Reaction versus diffusion: blow-up induced and inhibited by diffusivity
M. Fila, H. Ninomiya

This publication is cited in the following articles:

1. Hirokazu Ninomiya, “Pattern formation in two-component reaction-diffusion systems with equal diffusion coefficients”, Journal of Differential Equations, 482 (2026), 114709  
2. Hirokazu Ninomiya, “Example of Turing's instability by equal diffusion”, Journal of Differential Equations, 392 (2024), 255  
3. Nikos I. Kavallaris, Evangelos Latos, Takashi Suzuki, “Diffusion-Driven Blow-Up for a Nonlocal Fisher-KPP Type Model”, SIAM J. Math. Anal., 55:3 (2023), 2411  
4. [Anonymous], “Approximations of Kinetic Equations of Swarm Formation: Convergence and Exact Solutions”, Appl. Math. Comput., 417 (2022), 126778    
5. Tetsuya Ishiwata, Yukihiko Nakata, “A note on blow-up solutions for a scalar differential equation with a discrete delay”, Japan J. Indust. Appl. Math., 39:3 (2022), 959  
6. Kavallaris N.I., Barreira R., Madzvamuse A., “Dynamics of Shadow System of a Singular Gierer-Meinhardt System on An Evolving Domain”, J. Nonlinear Sci., 31:1 (2021), 5        
7. Wentz J.M., Bortz D.M., “Boundedness of a Class of Spatially Discrete Reaction-Diffusion Systems”, SIAM J. Appl. Math., 81:5 (2021), 1870–1892      
8. Aida Ch., Chen Ch.-N., Kuto K., Ninomiya H., “Bifurcation From Infinity With Applications to Reaction-Diffusion Systems”, Discret. Contin. Dyn. Syst., 40:6, SI (2020), 3031–3055      
9. O. N. Shablovskii, “Dinamika neustoichivykh reshenii volnovogo uravneniya s istochnikami”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 12:4 (2020), 51–61    
10. Kirane M., Alsaedi A., Ahmad B., “on Systems of Reaction-Diffusion Equations With a Balance Law: the Sequel”, Comput. Math. Appl., 78:5, SI (2019), 1244–1260      
11. Marciniak-Czochra A., Haerting S., Karch G., Suzuki K., “Dynamical Spike Solutions in a Nonlocal Model of Pattern Formation”, Nonlinearity, 31:4 (2018), 1757–1781          
12. Souplet Ph., “Global Existence For Reaction-Diffusion Systems With Dissipation of Mass and Quadratic Growth”, J. Evol. Equ., 18:4 (2018), 1713–1720          
13. Parshad R.D., Quansah E., Beauregard M.A., Kouachi S., “On “small” data blow-up in a three species food chain model”, Comput. Math. Appl., 73:4 (2017), 576–587          
14. Lachowicz M., Leszczynski H., Parisot M., “Blow-Up and Global Existence For a Kinetic Equation of Swarm Formation”, Math. Models Meth. Appl. Sci., 27:6, 1, SI (2017), 1153–1175        
15. Karch G., Suzuki K., Zienkiewicz J., “Finite-Time Blowup of Solutions To Some Activator-Inhibitor Systems”, Discret. Contin. Dyn. Syst., 36:9 (2016), 4997–5010        
16. Marciniak-Czochra A., Karch G., Suzuki K., Zienkiewicz J., “Diffusion-Driven Blowup of Nonnegative Solutions To Reaction-Diffusion-ODE Systems”, Differ. Integral Equ., 29:7-8 (2016), 715–730      
17. Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz, “Finite-time blowup of solutions to some activator-inhibitor systems”, DCDS-A, 36:9 (2016), 4997  
18. Suzuki R., Umeda N., “Blow-Up of Solutions of a Quasilinear Parabolic Equation”, Proc. R. Soc. Edinb. Sect. A-Math., 142:2 (2012), 425–448          
19. James C. Robinson, Mikołaj Sierżȩga, “A note on well-posedness of semilinear reaction–diffusion problem with singular initial data”, Journal of Mathematical Analysis and Applications, 2011      
20. Budac O., Fila M., “Stabilizing Effect of Diffusion and Dirichlet Boundary Conditions”, J. Math. Sci.-Univ. Tokyo, 17:3 (2010), 303–312      
21. Xiaowei An, Xianfa Song, “Blow-up rate estimates for a parabolic system with multiple nonlinearities”, Math. Meth. Appl. Sci., 33:5 (2009), 632–642      
22. Song Xianfa, “Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws”, Nonlinear Anal., 69:7 (2008), 1971–1980        
23. Birkhäuser Advanced Texts / Basler Lehrbücher, Superlinear Parabolic Problems, 2007, 377