GLOBAL EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 4, April 7, Pages 7 7 S ) Article electronically published on September 8, 6 GLOBAL EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM MEI ZHANG AND CHANGJIANG ZHU Communicated by Walter Craig) Abstract. In this paper, we investigate the global existence of solutions to a hyperbolic-parabolic model of chemotaxis arising in the theory of reinforced random walks. To get L -estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and aprioriassumptions.. Introduction and main result In this paper, we consider the following system: ut v.) x =, v t uv) x = v xx, with boundary conditions.) u,t)=u,t)=, t, and initial data.3) ux, ) = u x), vx, ) = v x) >, x [, ]. Here the compatibility conditions on u,v assume that u ),v )) =, ), which will be used in Section 3. Motivated by biological considerations and numerical computations carried out by Othmer and Stevens in [6] and Levine and Sleeman in [3], the system.) comes from:.4) p t = D x w = Rp, w), t p ln x )) p, x,l), t >, Φw) where px, t) is the particle density of a particular species, wx, t) is the concentration of the active agent, and D and B are positive constants. Received by the editors October 9, 5. Mathematics Subject Classification. Primary 35K, 35K55, 35L5. Key words and phrases. Hyperbolic-parabolic system, aprioriestimates, entropy-entropy flux, global existence. 7 c 6 American Mathematical Society Reverts to public domain 8 years from publication

2 8 MEI ZHANG AND CHANGJIANG ZHU In fact, as in [7], let.5) Φw) =w α, Rp, w) =λpw µw, where α and λ, µ are positive constants. Then the system.4) is transformed into the following form: p t = Dp xx + Dα p w ) x,.6) w x w t = λpw µw. Furthermore, set.7) q =lnw) x = w x w. Then the system.6) can be rewritten as: pt = Dp xx + Dαpq) x,.8) q t = λp x. Let.9) τ = At, ξ = lx, p = Bv, q = c u, where A, l and c are positive constants to be determined below. Then the system.8) becomes u τ = λlb v ξ, Ac.) v τ = Dl A v ξξ + Dαlc A uv) ξ. Choosing λlb =, Ac Dl A =, Dαlc A =, i.e., Bαλ Bλ.) A = Bαλ >, l = D >, c = αd >, then it is easy to verify that u, v satisfy uτ = v.) ξ, v τ = v ξξ +uv) ξ. If we replace ξ,τ) byx, t), then.) can be rewritten as.). The system.4) describes the model of chemotaxis in biology. Othmer and Stevens [6] have developed a number of mathematical models of chemotaxis to illustrate aggregation leading numerically) to nonconstant steady-states, blow-up resulting in the formation of singularities and collapse or the formation of a spatially uniform steady state. The models developed in [6] have been studied in depth by Levine and Sleeman [3]. They gave some heuristic understanding of some of these phenomena and investigated the properties of solutions of a system of chemotaxis

3 EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM 9 equations arising in the theory of reinforced random walks. Y. Yang, H. Chen and W.A. Liu [9] studied the global existence and blow-up in a finite time of solutions for the case considered in [3], respectively, and found that even at the same growth rate the behavior of the biological systems can be very different just because they started their action under different conditions. For the other results on the initial boundary value problem of.4) refer to [, 4]. In this paper, we will study the global existence of solutions to the initial boundary value problem.)-.3) in L [, ), H [, ])). To get L -estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and aprioriassumptions. The corresponding hyperbolic system of.) is ut v.3) x =, v t uv) x =. The eigenvalues of.3) are:.4) λ = u u +4v, λ = u + u +4v. Therefore, the system.3) is strictly hyperbolic when v>. Remark.. By the boundary conditions.) and.),wehave.5) u t,t)=u t,t)=v x,t)=v x,t)=. Notation. Throughout this paper, we denote positive constants by C. Moreover, the character C may differ in different places. L p = L p [, ]) p ) denotes the usual Lebesgue space with the norm ) f L p = fx) p p dx, p<, f =sup fx). [,] H l [, ]) l ) denotes the usual lth-order Sobolev space with the norm l f l = xf j, j= where = = L. For simplicity, f,t) L p and f,t) l are denoted by ft) L p and ft) l respectively. We will prove the following global existence theorem. Theorem.. Let u,v H [, ]) and let u + v be sufficiently small. Then there exists a unique global solution ux, t),vx, t)) of.)-.3) satisfying i) u, v L [, ),H [, ])), v x L [, ),H [, ]));.6) ii) ut) + vt) + t v x τ) dτ C u + v ).

4 MEI ZHANG AND CHANGJIANG ZHU. L -energy estimates In this section, we give L -energy estimates of the initial boundary value problem.),.) and.3) by a nonnegative convex entropy of the system of hyperbolic conservation laws.3). To do this, we first give the following relation on the entropy-entropy flux pair ηu, v),qu, v)) of.3) see [8]): qu = vη.) v, q v = η u uη v. Eliminating q from.), we have.) η uu + uη uv vη vv =. Next, we seek the entropy of.3) with the following form:.3) ηu, v) = u + av), where av) is a nonnegative convex function. Substituting.3) into.), we have va v) =, which implies.4) av) =vln v v + k v + k, where k,k are arbitrary constants. It is easy to get the flux corresponding to the entropy ηu, v) defined by.3) and.4), namely.5) qu, v) =uv ln v k uv + k 3, where k 3 is an arbitrary constant. In particular, if we take k = k 3 =, k =, we will get an entropy-entropy flux pair of.3): ηu, v) =.6) u + v ln v v +, qu, v) =uv ln v. In the next analysis, we devote ourselves to the estimates of the solution ux, t), vx, t)) of.),.) and.3) under the aprioriassumptions:.7) u ε, v, u x ε, v x ε, where <ε<<. Lemma.. The entropy ηu, v) defined by.6) satisfies for v,.8) u + 3 v ) ηu, v) u +v ). Proof. Let.9) a v) =v ln v v +. Then a ) = a ) =, 3 a v) = v,

5 EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM which implies.) 3 v ) a v) = a ξ)v ) v ), where ξ is between and v. From.6),.9) and.),.8) follows. This proves Lemma.. Lemma. L -energy estimates). Under the assumptions of Theorem. and the a priori assumptions.7), we have.) u + ) v ) dx + t ) v 3 3 xdxdτ u x)+v x) ) dx. Proof. Multiplying.) by η and integrating it, we have by the boundary conditions.) and.5), t v x.) ηx, t)dx + v dxds = ηx, )dx, i.e.,.3) ) t v ) u x + v ln v v + dx+ v dxds = u + v ln v v + dx. This proves Lemma. by Lemma. and the aprioriassumptions.7). 3. Higher energy estimates In this section, we will establish higher energy estimates. Lemma 3.. Under the assumptions of Theorem. and the a priori assumptions.7), we have 3.) u x + ) 3 v x dx + 4 t v 3 xxdxds C u + v ), where C is a positive constant. Proof. Differentiating.) with respect to x, wehave uxt v 3.) xx =, v xt uv) xx = v xxx. For any smooth function g v), take 3.) u x +3.) g v)v x, and integrate it to get d u dt xdx + d g v)v dt xdx 3.3) = + u x v xx dx + g v)u x v xdx + g v)v xv t dx + g v)u xx vv x dx + g v)uv x v xx dx g v)v x v xxx dx.

6 MEI ZHANG AND CHANGJIANG ZHU Next, we estimate the terms in the right side of 3.3) as follows: 3.4) 3.5) 3.6) and 3.7) g v)v xv t dx = g v)uv x v xx dx = = = 6 g v)vv x u xx dx = g v)v x v xxx dx = + g v)v xuv) x dx + g v)vxdx 4 + g 6 v)vx) 3 g v)uv 3 xdx + g v)u) x v xdx + g v)u x v xdx = = 3 3 g v)uv x) g v)v xv xx dx g v)u x vv xdx, g v)uv 3 xdx + g v)vv x ) x u x dx +g v)vv x u x ) g v)vv xu x dx g v)v xu x dx g v)vv xx u x dx +g v)vv x u x ), g v)v x ) x v xx dx +g v)v x v xx ) g v)v 4 xdx g v)v xxdx g v)v 3 x) +g v)v x v xx ). g v)uv x), Substituting 3.4)-3.7) into 3.3) and using the boundary conditions.) and.5), we obtain 3.8) d dt = 6 Choosing u x + g v)vx) dx + g v)vxxdx g v)v 4 xdx g v)v )u x v xx dx 3.9) g v) = v >, g v)v g v))u x v xdx. we have 3.) d dt ) u x + v x vxx dx + v v dx = v 4 x 3 v 3 dx + u x vx v dx.

7 EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM 3 From the aprioriassumptions.7), we have d ) u x + v x vxx dx + dt v v dx 3.) Integrating 3.) in t over [,t], we can obtain ) t u x + v x vxx dx + v v dxds 3.) u x + v x v ) dx ε = 4 3 ε + ε vx 4 3 ε + ε ) t v dx + ε ) which implies 3.) by.) and the aprioriassumptions.7). v x v dx. v x v dxds, v x v dx The proof of Lemma 3. is completed. Lemma 3.. Under the assumptions of Theorem. and the a priori assumptions.7), we have 3.3) u xx + ) 3 v xx dx + t v 3 xxxdxds C u + v ), where C is a positive constant. Before proving Lemma 3., we give the following result. Proposition 3.3. The smooth function vx, t) obtained by Theorem. satisfies the following properties: 3.4) v v3 xxdx v v xxxdx + C v v xdx, where C is a positive constant. Proof. From the Gagliardo-Nirenberg-Moser inequality, we have 3.5) v xx t) 3 L C v xt) 3 3 L 6 v xxx t) 3 L, where C is a positive constant. 3.5) and Young s inequality show that 3.6) v xx t) 3 L 3 C v xt) 6 L v xxxt) L. Therefore, we have from the aprioriassumptions.7), v v3 xxdx C v x t) 6 L v xxxt) L C v x 4 v x L v dx + v v xxxdx v x 3.7) C v dx + v v xxxdx. This proves Proposition 3.3.

8 4 MEI ZHANG AND CHANGJIANG ZHU Proof of Lemma 3.. Differentiating 3.) with respect to x, wehave 3.8) uxxt v xxx =, v xxt uv) xxx = v xxxx. For any smooth function g v), taking 3.8) u xx +3.8) g v)v xx, and integrating it, we have 3.9) d u dt xx + g v)vxx) dx = u xx v xxx dx + g v)v tvxx dx + g v)v xx uv) xxx dx + Next, we estimate the terms in the right side of 3.9) as follows: g v)v xx v xxxx dx. 3.) 3.) g v)v t v xxdx = g v)v xx uv) xxx dx = = = + g v)[uv) x + v xx ]v xxdx g v)uv x v xxdx + g v)v 3 xxdx, g v)u x vv xxdx uv) xx g v)v xx ) x dx +uv) xx g v)v xx ) ug v)v xx v xxx dx vg v)u xx v xxx dx g v)u x v xv xx dx + g v)u x v x v xxx dx ug v)v x v xxdx vg v)v x u x v xx ) + ug v)v xx) vg v)v x v xx ) x u x dx + g v)u x v x v xx ) +vg v)u xx v xx ), and 3.) g v)v xx v xxxx dx = = g v)v xx ) x v xxx dx +g v)v xx v xxx ) g v)v xxxdx +g v)v xx v xxx ). g v)v x v xx v xxx dx

9 EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM 5 Substituting 3.)-3.) into 3.9), we obtain by the boundary conditions.) and.5), 3.3) d u dt xx + g v)vxx) dx + g v)vxxxdx = u xx v xxx dx g v)u x vv xxdx + g v)u x v x v xxx dx g v)v xv xx u x dx + vg v)v x v xxx u x dx vg v)u xx v xxx dx + +g v)v xx vu xx + v xxx )). If choosing g v) = v,thenweget u xx + v v xx = d dt u v v xv xxdx u v v xxv xxx dx 3 g v)v 3 xxdx ug v)v x v xxdx vg v)v xv xx u x dx + g v)uv x v xxdx g v)v x v xx v xxx dx ) dx + v v xxxdx v u xvxxdx v u xv x v xxx dx +3 ug v)v xx v xxx dx g v)u x v xv xx dx vg v)v xxu x dx v v3 xxdx v v xu x v xx dx vxx v vu xx + v xxx ) 3.4) v u xvxxdx + v v xv xx v xxx dx + From 3.) and the boundary conditions.) and.5), we have Thus 3.5) v xxx = uv) xx ) = uv xx ) u x v x ) u xx v) = u xx v). vxx ) v vu xx + v xxx ) =. By 3.4), 3.5), 3.4) and using the Cauchy-Schwarz inequality, we have d u xx + ) dt v v xx dx + v v xxxdx ). 3.6) 3 4 v v xxxdx + C v v x + vxx)dx. Integrating 3.6) in t over [,t] and using Lemmas. and 3., and the apriori assumptions.7), we get 3.3). This proves Lemma 3..

10 6 MEI ZHANG AND CHANGJIANG ZHU 4. The proof of Theorem. The global existence in Theorem. follows from a local existence theorem see [, 5]) and the aprioriestimate.6) obtained by.), 3.) and 3.3). Now, we have to show that the aprioriassumptions.7) can be closed since, under the aprioriassumptions.7), we have proved that.6) holds. In fact, by Sobolev s embedding theorem W, [, ]) L [, ]) and Hölder s inequality, we have vx, t) C vx, t) dx + C vx, t) ) x dx ) C vx, t) ) ) dx + C vxx, t)dx C u + v ), which implies 4.) vx, t) C u + v ). Similarly, we have 4.) ux, t), u x x, t), v x x, t) C u + v ). By 4.) and 4.), it is easy to see that the aprioriassumptions.7) hold provided u + v is sufficiently small. Therefore, the aprioriassumptions.7) arealwaystrueprovided u + v is sufficiently small. Acknowledgement The research was supported by the Program for New Century Excellent Talents in University #NCET-4-745, the Key Project of the National Natural Science Foundation of China #436. Special thanks go to the anonymous referee for his/her helpful comments on the draft version of this manuscript. References [] T. Hillen, A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile, Math. Methods Appl. Sci., 74), MR8797 5e:35) [] S. Kato, On local and global existence theorems for a nonautonomous differential equation in a Banach space, Funkcial. Ekvac., 9976), MR :39) [3] H.A. Levine, B.D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57997), MR g:356) [4] H.A. Levine, B.D. Sleeman, M.N. Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 4), MR8885 3b:93) [5] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math., 978. MR :69) [6] H.G. Othmer, A. Stevens, Aggregation, blowup, and collapse: the ABC s of taxis in reinforced random walks, SIAM J. Appl. Math., 57997), MR465 99b:9) [7] B.D. Sleeman, H.A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Methods. Appl. Sci., 4), MR8934 c:99)

11 EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM 7 [8] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York- Berlin, 983. MR d:35) [9] Y. Yang, H. Chen, W.A. Liu, On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis, SIAM J. Math. Anal., 33), MR8847 3d:3559) Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 4379, People s Republic of China Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 4379, People s Republic of China address: cjzhu@mail.ccnu.edu.cn

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