, I c c B () (x, t)ηq (t)dxdt u(x, t)φ q (x, t)ηq (t)dxdt. (2.5) u(x, t)ψ q (x, t)dxdt B ()\B 2 () Using Hölder inequality, we can deduce from (2.6) that u(x, t)ψ q (x, t)dxdt. (2.6)

9 On the existence and nonexistence of global sign changing solutions 297 I c [ B () + c u p (x, t)ψ p(q ) (x, t)dxdt B ()\B 2 () B ()\B 2 () dxdt u p (x, t)ψ p(q ) q. ] p [ (x, t)dxdt p B () dxdt ] q Therefore, as p + q =, we have [ 8 2 I c + c B () u p (x, t)ψ q (x, t)dxdt ] B ()\B 2 () p 2+α q 2 u p (x, t)ψ q (x, t)dxdt p 2+α q 2, which yields p I c I 2+α q 2. (2.7) Step 3. (Conclusion) From (2.7), we obtain q I c 2+α q and so q I c (2+α) 2q c k. (2.8) On the other hand, from (2.4), (2.5), in Step and (2.2), we have for sufficiently large, I u(x, t) p dxdt ω(x, t) p dxdt c α+2 αp then 7 2 B () B () I c k 2. (2.9) Since p < + 2 α, so k 2 >. Also p < + 2 α+2 α, then q > 2. Therefore k <. If the solution u is global then (2.8) and (2.9) lead to a contradiction, when is large. So u cannot be global. Theorem.9 is proved.

10 298 S. Chemikh, M. Dammak and I. Dridi 3. Existence of global solutions In this section, we prove the Theorem.. Proof of Theorem.. As in [7], we define the Green s function Γ(x, y) = + We have Γ(x, y) >, Γ = δ x (y) and t G(x, y, t s)ds = t G(x, y, ω)dω < G(x, y, s)ds. G(x, y, ω)dω = Γ(x, y). (3.) Let Γ the integral operator defined by t Γu(x, t) = G(x, y, t)u (y)dy + G(x, y, t s) u p (y, s)dyds, (3.2) M n M n for u(x, t) L loc (M n [, + ), ). For N (, ), we denote H N = u(x, t) C(M n (, + ), )/ u(x, t) N + d(x, x ) α }. (3.3) Next, we show that the operator Γ has a fixed point in H N, for some N. For ε > > and δ > to be chosen later, select u satisfying u (x) N. (3.4) + d(x, x ) α+δ By Lemma.6 G(x, y, t)u (y)dy ε G(x, y, t) M n M n + d(x, x ) α+δ dy εc 4 + d(x, x ) α. (3.5) By assumption (3.3), we have for u(x, t) H N, t G(x, y, t s) u(y, s) p dyds M n t N p G(x, y, t s) M n ( + d(y, x ) α ) p dyds. Since p > + 2 α, we can find c 5 > and δ > such that (3.6) ( + d(y, x ) α ) p N p c 5 (3.7) + d(y, x ) α+2+δ

11 On the existence and nonexistence of global sign changing solutions 299 Substituting (3.7) on the right hand side of (3.6) and by Lemma.5 t G(x, y, t s) u(y, s) p dyds M n N p c 5 Γ(x, y) dy M n + d(y, x ) α+2+δ N p c 3 c 5 d(x, y) α, when ε and N are sufficiently small, we have Γu(x, y) (3.8) N + d(x, x ) α (3.9) This shows that ΓH N H N. To obtain the global existence of solution to (.), we check that Γ is contractive operator. Let u i (x, t) H N, i =, 2, then noticing that a b a b and } u p (x, t) up 2 (x, t) p u (x, t) u 2 (x, t) max u p (x, t), u p 2 (x, t). (3.) We have Γu (x, t) Γu 2 (x, t) t G(x, y, t s) u p (y, s) up 2 (y, s) dyds M n t N p G(x, y, t s)( M n + d(y, x ) α )p u p (y, s) up 2 (y, s) dyds. Denoting. = Γu (x, t) Γu 2 (x, t) pn p u u 2 max., we have x M n,t> t (3.) G(x, y, t s) M n [ + d(y, x ) α ] p dyds. (3.2) Since p > + 2 α, we can select δ > such that (p )α 2 + δ and then, there exist a constant c 6 > such that [ + d(y, x ) α ]) p c 6. (3.3) + d(y, x ) 2+δ From (3.2), (3.3) and (3.), we obtain Γu (x, t) Γu 2 (x, t) c 6 pn p u u 2 Γ(x, y) M n + d(y, x ) 2+δ dy. (3.4)

12 3 S. Chemikh, M. Dammak and I. Dridi By Li and Yau in [7], there is a positive constant λ > such that Γ(x, y) +d(x,y) α 2, when d(x, y) > λ, then we have for some constant c 7 > Γu (x, t) Γu 2 (x, t) pc 6 c 7 N p u u 2 M n d(x, y) α 2 [ + d(y, x ) 2+δ ] dy pc 6 c 7 c N p u u 2, by Lemma.3. If N is small enough so that pc c 6 c 7 N p < and δ good chosen, then Γ is contractive. Hence problem (.) has a global solution and the proof of Theorem. is completed. Acknowledgments: The authors thank Professor Tej eddine Ghool who brought the writer s attention to the present subject and encouraged them through valuable discussions. eferences [] D.G. Aronson and F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 3 (987), [2] K. Deng and H.A. Levine, The role of critical exponents in blow up theorems, J. Math. Anal. Appl., 243 (2), [3] H. Fujita, On the blow up of solutions of the Cauchy problem for u t = u + u +α, J. Fac. Sci. Univ., Tokyo, Sect., I(3) (966), [4] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differntiel equations, Proc. Japan Acad., 49 (973), [5] K. Kobayashi, T. Sirao and H. Tanaka, On the growing problem for semilinear heat equations, J. Math. Soc. Japan, 29 (977), [6] H.A. Levine, The role of critical exponents in blowup theorems, SIAM ev., 32 (99), [7] P. Li and S.T. Yau, On The parabolic kernel of the schrödinger operator, Acta Math., 56 (986), [8] N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation, Math. Ann., 37 (997), [9] Q. u, A critical exponents of Fujita type for a nonlinear reaction-diffusion system on a iemannian manifold, J. Math. Anal. Appl., 4 (23), [] Q.S. Zhang, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal. Appl., 29 (998), [] Q.S. Zhang, Blow up results for nonlinear parabolic problems on manifolds, Duke Math. J., 97 (999), [2] Q.S. Zhang, A critical behavior for some semilinear parabolic equations involving sign changing solutions, Nonlinear Anal., 5 (22),