Abstract:

In the fields of population biology and cell biology concentration phenomena are often observed by aggregation of species and chemical substances respectively. One of the well-known models is a Keller-Segel chemotaxis model [21]in which spiky patterns appears by the aggregation of cellular slime mold, though it blows up in a higher dimensional domain (for instance, see [17], [5], [23], [20], [25]and the references therein). In this model the total mass of the slime mold is conserved in a reasonable setting. On the other hand in a study for the cell polarity the authors [19] and [7] proposed simple conceptual models to describe the concentration phenomenon induced by a different mechanism from the chemotaxis model, though the mass conservation property shares in the both models. After their contribution, mathematical studies for the conceptual models are developed in [16], [15], [8], [10]and [9] (see also [13], [14], [11]and [12]). In particular, it is shown in [16], [15]and [8] that the spiky pattern is certainly stable in their model equations.

Motivated by those studies, we are concerned with stationary problem of a reaction–diffusion system with a conservation law under the Neumann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coefficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling withκthe asymptotic profile exhibits a parabola in the non-vanishing region. We will also study the existence of an unstable monotone solution when the mass is small.

By the way, in this course, we will also discuss some interesting open problems in high dimensions.

References:

[1] S. Boussaïd, D. Hilhorst, T.N. Nguyen, Convergence to steady state for the solutions of a nonlocal reaction–diffusion equation, Evol. Equ. and Control Theory 4 (2015) 39–59.

[2] J. Carr, M.E. Gurtin, M. Slemrod, Structured phase transitions on a finite interval, Arch. Ration. Mech. Anal. 86 (1984) 317–351.

[3] J.-L. Chern, Y. Morita, T.-T. Shieh, Asymptotic behavior of equilibrium states of reaction-diffusion systems with mass conservation.**J. Differential Equations****264 ****(2018), ****no. 2,****550–574.**

[4] M.E. Gurtin, H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 156 (1988) 301–317.

[5] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol.25, American Mathematical Society, Providence, RI, 1988.

[6] M. Herrero, J. Velzquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583–623.

[7] S.-Z. Huang, Gradient Inequalities: With Applications to Asymptotic Behavior and Stability of Gradient-Like Systems, Math. Surveys Monogr., vol.126, American Mathematical Society, Providence, RI, 2006.

[8] S. Ishihara, M. Otsuji, A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E 75 (2007), 015203(R).

[9] S. Jimbo, Y. Morita, Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation, J. Differential Equations 255 (2013) 1657–1683.

[10] E. Latos, Y. Morita, T. Suzuki, Stability and spectral comparison of a reaction-diffusion system with mass conservation, preprint.

[11] E. Latos, T. Suzuki, Global dynamics of a reaction–diffusion system with mass conservation, J. Math. Anal. Appl. 411 (2014) 107–118.

[12] T. Mori, K. Kuto, T. Tsujikawa, M. Nagayama, S. Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction–diffusion model for cell polarization, in: Dynamical Systems, Differential Equations and Applications AIMS Proceedings, 2015, pp.861–877.

[13] T. Mori, K. Kuto, T. Tsujikawa, S. Yotsutani, Exact multiplicity of stationary limiting problem of a cell polarization model, Discrete Contin. Dyn. Syst. Ser. A 36 (2016) 5627–5655.

[14] Y. Mori, Y. Jilkine, L. Edelstein-Keshet, Wave-pinning and cell polarity from bistable reaction–diffusion system, Biophys. J. 94 (2008) 3684–3697.

[15] Y. Mori, Y. Jilkine, L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction–diffusion model for cell polarization, SIAM J. Appl. Math. 71 (2011) 1401–1427.

[16] Y. Morita, Spectrum comparison for a conserved reaction–diffusion system with a variational property, J. Comput. Anal. Appl. 2 (2012) 57–71.

[17] Y. Morita, T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction–diffusion system with conservation of mass, Nonlinearity 23 (2010) 1387–1411.

[18] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995) 581–601.

[19] A. Novick-Cohen, On the viscous Chan–Hilliard equation, in: J.M. Ball (Ed.), Material Instabilities in Continuum Mechanics and Related Mathematical Problems, Clarendon Press, Oxford, 1988, pp.329–342.

[20] M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki, S. Kuroda, A mass conserved reaction–diffusion system captures properties of cell polarity, PLoS Comput. Biol. 3 (2007) 1040–1054.

[21] T. Hillen, K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217.

[22] E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415.

[23] K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H.M. Byrne, V. Cristini, J. Lowengrub, Density-dependent quiescence in glioma invasion: instability in a simple reaction–diffusion model for the migration/proliferation dichotomy, J.Biol. Dyn. 6 (2012) 54–71.

[24] T. Senba, T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal. 8 (2001) 349–367.

[25] T. Suzuki, S. Tasaki, Stationary Fix–Caginalp equation with non-local term, Nonlinear Anal. 71 (2009) 1329–1349.

[26] X. Wang, Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, J. Math. Biol. 66 (2013) 1241–1266.