- Email: [email protected]
Do some chemotaxis-growth models possess Lyapunov functionals?
Accepted Manuscript Do some chemotaxis-growth models possess Lyapunov functionals? Dirk Horstmann PII: DOI: Reference:
S0893-9659(15)00295-5 http://dx.doi.org/10.1016/j.aml.2015.10.007 AML 4877
To appear in:
Applied Mathematics Letters
Received date: 14 September 2015 Revised date: 13 October 2015 Accepted date: 14 October 2015 Please cite this article as: D. Horstmann, Do some chemotaxis-growth models possess Lyapunov functionals?, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.10.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Do some Chemotaxis-Growth Models possess Lyapunov Functionals? Dirk Horstmann∗ Mathematisches Institut der Universit¨ at zu K¨ oln, Weyertal 86 -90, D-50931 K¨ oln, Germany
Abstract In spirit of a result by W. Alt from 1980 we give some sufficient criteria that guarantee the existence of Lyapunov functionals for parabolic cross-diffusion models including chemotaxis-growth models with non-diffusive chemotactic signals (resp. with non-diffusive memory). Keywords: Lyapunov functional, cross-diffusion systems, non-diffusive memory, chemotaxis, Othmer-Stevens model
1. Introduction Sufficient conditions that guarantee the existence of Lyapunov functionals for cross-diffusion models including chemotaxis models without growth are wellknown in literature (compare for instance [1, 3, 4, 6, 7, 8] and [11]). The reverse is true for the case of chemotaxis growth models. Besides the following result by W. Alt from 1980 there seem to be no case known, where a chemotaxis-growth model processes a Lyapunov functional. 1,1 Theorem 1.1 (compare [1] or [7]). Let p ≥ 0 be in Cloc (IRM ) and let M be M defined as the set M := {z ∈ IR | p(z) = 0} = 6 ∅. Furthermore let us assume that u is a weak solution of the parabolic problem N P N ∂ ∂ ut = A(u) u + F (u), in Ω × (0, τ ) (for Ω ⊂ IR ) ∂xj ∂xi j=1 = ψ(u), on ∂ΩN × [0, τ ], u(x, 0) = u0 (x), on ∂ΩD × [0, τ ], A(u) ∂u ∂n (1) where ∂ΩN and ∂ΩD denote disjunct subsets of ∂Ω. For the boundary conditions we assume that there exists a continuous family of symmetric M × M -matrices A∗ (z) ≥ 0, z ∈ IRM and that there exist vector functions θj : ∂ΩN × IRM → IR such that ψj (x, z) = A∗ (z)θj (x, z) and θjT (x, z) · A∗ (z) · θj (x, z) ≤ constK for all ∗ Dirk
Horstmann Email address: [email protected] (Dirk Horstmann)
Preprint submitted to Applied Mathematics Letters
October 13, 2015
z in a compact subset K of IRM and x ∈ ∂ΩN , where ψ0 = ψ and ψj = ∂z∂ j ψ, for j = 1, ..., M . Furthermore we suppose that there exists a neighbourhood U of M such that ∇p · ψ ≤ 0, ∇p · F ≤ 0 as well as ∇2 p · A ≥ 0 holds on U \ M, if ∇2 p exists. Then u(·, 0) ⊂ M impliesRu(·, t) ⊂ M for all t ∈ [0, τ ]. The Dini-derivative of the functional E(t) := p(u(t, x))dx, t ∈ [0, τ ] satisfies for Ω
all weak solution of problem (1) which have values in U the inequality d E(t) ≤ dt
Z X N ∂ ∂ 2 − u ∇ p(u)A(u) u dx ∂xj ∂xj j=1 Ω Z Z + ∇p(u) · F (u)dx + ∇p(u) · ψ(u)dS Ω
∂Ω
almost everywhere in [0, τ ]. If for all z ∈ U \ M either ∇p · ψ < 0 or ∇p · F < 0 d E(t) < 0 as long as holds, then E is a Lyapunov functional for M, i.e. dt E(t) > 0. If for all z ∈ U \ M either ∇p · ψ < 0 or ∇p · F < 0 or ∇2 p · A < 0 holds, then E is a Lyapunov functional for M ∪ N0 , where N0 contains all constants z0 ∈ IRM , that are zeros of ∇p · ψ and ∇p · F . We refer to [1, Beispiel 1.41, page 38] and [7, page 48 f.] for some applications of Theorem 1.1 and concrete examples. However, chemotaxis models with nondiffusive memory do not belong to the class of systems that W. Alt analyzed in [1]. But there are some situations where Lyapunov functionals (in the same spirit as presented in Theorem 1.1 but without the requirement that the functional has always to be positive) also exist for such kind of systems. Lyapunov functionals are useful tools for analyzing the time asymtotic behavior of the solutions and can help to prove the convergence of the solution to some steady state solution or the blow-up of the solution in either finite or in infinite time. However, the fact that there are Lyapunov-functions for chemotaxis-growth models if the chemotactic signal (resp. the “memory”) is non-diffusive has (to my knowledge) not been recognized in literature. Therefore, the short note raises the question phrased in the title and gives the following positive answer: Do chemotaxisgrowth models possess Lyapunov functionals? Yes, some and especially those with non-diffusive memory. 2. Chemotaxis-growth models with non-diffusive memory For notational reasons we set Ω × (0, T ) = QT and ∂Ω × (0, T ) = ΓT . In the following we will focus on systems of the following type: (ui )t = ∇(∇k1i (ui ) − k2i (ui )∇χi (vj )) + fi (u1 , ..., ul , vj ), (x, t) ∈ QT (vj )t = gj (u1 , ..., ul , vj ), (x, t) ∈ QT ∂vj 0 0 i 0 = k1i (ui ) ∂u (x, t) ∈ ΓT ∂ν − k2i (ui )χi (vj ) ∂ν , ui (0, x) = (ui )0 (x), vj (0, x) = (vj )0 (x), x∈Ω (2) 2
where i ∈ {1, ..., l}, l ≥ 1 and j ∈ {1, ..., r}, r ≥ 1. In particular there are l different chemotactic sensitivity χ0i and r different chemotactic signals vj . As a consequence there are r set Nq of index-pairs (i, j) with ]Nq = lq such that ui r r P P ]Nq = lq . reacts in a chemotactic way to vj and l = q=1
q=1
Models of this kind for single and several species have been studied for instance in [2, 3, 5, 9, 10] and [12]. Here we are not concerned about the question under which conditions this problem is well-posed and whether there are some conditions that guarantee global in time solutions as it has been done in the just mentioned papers. We want to focus on the questions whether this kind of models have Lyapunov functionals or not. Therefore, we give some sufficient conditions for the existence in the upcoming Lemma. Lemma 2.1. Let (u1 , ..., ul , v1 , ..., vr ) be a (weak) solution of system (2). Let us assume that χi ∈ C 1 (IR) for all i ∈ {1, ..., l} satisfies χ0i (v) > 0 for all v ∈ IR+ . Furthermore, we assume that there are l functions kl ∈ C 2 (IR) satisfying 0 ki00 (u) > 0, k2i (u)/ki0 (u) > 0 and ki00 (u) = k1i (u)ki0 (u)/k2i (u) for all u ∈ IR+ . If 0 0 ki (ui )fi (u1 , ..., ul , vj ) ≤ gj (u1 , ..., ul , vj )χi (vj )ki (ui ) for (i, j) ∈ Nq for some q ∈ D ⊂ {1, ..r} resp. ki0 (ui )fi (u1 , ..., ul , vj ) < 0 in the special situation that Pfor some j ∈ {1, ..., r} the function gj is given by gj (u1 , ..., ul , vj ) = γj (vj )+ ki (ui ) (i,j)∈Nq
for (i, j) ∈ Nq and χi (vj ) = χy (vj ) for all i 6= y s.t. (i, j), (y, j) ∈ Nq with q ∈ R := {1, ..., r} \ D, then E(t) := E1 (t) + E2 (t) with Z X X E1 (t) := ki (ui )e−χi (vj ) dx and Ω q∈D (i,j)∈Nq
E2 (t) :=
Z
Ω
X
X
q∈R (i,j)∈Nq
vj Z −χ0i (s)γj (s)e−χi (s) ds + ki (ui )e−χi (vj ) dx.
is a Lyapunov functional for the given system (2), i.e.
d dt Ep (t)
< 0 for p = 1, 2.
Remark 2.1. We point out that if (i, j) ∈ Nq with q ∈ R, then all ki ≡ 0 for those i for which (i, j) 6∈ Nq and all χi (vj ) = χ(vj ) for those i for which (i, j) ∈ Nq , where χ ∈ C 1 (IR) with χ0 (v) > 0 for all v ∈ IR+ . Proof: We see, that: Z X X d E1 (t) = − (vj )t χ0i (vj )ki (ui )e−χi (vj ) dx dt Ω q∈D (i,j)∈Nq Z X X + (ui )t ki0 (ui )e−χi (vj ) dx Ω q∈D (i,j)∈Nq
=
−
Z X
X
Ω q∈D (i,j)∈Nq
k2i (ui )e−χi (vj ) 00 2 |ki (ui )∇ui − ki0 (ui )∇χi (vj )| dx ki0 (ui ) 3
+
Z X
X
ki0 (ui )fi (u1 , ..., ul , vj )e−χi (vj ) dx
Ω q∈D (i,j)∈Nq
−
Z X
X
gj (u1 , ..., ul , vj )χ0i (vj )ki (ui )e−χi (vj ) dx < 0
Ω q∈D (i,j)∈Nq
and for E2 d E2 (t) dt
= − +
Z X
X
Ω q∈R (i,j)∈Nq
Z X
X
Ω q∈R (i,j)∈Nq
= − − +
(vj )t χ0i (vj )gj (u1 , ..., ul , vj )e−χi (vj ) dx
Z X
X
Ω q∈R (i,j)∈Nq
Z X
X
Ω q∈R (i,j)∈Nq
Z X
X
(ui )t ki0 (ui )e−χi (vj ) dx 4 −χ(vj )/2 2 e dx χ0 (vj ) t
k2i (ui )e−χ(vj ) 00 2 |ki (ui )∇ui − ki0 (ui )∇χ(vj )| dx 0 ki (ui ) ki0 (ui )e−χ(vj ) fi (u1 , ..., ul , vj )dx < 0.
Ω q∈R (i,j)∈Nq
Remark 2.2. The given conditions are sufficient to guarantee the existence of Lypaunov functionals also in the absence of growth as we will see in Example 3.2. However, in the absence of growth one can also apply the results in [3, 6, 8] to find other conditions that ensure the existence of Lyapunov functionals for even a larger class of cross-diffusion systems. However, the results presented here are different from those known so far. Furthermore, we want to point out that the condition stated above do not require that the function p(u(t, x)) is positive, as it was necessary in Theorem 1.1 (see for instance Example 3.3). 3. Examples and Applications In this section we want to give four concrete examples for the result stated above. The first two examples will present Lyapunov functionals for systems, where one can find global existence results in the literature. The other two examples will present some models for which the well-posedness and the time asymptotic behavior of a solution has not been studied so far (as far as I know). However, these two examples will present some systems with either logistic or Fisher-KPP type growth where Lyapunov functionals exist. Example 3.1. In [2] the authors study the following model with reproduction
4
term:
u ut = ∇ (∇u − u∇ ln (α + βv)) + u a − b α+βv , (x, t) ∈ QT vt = µu − δv, (x, t) ∈ QT ∂ u 0 = ∂u ∂ν − u ∂ν ln α+βv , u(0, x) = u0 (x), v(0, x) = v0 (x),
(x, t) ∈ ΓT x ∈ Ω.
(3)
In [2, Theorem 1] the authors state that there is unique solution for a = 0 = δ and µβ + b > 0 that exists globally in time. In fact if µβ + 2b > 0, then there exists the Lyapunov functional Z Z u2 2 − ln(α+βv) dx ≥ 0 E(t) = u e dx = α + βv Ω
Ω
for the given system (3), if α > 0, β > 0. Thus we conclude that Z Z 2 2 |∇u − u∇ ln(α + βv)| (2b + µβ)u3 d E(t) = − dx − dx → 0 dt α + βv (α + βv)2 Ω
Ω
as t → ∞.
Example 3.2. In [12] the authors proved for 1 > β > 0 the global existence of a unique solution of the system ut = ∇ (∇u − u∇ ln (v)) and vt = uv β , (x, t) ∈ QT ∂ (4) 0 = ∂u (x, t) ∈ ΓT ∂ν − u ∂ν ln (v) , u(0, x) = u0 (x), v(0, x) = v0 (x), x ∈ Ω. R Here E(t) = u2 v −1 dx ≥ 0 is a Lyapunov functional for the given system. Ω
Now, we give two examples for models where one has either a logistic or a Fisher-KPP type source, but that have not been studied so far.
Example 3.3. Suppose there exists a solution to the following system: ut = ∇ (∇u − (u − 1)∇v) + um (1 − u), m ≥ 1 (x, t) ∈ QT vt = γ(v) + 21 u2 − u, (x, t) ∈ QT (5) ∂ 0 = ∂u (x, t) ∈ ΓT ∂ν − (u − 1) ∂ν ln (v) , u(0, x) = u0 (x), v(0, x) = v0 (x), x ∈ Ω. v R R Then E(t) = −γ(s)e−s ds + 21 u2 − u e−v dx is a Lyapunov functional. Ω
Example 3.4. Suppose there exists a solution to the system:
(x, t) ∈ QT (x, t) ∈ ΓT x ∈ Ω, (6) where 0 < Ra < 1. If γ · θ > 2(1 + a) then a Lyapunov functional for (6) is given by E(t) = u2 v −θ dx ≥ 0. ut = ∇ (∇u − θu∇ ln(v)) + u(u − a)(1 − u) and vt = γuv, ∂ 0 = ∂u ∂ν − θu ∂ν ln (v) , u(0, x) = u0 (x), v(0, x) = v0 (x),
Ω
5
References [1] W. Alt, Vergleichs¨ atze f¨ ur quasilineare elliptisch-parabolische Systems partieller Differentialgleichungen, Habilitation, Ruprecht-Karl-Universit¨ at Heidelberg, 1980. [2] X. Chen & W. Liu, Behaviout of the solutions to an Othmer-Stevens chemotaxis Model with Reproduction Term, Wuhan University Journal of Natural sciences 15 (2010), no. 4, pp.m 277–282. [3] L. Corrias, B. Perthame & H. Zaag, A chemotaxis model motivated by angiogenesis. C. R. Math. Acad. Sci. Paris 336 (2003), no. 2, pp. 141-146. [4] H. Gajewski & K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 195 (1998), pp. 77 – 114. [5] F. R. Guarguaglini & R. Natalini, Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Commun. Pure Appl. Anal. 6 (2007), no. 1, pp. 287-309. [6] D. Horstmann, Lyapunov functions and Lp -estimates for a class of reaction-diffusion systems, Colloquium Mathematicum 87 (2001), pp. 113 – 127. [7] D. Horstmann,From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV 105 (2003), 103–165. [8] D. Horstmann, Generalizing Keller-Segel: Lyapunov functionals, steady state analysis and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Non. Science 21 (2011), pp. 231–270. [9] M. Negreanu & J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant. J. Differential Equations 258 (2015), no. 5, pp. 1592-1617. [10] H. G. Othmer & A. Stevens, Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997), no4, pp. 1044–1081. [11] K. Post, A non-linear parabolic system modeling chemotaxis with sensitivity functions, Dissertation, Humboldt-Universit¨ at zu Berlin, Institut f¨ ur Mathematik 1999. [12] Y. Sugiyama, Y. Tsutsui & J. J. L. Vel´ azquez, Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl. 410 (2014), pp. 908–917.
6
S0893-9659(15)00295-5 http://dx.doi.org/10.1016/j.aml.2015.10.007 AML 4877
To appear in:
Applied Mathematics Letters
Received date: 14 September 2015 Revised date: 13 October 2015 Accepted date: 14 October 2015 Please cite this article as: D. Horstmann, Do some chemotaxis-growth models possess Lyapunov functionals?, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.10.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Do some Chemotaxis-Growth Models possess Lyapunov Functionals? Dirk Horstmann∗ Mathematisches Institut der Universit¨ at zu K¨ oln, Weyertal 86 -90, D-50931 K¨ oln, Germany
Abstract In spirit of a result by W. Alt from 1980 we give some sufficient criteria that guarantee the existence of Lyapunov functionals for parabolic cross-diffusion models including chemotaxis-growth models with non-diffusive chemotactic signals (resp. with non-diffusive memory). Keywords: Lyapunov functional, cross-diffusion systems, non-diffusive memory, chemotaxis, Othmer-Stevens model
1. Introduction Sufficient conditions that guarantee the existence of Lyapunov functionals for cross-diffusion models including chemotaxis models without growth are wellknown in literature (compare for instance [1, 3, 4, 6, 7, 8] and [11]). The reverse is true for the case of chemotaxis growth models. Besides the following result by W. Alt from 1980 there seem to be no case known, where a chemotaxis-growth model processes a Lyapunov functional. 1,1 Theorem 1.1 (compare [1] or [7]). Let p ≥ 0 be in Cloc (IRM ) and let M be M defined as the set M := {z ∈ IR | p(z) = 0} = 6 ∅. Furthermore let us assume that u is a weak solution of the parabolic problem N P N ∂ ∂ ut = A(u) u + F (u), in Ω × (0, τ ) (for Ω ⊂ IR ) ∂xj ∂xi j=1 = ψ(u), on ∂ΩN × [0, τ ], u(x, 0) = u0 (x), on ∂ΩD × [0, τ ], A(u) ∂u ∂n (1) where ∂ΩN and ∂ΩD denote disjunct subsets of ∂Ω. For the boundary conditions we assume that there exists a continuous family of symmetric M × M -matrices A∗ (z) ≥ 0, z ∈ IRM and that there exist vector functions θj : ∂ΩN × IRM → IR such that ψj (x, z) = A∗ (z)θj (x, z) and θjT (x, z) · A∗ (z) · θj (x, z) ≤ constK for all ∗ Dirk
Horstmann Email address: [email protected] (Dirk Horstmann)
Preprint submitted to Applied Mathematics Letters
October 13, 2015
z in a compact subset K of IRM and x ∈ ∂ΩN , where ψ0 = ψ and ψj = ∂z∂ j ψ, for j = 1, ..., M . Furthermore we suppose that there exists a neighbourhood U of M such that ∇p · ψ ≤ 0, ∇p · F ≤ 0 as well as ∇2 p · A ≥ 0 holds on U \ M, if ∇2 p exists. Then u(·, 0) ⊂ M impliesRu(·, t) ⊂ M for all t ∈ [0, τ ]. The Dini-derivative of the functional E(t) := p(u(t, x))dx, t ∈ [0, τ ] satisfies for Ω
all weak solution of problem (1) which have values in U the inequality d E(t) ≤ dt
Z X N ∂ ∂ 2 − u ∇ p(u)A(u) u dx ∂xj ∂xj j=1 Ω Z Z + ∇p(u) · F (u)dx + ∇p(u) · ψ(u)dS Ω
∂Ω
almost everywhere in [0, τ ]. If for all z ∈ U \ M either ∇p · ψ < 0 or ∇p · F < 0 d E(t) < 0 as long as holds, then E is a Lyapunov functional for M, i.e. dt E(t) > 0. If for all z ∈ U \ M either ∇p · ψ < 0 or ∇p · F < 0 or ∇2 p · A < 0 holds, then E is a Lyapunov functional for M ∪ N0 , where N0 contains all constants z0 ∈ IRM , that are zeros of ∇p · ψ and ∇p · F . We refer to [1, Beispiel 1.41, page 38] and [7, page 48 f.] for some applications of Theorem 1.1 and concrete examples. However, chemotaxis models with nondiffusive memory do not belong to the class of systems that W. Alt analyzed in [1]. But there are some situations where Lyapunov functionals (in the same spirit as presented in Theorem 1.1 but without the requirement that the functional has always to be positive) also exist for such kind of systems. Lyapunov functionals are useful tools for analyzing the time asymtotic behavior of the solutions and can help to prove the convergence of the solution to some steady state solution or the blow-up of the solution in either finite or in infinite time. However, the fact that there are Lyapunov-functions for chemotaxis-growth models if the chemotactic signal (resp. the “memory”) is non-diffusive has (to my knowledge) not been recognized in literature. Therefore, the short note raises the question phrased in the title and gives the following positive answer: Do chemotaxisgrowth models possess Lyapunov functionals? Yes, some and especially those with non-diffusive memory. 2. Chemotaxis-growth models with non-diffusive memory For notational reasons we set Ω × (0, T ) = QT and ∂Ω × (0, T ) = ΓT . In the following we will focus on systems of the following type: (ui )t = ∇(∇k1i (ui ) − k2i (ui )∇χi (vj )) + fi (u1 , ..., ul , vj ), (x, t) ∈ QT (vj )t = gj (u1 , ..., ul , vj ), (x, t) ∈ QT ∂vj 0 0 i 0 = k1i (ui ) ∂u (x, t) ∈ ΓT ∂ν − k2i (ui )χi (vj ) ∂ν , ui (0, x) = (ui )0 (x), vj (0, x) = (vj )0 (x), x∈Ω (2) 2
where i ∈ {1, ..., l}, l ≥ 1 and j ∈ {1, ..., r}, r ≥ 1. In particular there are l different chemotactic sensitivity χ0i and r different chemotactic signals vj . As a consequence there are r set Nq of index-pairs (i, j) with ]Nq = lq such that ui r r P P ]Nq = lq . reacts in a chemotactic way to vj and l = q=1
q=1
Models of this kind for single and several species have been studied for instance in [2, 3, 5, 9, 10] and [12]. Here we are not concerned about the question under which conditions this problem is well-posed and whether there are some conditions that guarantee global in time solutions as it has been done in the just mentioned papers. We want to focus on the questions whether this kind of models have Lyapunov functionals or not. Therefore, we give some sufficient conditions for the existence in the upcoming Lemma. Lemma 2.1. Let (u1 , ..., ul , v1 , ..., vr ) be a (weak) solution of system (2). Let us assume that χi ∈ C 1 (IR) for all i ∈ {1, ..., l} satisfies χ0i (v) > 0 for all v ∈ IR+ . Furthermore, we assume that there are l functions kl ∈ C 2 (IR) satisfying 0 ki00 (u) > 0, k2i (u)/ki0 (u) > 0 and ki00 (u) = k1i (u)ki0 (u)/k2i (u) for all u ∈ IR+ . If 0 0 ki (ui )fi (u1 , ..., ul , vj ) ≤ gj (u1 , ..., ul , vj )χi (vj )ki (ui ) for (i, j) ∈ Nq for some q ∈ D ⊂ {1, ..r} resp. ki0 (ui )fi (u1 , ..., ul , vj ) < 0 in the special situation that Pfor some j ∈ {1, ..., r} the function gj is given by gj (u1 , ..., ul , vj ) = γj (vj )+ ki (ui ) (i,j)∈Nq
for (i, j) ∈ Nq and χi (vj ) = χy (vj ) for all i 6= y s.t. (i, j), (y, j) ∈ Nq with q ∈ R := {1, ..., r} \ D, then E(t) := E1 (t) + E2 (t) with Z X X E1 (t) := ki (ui )e−χi (vj ) dx and Ω q∈D (i,j)∈Nq
E2 (t) :=
Z
Ω
X
X
q∈R (i,j)∈Nq
vj Z −χ0i (s)γj (s)e−χi (s) ds + ki (ui )e−χi (vj ) dx.
is a Lyapunov functional for the given system (2), i.e.
d dt Ep (t)
< 0 for p = 1, 2.
Remark 2.1. We point out that if (i, j) ∈ Nq with q ∈ R, then all ki ≡ 0 for those i for which (i, j) 6∈ Nq and all χi (vj ) = χ(vj ) for those i for which (i, j) ∈ Nq , where χ ∈ C 1 (IR) with χ0 (v) > 0 for all v ∈ IR+ . Proof: We see, that: Z X X d E1 (t) = − (vj )t χ0i (vj )ki (ui )e−χi (vj ) dx dt Ω q∈D (i,j)∈Nq Z X X + (ui )t ki0 (ui )e−χi (vj ) dx Ω q∈D (i,j)∈Nq
=
−
Z X
X
Ω q∈D (i,j)∈Nq
k2i (ui )e−χi (vj ) 00 2 |ki (ui )∇ui − ki0 (ui )∇χi (vj )| dx ki0 (ui ) 3
+
Z X
X
ki0 (ui )fi (u1 , ..., ul , vj )e−χi (vj ) dx
Ω q∈D (i,j)∈Nq
−
Z X
X
gj (u1 , ..., ul , vj )χ0i (vj )ki (ui )e−χi (vj ) dx < 0
Ω q∈D (i,j)∈Nq
and for E2 d E2 (t) dt
= − +
Z X
X
Ω q∈R (i,j)∈Nq
Z X
X
Ω q∈R (i,j)∈Nq
= − − +
(vj )t χ0i (vj )gj (u1 , ..., ul , vj )e−χi (vj ) dx
Z X
X
Ω q∈R (i,j)∈Nq
Z X
X
Ω q∈R (i,j)∈Nq
Z X
X
(ui )t ki0 (ui )e−χi (vj ) dx 4 −χ(vj )/2 2 e dx χ0 (vj ) t
k2i (ui )e−χ(vj ) 00 2 |ki (ui )∇ui − ki0 (ui )∇χ(vj )| dx 0 ki (ui ) ki0 (ui )e−χ(vj ) fi (u1 , ..., ul , vj )dx < 0.
Ω q∈R (i,j)∈Nq
Remark 2.2. The given conditions are sufficient to guarantee the existence of Lypaunov functionals also in the absence of growth as we will see in Example 3.2. However, in the absence of growth one can also apply the results in [3, 6, 8] to find other conditions that ensure the existence of Lyapunov functionals for even a larger class of cross-diffusion systems. However, the results presented here are different from those known so far. Furthermore, we want to point out that the condition stated above do not require that the function p(u(t, x)) is positive, as it was necessary in Theorem 1.1 (see for instance Example 3.3). 3. Examples and Applications In this section we want to give four concrete examples for the result stated above. The first two examples will present Lyapunov functionals for systems, where one can find global existence results in the literature. The other two examples will present some models for which the well-posedness and the time asymptotic behavior of a solution has not been studied so far (as far as I know). However, these two examples will present some systems with either logistic or Fisher-KPP type growth where Lyapunov functionals exist. Example 3.1. In [2] the authors study the following model with reproduction
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term:
u ut = ∇ (∇u − u∇ ln (α + βv)) + u a − b α+βv , (x, t) ∈ QT vt = µu − δv, (x, t) ∈ QT ∂ u 0 = ∂u ∂ν − u ∂ν ln α+βv , u(0, x) = u0 (x), v(0, x) = v0 (x),
(x, t) ∈ ΓT x ∈ Ω.
(3)
In [2, Theorem 1] the authors state that there is unique solution for a = 0 = δ and µβ + b > 0 that exists globally in time. In fact if µβ + 2b > 0, then there exists the Lyapunov functional Z Z u2 2 − ln(α+βv) dx ≥ 0 E(t) = u e dx = α + βv Ω
Ω
for the given system (3), if α > 0, β > 0. Thus we conclude that Z Z 2 2 |∇u − u∇ ln(α + βv)| (2b + µβ)u3 d E(t) = − dx − dx → 0 dt α + βv (α + βv)2 Ω
Ω
as t → ∞.
Example 3.2. In [12] the authors proved for 1 > β > 0 the global existence of a unique solution of the system ut = ∇ (∇u − u∇ ln (v)) and vt = uv β , (x, t) ∈ QT ∂ (4) 0 = ∂u (x, t) ∈ ΓT ∂ν − u ∂ν ln (v) , u(0, x) = u0 (x), v(0, x) = v0 (x), x ∈ Ω. R Here E(t) = u2 v −1 dx ≥ 0 is a Lyapunov functional for the given system. Ω
Now, we give two examples for models where one has either a logistic or a Fisher-KPP type source, but that have not been studied so far.
Example 3.3. Suppose there exists a solution to the following system: ut = ∇ (∇u − (u − 1)∇v) + um (1 − u), m ≥ 1 (x, t) ∈ QT vt = γ(v) + 21 u2 − u, (x, t) ∈ QT (5) ∂ 0 = ∂u (x, t) ∈ ΓT ∂ν − (u − 1) ∂ν ln (v) , u(0, x) = u0 (x), v(0, x) = v0 (x), x ∈ Ω. v R R Then E(t) = −γ(s)e−s ds + 21 u2 − u e−v dx is a Lyapunov functional. Ω
Example 3.4. Suppose there exists a solution to the system:
(x, t) ∈ QT (x, t) ∈ ΓT x ∈ Ω, (6) where 0 < Ra < 1. If γ · θ > 2(1 + a) then a Lyapunov functional for (6) is given by E(t) = u2 v −θ dx ≥ 0. ut = ∇ (∇u − θu∇ ln(v)) + u(u − a)(1 − u) and vt = γuv, ∂ 0 = ∂u ∂ν − θu ∂ν ln (v) , u(0, x) = u0 (x), v(0, x) = v0 (x),
Ω
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References [1] W. Alt, Vergleichs¨ atze f¨ ur quasilineare elliptisch-parabolische Systems partieller Differentialgleichungen, Habilitation, Ruprecht-Karl-Universit¨ at Heidelberg, 1980. [2] X. Chen & W. Liu, Behaviout of the solutions to an Othmer-Stevens chemotaxis Model with Reproduction Term, Wuhan University Journal of Natural sciences 15 (2010), no. 4, pp.m 277–282. [3] L. Corrias, B. Perthame & H. Zaag, A chemotaxis model motivated by angiogenesis. C. R. Math. Acad. Sci. Paris 336 (2003), no. 2, pp. 141-146. [4] H. Gajewski & K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 195 (1998), pp. 77 – 114. [5] F. R. Guarguaglini & R. Natalini, Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Commun. Pure Appl. Anal. 6 (2007), no. 1, pp. 287-309. [6] D. Horstmann, Lyapunov functions and Lp -estimates for a class of reaction-diffusion systems, Colloquium Mathematicum 87 (2001), pp. 113 – 127. [7] D. Horstmann,From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV 105 (2003), 103–165. [8] D. Horstmann, Generalizing Keller-Segel: Lyapunov functionals, steady state analysis and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Non. Science 21 (2011), pp. 231–270. [9] M. Negreanu & J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant. J. Differential Equations 258 (2015), no. 5, pp. 1592-1617. [10] H. G. Othmer & A. Stevens, Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997), no4, pp. 1044–1081. [11] K. Post, A non-linear parabolic system modeling chemotaxis with sensitivity functions, Dissertation, Humboldt-Universit¨ at zu Berlin, Institut f¨ ur Mathematik 1999. [12] Y. Sugiyama, Y. Tsutsui & J. J. L. Vel´ azquez, Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl. 410 (2014), pp. 908–917.
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