June  2020, 40(6): 3595-3627. doi: 10.3934/dcds.2020170

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

1. 

Institute of Applied Mathematics and Bioquant, Heidelberg University, Heidelberg, 69120, Germany

2. 

Institute of Applied Mathematics, Bioquant and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, 69120, Germany

3. 

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

4. 

Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

* Corresponding author: Anna Marciniak-Czochra

Received  May 2019 Revised  January 2020 Published  March 2020

Fund Project: This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Collaborative Research Center 1324 (SFB1324, project B6). IT has been supported in part by JSPS Kakenhi, Grant Numbers 16KT0128 and 19K03557

The paper is devoted to analysis of far-from-equilibrium pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.

Citation: Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
References:
[1]

D. Angeli, J. E. Ferrell and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, PNAS, 101 (2004), 1822–1827. doi: 10.1073/pnas.0308265100.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

D. G. AronsonA. Tesei and H. Weinberger, A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., 152 (1988), 259-280.  doi: 10.1007/BF01766153.  Google Scholar

[4]

J. E. Ferrell and W. Xiong, Bistability in cell signaling: How to make continuos processes discontinous, and reversible processes irreversible, Chaos, 11 (2001), 227-236.  doi: 10.1063/1.1349894.  Google Scholar

[5]

T. GregorE. F. WieschausA. P. McGregorW. Bialek and D. W. Tank, Stability and nuclear dynamics of the bicoid morphogen gradient, Cell, 130 (2007), 141-152.  doi: 10.1016/j.cell.2007.05.026.  Google Scholar

[6]

S. HärtingA. Marciniak-Czochra and I. Takagi, Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst. Ser. A., 37 (2017), 757-800.  doi: 10.3934/dcds.2017032.  Google Scholar

[7]

S. Härting and A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Meth. Appl. Sci., 37 (2014), 1377-1391.   Google Scholar

[8]

S. Hock, Y. Ng, J. Hasenauer, D. Wittmann, D. Lutter, D. Trümbach, W. Wurst, N. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Syst. Biol., 7 (2013), 48. doi: 10.1186/1752-0509-7-48.  Google Scholar

[9]

J. Jaros and T. Kusano, A picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian, 68 (1999), 137-151.   Google Scholar

[10]

V. KlikaR. BakerD. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organization, Bulletin of Mathematical Biology, 74 (2012), 935-957.  doi: 10.1007/s11538-011-9699-4.  Google Scholar

[11]

S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.  doi: 10.1126/science.1179047.  Google Scholar

[12]

K. KorvasováE. A. GaffneyP. K. MainiM. A. Ferreira and V. Klika, Investigating the turing conditions for diffusion-driven instability in the presence of a binding immobile substrate, J. Theor. Biol., 367 (2015), 286-295.  doi: 10.1016/j.jtbi.2014.11.024.  Google Scholar

[13]

Y. LiA. Marciniak-CzochraI. Takagi and B. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J., 47 (2017), 217-247.  doi: 10.32917/hmj/1499392826.  Google Scholar

[14]

W. S. Loud, "Periodic solutions of $x'' +cx' +g(x) = \epsilon f(t)''$, Mem. Amer. Math. Soc., 31 1959, 58 pp.  Google Scholar

[15]

A. Marasco and et al., Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback, Bull. Math. Biol., 76 (2014), 2866-2883.  doi: 10.1007/s11538-014-0036-6.  Google Scholar

[16]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Sys., 11 (2003), 293-324.  doi: 10.1142/S0218339003000889.  Google Scholar

[17]

A. Marciniak-Czochra, Receptor-based models with hysteresis for pattern formation in hydra, Math. Biosci., 199 (2006), 97-119.  doi: 10.1016/j.mbs.2005.10.004.  Google Scholar

[18]

A. Marciniak-Czochra, Strong two-scale convergence and corrector result for the receptor-based model of the intercellular communication, IMA J. Appl. Math., 77 (2012), 855-868.  doi: 10.1093/imamat/hxs052.  Google Scholar

[19]

A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol. 74 (2017), 583-618. doi: 10.1007/s00285-016-1035-z.  Google Scholar

[20]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011.  Google Scholar

[21]

A. Marciniak-Czochra and M. Kimmel, Modeling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci., 17 (2007), 1693-1719.  doi: 10.1142/S0218202507002443.  Google Scholar

[22]

A. Marciniak-CzochraM. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694.   Google Scholar

[23]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques., SIAM J. Math. Anal., 40 (2008), 215-237.  doi: 10.1137/050645269.  Google Scholar

[24]

M. MimuraM. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631.  doi: 10.1137/0511057.  Google Scholar

[25]

C. Niehrs, The Spemann organizer and embryonic head induction, EMBO J., 20 (2001), 631-637.   Google Scholar

[26]

K. PhamA. ChauviereH. HatzikirouX. LiH.M.. ByrneV. Cristini and 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.  doi: 10.1080/17513758.2011.590610.  Google Scholar

[27]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, 1984. doi: 10.1007/BFb0099278.  Google Scholar

[28]

R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458, Springer, 1990. doi: 10.1007/BFb0098346.  Google Scholar

[29]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 258, Springer, New York; Heidelberg; Berlin, 1983.  Google Scholar

[30]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[31]

D. M. UmulisM. SerpeM. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo,, Proc. Nat. Ac. Sci., 103 (2006), 11613-11618.  doi: 10.1073/pnas.0510398103.  Google Scholar

show all references

References:
[1]

D. Angeli, J. E. Ferrell and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, PNAS, 101 (2004), 1822–1827. doi: 10.1073/pnas.0308265100.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

D. G. AronsonA. Tesei and H. Weinberger, A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., 152 (1988), 259-280.  doi: 10.1007/BF01766153.  Google Scholar

[4]

J. E. Ferrell and W. Xiong, Bistability in cell signaling: How to make continuos processes discontinous, and reversible processes irreversible, Chaos, 11 (2001), 227-236.  doi: 10.1063/1.1349894.  Google Scholar

[5]

T. GregorE. F. WieschausA. P. McGregorW. Bialek and D. W. Tank, Stability and nuclear dynamics of the bicoid morphogen gradient, Cell, 130 (2007), 141-152.  doi: 10.1016/j.cell.2007.05.026.  Google Scholar

[6]

S. HärtingA. Marciniak-Czochra and I. Takagi, Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst. Ser. A., 37 (2017), 757-800.  doi: 10.3934/dcds.2017032.  Google Scholar

[7]

S. Härting and A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Meth. Appl. Sci., 37 (2014), 1377-1391.   Google Scholar

[8]

S. Hock, Y. Ng, J. Hasenauer, D. Wittmann, D. Lutter, D. Trümbach, W. Wurst, N. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Syst. Biol., 7 (2013), 48. doi: 10.1186/1752-0509-7-48.  Google Scholar

[9]

J. Jaros and T. Kusano, A picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian, 68 (1999), 137-151.   Google Scholar

[10]

V. KlikaR. BakerD. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organization, Bulletin of Mathematical Biology, 74 (2012), 935-957.  doi: 10.1007/s11538-011-9699-4.  Google Scholar

[11]

S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.  doi: 10.1126/science.1179047.  Google Scholar

[12]

K. KorvasováE. A. GaffneyP. K. MainiM. A. Ferreira and V. Klika, Investigating the turing conditions for diffusion-driven instability in the presence of a binding immobile substrate, J. Theor. Biol., 367 (2015), 286-295.  doi: 10.1016/j.jtbi.2014.11.024.  Google Scholar

[13]

Y. LiA. Marciniak-CzochraI. Takagi and B. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J., 47 (2017), 217-247.  doi: 10.32917/hmj/1499392826.  Google Scholar

[14]

W. S. Loud, "Periodic solutions of $x'' +cx' +g(x) = \epsilon f(t)''$, Mem. Amer. Math. Soc., 31 1959, 58 pp.  Google Scholar

[15]

A. Marasco and et al., Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback, Bull. Math. Biol., 76 (2014), 2866-2883.  doi: 10.1007/s11538-014-0036-6.  Google Scholar

[16]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Sys., 11 (2003), 293-324.  doi: 10.1142/S0218339003000889.  Google Scholar

[17]

A. Marciniak-Czochra, Receptor-based models with hysteresis for pattern formation in hydra, Math. Biosci., 199 (2006), 97-119.  doi: 10.1016/j.mbs.2005.10.004.  Google Scholar

[18]

A. Marciniak-Czochra, Strong two-scale convergence and corrector result for the receptor-based model of the intercellular communication, IMA J. Appl. Math., 77 (2012), 855-868.  doi: 10.1093/imamat/hxs052.  Google Scholar

[19]

A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol. 74 (2017), 583-618. doi: 10.1007/s00285-016-1035-z.  Google Scholar

[20]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011.  Google Scholar

[21]

A. Marciniak-Czochra and M. Kimmel, Modeling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci., 17 (2007), 1693-1719.  doi: 10.1142/S0218202507002443.  Google Scholar

[22]

A. Marciniak-CzochraM. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694.   Google Scholar

[23]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques., SIAM J. Math. Anal., 40 (2008), 215-237.  doi: 10.1137/050645269.  Google Scholar

[24]

M. MimuraM. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631.  doi: 10.1137/0511057.  Google Scholar

[25]

C. Niehrs, The Spemann organizer and embryonic head induction, EMBO J., 20 (2001), 631-637.   Google Scholar

[26]

K. PhamA. ChauviereH. HatzikirouX. LiH.M.. ByrneV. Cristini and 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.  doi: 10.1080/17513758.2011.590610.  Google Scholar

[27]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, 1984. doi: 10.1007/BFb0099278.  Google Scholar

[28]

R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458, Springer, 1990. doi: 10.1007/BFb0098346.  Google Scholar

[29]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 258, Springer, New York; Heidelberg; Berlin, 1983.  Google Scholar

[30]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[31]

D. M. UmulisM. SerpeM. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo,, Proc. Nat. Ac. Sci., 103 (2006), 11613-11618.  doi: 10.1073/pnas.0510398103.  Google Scholar

Figure 1.  Typical configurations of the zero sets of the kinetic functions
Figure 2.  (A) Phase plane of $\gamma^{-1}U_{xx}+q \bar{u}(U)=0$. The blue trajectory $(U(x),U_x(x))$ connects the points $(u_0,0)$ and $(u_e,0)$ and is a solution of the boundary value problem satisfying $U_x(0)=U_x(1)=0$. (B) A monotone increasing stationary solution with jump at $\bar{u}$ and layer position at $\bar{x}$
Figure 3.  The time-maps for the kinetic functions $f(u,v)=1.4v-u$, $g(u,v)=u-(v^3-6.3v^2+10v)$ and the jump $\bar{u}=3.1$. Here, it holds $Q \bar{u}(u_2)<0$ and umin=0.8957. We see that Tu1 is decreasing, whereas $T_{\bar{u}}^{2}$ is increasing, which leads to $\frac{\mathrm{d}}{\mathrm{d} u_{0}} T_{\bar{u}}(u_0)<0$. Furthermore, we observe that $\lim_{u_0\to {\bar{u}}}T_{\bar{u}}{}(u_0)=0$ and $\lim_{u_0\to u_{\rm{min}}}T_{\bar{u}}{}(u_0)=\infty$
Figure 4.  Simulations of the generic model with hysteresis for different types of perturbations of a stationary solution. The plots show initial conditions (dotted lines) and the approached stationary solution (continuous lines) after a sufficiently large time $ t_{end} $
Figure 5.  The layer position $ \bar{x}( \bar{u}) $ and the Interval $ I^0 $. (A) and (B) Plots for the kinetic functions eq. (60) and for diffusion coefficients $ 1/\gamma $ with $ \gamma=50 $ in A and $ \gamma=200 $ in B. (C) Plots for the kinetic functions eq. (61) and the diffusion coefficient $ 1/200 $
Figure 6.  The phase planes of $\frac{1}{\gamma} U_{x x}+q_{H}(U)=0$ and $\frac{1}{\gamma} U_{x x}+q{T}(U)=0$ are overlapping. In blue we see a periodic solution with jump at u. We cannot determine the mode of a periodic solution in the phase plane. It corresponds to how often the trajectory has been traveled through. In red we see a irregular solution with three different jumps
Figure 7.  An irregular solution $ \big(U(x), V(x)\big) $ with three jumps $ \bar{u}^{{1}}, \bar{u}^{{2}}, \bar{u}^{{3}} $, which is monotone increasing restricted to $ [0, x^1] $. We see that for continuity of $ U(x) $ we need to have $ u_{e}^1=u_0^2 $ and $ u_{e}^2=u_0^3 $ fulfilled. Furthermore, we see how the partition of the interval is determined: $ x^1=T(\bar{u}^{{1}}, u_0^1), x^2=x^1+T(\bar{u}^{{2}}, u_0^2) $ and $ 1=x^2+T(\bar{u}^{{3}}, u_0^3) $. The layer positions are given by $ \bar{x}^{{1}}=T_1(\bar{u}^{{1}}, u_0^1) $ and $ \bar{x}^{{3}}=x^2+T_1(\bar{u}^{{3}}, u_0^3) $, because $ U(x) $ is increasing on the corresponding subintervals and $ \bar{x}^{{2}}=x^1+T_2(\bar{u}^{{2}}, u_0^2)=x^2-T_1(\bar{u}^{{2}}, u_0^3) $
Figure 8.  Simulations of model (1)-(3) for admissible kinetic functions, diffusion coefficient $ 1/\gamma=1/1000 $ and initial conditions of type (59) having four discontinuities. The $ u $-component is plotted in blue, whereas the $ v $-component is red. The initial condition $ \big(u(0, x), v(0, x)\big)=\big(u_0(x), v_0(x)\big) $ is indicated by dotted lines and the stationary solution $ \big(u(t_{end}, x), v(t_{end}, x)\big) $ is indicated by continuous bold lines. Here, $ t_{end} $ is a sufficiently large timepoint, such that the solution $ \big(u(t, x), v(t, x)\big) $ does not change in time anymore. A-C the kinetic functions are given by (60) D the kinetic functions are given by (62)
[1]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[2]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[3]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[4]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[5]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[6]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[7]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[8]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[9]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[10]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[11]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[12]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[13]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[14]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[15]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210

[16]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[17]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[18]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[19]

Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511

[20]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (153)
  • HTML views (292)
  • Cited by (2)

[Back to Top]