Marek FILA

Department of Applied Mathematics and Statistics, Comenius University, Mlynska dolina, SK-84248 Bratislava
E-mail: fila@fmph.uniba.sk
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List of publications
  1. Marek Fila, Apriornaja ocenka proizvodnoj $u_x$ rešenija kvazilinejnogo paraboličeskogo uravnenija, Acta Mathematica Universitatis Comenianae, Vol. 44–45, Bratislava: Alfa, 1984 S. 251–256.
  2. Marek Fila, On some properties of solutions of the Cauchy problem for a quasilinear parabolic equation, Časopis pro pěstování matematiky, Roč. 109, č. 3 (1984), s. 268–276.
  3. Marek Fila, Stabilization of solutions of a reaction-diffusion equation, Partial Differential Equations, Warszawa: PWN, 1987 S. 89–93.
  4. Marek Fila, Ján Filo, Stabilization of solutions of certain one-dimensional degenerate diffusion equations, Mathematica Slovaca, Vol. 37, No. 2 (1987), s. 217–229.
  5. Marek Fila, Ján Filo, Blow up above stationary solutions of certain nonlinear parabolic equations, Commentationes Mathematicae Universitatis Carolinae, Vol. 29, No. 1 (1988), s. 179–193.
  6. Marek Fila, Boundedness of global solutions for the heat equation with nonlinear boundary condition, Commentationes Mathematicae Universitatis Carolinae, Vol. 30, No. 3 (1989), s. 479–484.
  7. Marek Fila, František Marko, A remark on a flow with a homoclinic trajectory, Acta Mathematica Universitatis Comenianae, Vol. 54–55, Bratislava: Alfa, 1989 S. 15–19.
  8. Marek Fila, Ján Filo, A blow-up result for nonlinear diffusion equations, Mathematica Slovaca, Vol. 39, No. 3 (1989), s. 331–346.
  9. Marek Fila, Ján Filo, Kvalitatívna analýza vybraných úloh nelineárnej difúzie, Sborník referátů 13. semináře parciálních diferenciálních rovnic, Praha: JČSMF, 1989 S. 5–76.
  10. Marek Fila, Ján Filo, Blow up for a nonlinear degenerate parabolic equation, EQUADIFF 7: Proceedings of the 7th Czechoslovak Conference on Differential Equations and their Applications, Leipzig: Teubner, 1990 S. 163–166.
  11. Marek Fila, Bernd Kawohl, Is quenching in infinite time possible?, Quarterly of Applied Mathematics, Vol. 48, No. 3 (1990), s. 531–534.
  12. Marek Fila, Ján Filo, Global behaviour of solutions to some nonlinear diffusion equations, Czechoslovak Mathematical Journal, Vol. 40 (115), No. 2 (1990), s. 226–238.
  13. Marek Fila, Pavol Quittner, The blow-up rate for the heat equation with a non-linear boundary condition, Mathematical Methods in the Applied Sciences, Vol. 14, No. 3 (1991), s. 197–205.
  14. Marek Fila, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proceedings of the American Mathematical Society, Vol. 111, No. 3 (1991), s. 795–801.
  15. Marek Fila, Josephus Hulshof, A note on the quenching rate, Proceedings of the American Mathematical Society, Vol. 112, No. 2 (1991), s. 473–477.
  16. Michel Chipot, Marek Fila, Pavol Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Mathematica Universitatis Comenianae—New Series, Vol. 60, No. 1 (1991), s. 35–103.
  17. Marek Fila, Josephus Hulshof, Pavol Quittner, The quenching problem on the n-dimensional ball, Nonlinear Diffusion Equations and their Equilibrium States, Boston: Birkhäuser, 1992 S. 183–196.
  18. Marek Fila, Bernd Kawohl, Howard A. Levine, Quenching for quasilinear equations, Communications in Partial Differential Equations, Vol. 17, No. 3 (1992), s. 593–614.
  19. Marek Fila, Bernhard Kawohl, Asymptotic analysis of quenching problems, Rocky Mountain Journal of Mathematics, Vol. 22, No. 2 (1992), s. 563–577.
  20. Marek Fila, Boundedness of global solutions of nonlinear diffusion equations, Journal of Differential Equations, Vol. 98, No. 2 (1992), s. 226–240.
  21. Marek Fila, Howard A. Levine, Quenching on the boundary, Nonlinear Analysis—Theory, Methods & Applications, Vol. 21, No. 10 (1993), s. 795–802.
  22. Marek Fila, Howard A. Levine, Juan Luis Vazquez, Stabilization of solutions of weakly singular quenching problems, Proceedings of the American Mathematical Society, Vol. 119, No. 2 (1993), s. 555–559.
  23. Marek Fila, Pavol Quittner, Radial positive solutions for a semilinear elliptic equation with a gradient term, Advances in Mathematical Sciences and Applications, Vol. 2, No. 1 (1993), s. 39–45.
  24. Keng Deng, Marek Fila, Howard A. Levine, On critical exponents for a system of heat equations coupled in the boundary conditions, Acta Mathematica Universitatis Comenianae—New Series, Vol. 63, No. 2 (1994), s. 169–192.
  25. Marek Fila, Global existence of small solutions of a parabolic system via linearized stability, Progress in partial Differential Equations: the Metz Surveys 3, Harlow: Longman Scientific & Technical, 1994 S. 215–220.
  26. Marek Fila, Howard A. Levine, Yoshitaka Uda, A Fujita type global nonexistence theorem for a system of reaction diffusion equations with differing diffusivities, Mathematical Methods in the Applied Sciences, Vol. 17, No. 10 (1994), s. 807–835.
  27. Marek Fila, Gary M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential and Integral Equations, Vol. 7, No. 3–4 (1994), s. 811–821.
  28. Sigurd B. Angenent, Marek Fila, Interior gradient blow-up in a semilinear parabolic equation, Differential and Integral Equations, Vol. 9, No. 5 (1996), s. 865–877.
  29. Marek Fila, Paul Sakcks, The transition from decay to blow-up in some reaction-diffusion-convection equations, World congress of nonlinear analysts ´92: Proceedings, Vol. 1, Berlin: Walter de Gruyter, 1996 S. 455–463.
  30. Marek Fila, Howard A. Levine, On critical exponents for a seminilear parabolic system coupled in an equation and a boundary condition, Journal of Mathematical Analysis and Applications, Vol. 204, No. 2 (1996), s. 494–521.
  31. Marek Fila, Ján Filo, Blow-up on the boundary: a survey, Singularities and Differential Equations, Warsaw: Stefan Banach International Mathematical Center, 1996 S. 67–78.
  32. Michel Chipot, Itai Shafrir, Marek Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Advances in Differential Equations, Vol. 1, No. 1 (1996), s. 91–110.
  33. Herbert Amann, Marek Fila, A Fujita-type theorem for the Laplace equation with a dynamical boundary condition, Acta Mathematica Universitatis Comenianae—New Series, Vol. 66, No. 2 (1997), s. 321–328.
  34. Marek Fila, Howard A. Levine, On the boundedness of global solutions of abstract semilinear parabolic equations, Journal of Mathematical Analysis and Applications, Vol. 216, No. 2 (1997), s. 654–666.
  35. Marek Fila, Boundedness of global solutions of nonlocal parabolic equations, Nonlinear Analysis—Theory, Methods & Applications, Vol. 30, No. 2 (1997), s. 877–885.
  36. Marek Fila, Pavol Quittner, Global solutions of the Laplace equation with a nonlinear dynamical boundary condition, Mathematical Methods in the Applied Sciences, Vol. 20, No. 15 (1997), s. 1325–1333.
  37. Michel Chipot, Miroslav Chlebík, Marek Fila, Itai Shafrir, Existence of positive solutions of a semilinear elliptic equation in $R^N_+$ with a nonlinear boundary condition, Journal of Mathematical Analysis and Applications, Vol. 223, No. 2 (1998), s. 429–471.
  38. Marek Fila, Pavol Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition, Topics in nonlinear analysis—The Herbert Amann anniversary volume, Basel: Birkhäuser, 1999 S. 251–272.
  39. Marek Fila, Peter Poláčik, Global nonexistence without blow-up for an evolution problem, Mathematische Zeitschrift, Vol. 232, No. 3 (1999), s. 531–545.
  40. Marek Fila, Pavol Quittner, The blow-up rate for a semilinear parabolic system, Journal of Mathematical Analysis and Applications, Vol. 238, No. 2 (1999), s. 468–476.
  41. Marek Fila, Peter Poláčik, Global solutions of a semilinear parabolic equation, Advances in Differential Equations, Vol. 4, No. 2 (1999), s. 163–196.
  42. Miroslav Chlebík, Marek Fila, From critical exponents to blow-up rates for parabolic problems, Rendiconti di Matematica e delle Sue Applicazioni, Serie VII, Vol. 19, No. 4 (1999), s. 449–470.
  43. Marek Fila, Hiroshi Matano, Connecting equilibria by blow-up solutions, EQUADIFF 1999, Vol. 1, Singapore: World Scientific, 2000 S. 741–743.
  44. Marek Fila, Hiroshi Matano, Connecting equilibria by blow-up solutions, Discrete and Continuous Dynamical Systems, Vol. 6, No. 1, (2000), s. 155–164.
  45. Marek Fila, Ján Filo, Gary M. Lieberman, Blow-up on the boundary for the heat equation, Calculus of Variation and Partial Differential Equations, Vol. 10, No. 1 (2000), s. 85–99.
  46. Miroslav Chlebík, Marek Fila, Some recent results on blow-up on the boundary for the heat equation, Evolution equations: existence, regularity and singularities. Warszawa: Instytut Matematyczny PAN, 2000, S. 61–71.
  47. Miroslav Chlebík, Marek Fila, On the blow-up rate for the heat equation with a nonlinear boundary condition, Mathematical Methods in the Applied Sciences, Vol. 23, No. 15 (2000), s. 1323–1330.
  48. Marek Fila, Philippe Souplet, Fred B. Weissler, Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions, Mathematische Annalen, Vol. 320, No. 1 (2001), s. 87–113.
  49. Marek Fila, Philippe Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces and Free Boundaries, Vol. 3, No. 3 (2001), s. 337–344.
  50. Marek Fila, Philippe Souplet, The blow-up rate for semilinear parabolic problems on general domains, NoDEA—Nonlinear Differential Equations and Applications, Vol. 8, No. 4 (2001), s. 473–480.
  51. Marek Fila, Bernd Kawohl, Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition, Advances in Mathematical Sciences and Applications, Vol. 11, No. 1 (2001), s. 113–126.
  52. Marek Fila, Hiroshi Matano, Blow-up in nonlinear heat equations from the dynamical systems point of view, Handbook of Dynamical Systems, Vol. 2, Amsterdam: Elsevier Science, 2002 S. 723–758.
  53. Marek Fila, Universal bounds for global solutions of nonlinear parabolic equations, Nonlinear Boundary Value Problems, Vol. 12 (2002), s. 179–188.
  54. Marek Fila, Hiroshi Matano, Peter Poláčik, Existence of L1-connections between equilibria of a semilinear parabolic equation, Journal of Dynamics and Diffierential Equations, Vol. 14, No. 3 (2002), s. 463–491.
  55. Marek Fila, Boundedness of global solutions of nonlinear parabolic equations, Elliptic and Parabolic Problems: Proceedings of the 4th European Conference, Singapore: World Scientific, 2002 S. 88–102.
  56. Marek Fila, Jong-Shenq Guo, Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition, Nonlinear Analysis—Theory, Methods & Applications, Vol. 48, No. 7 (2002), s. 995–1002.
  57. Miroslav Chlebík, Marek Fila, Wolfgang Reichel, Positive solutions of linear elliptic equations with critical growth in the Neumann boundary, NoDEA—Nonlinear Differential Equations and Applications, Vol. 10, No. 3 (2003), s. 329–346.
  58. Miroslav Chlebík, Marek Fila, Pavol Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term, Dynamics of Continuous, Discrete and Impulsive Systems—Series A: Mathematical Analysis, Vol. 10, No. 4 (2003), s. 525–537.
  59. Marek Fila, Michal Winkler, Eiji Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, Journal of Differential Equations, Vol. 205, No. 2 (2004), s. 365–389.
  60. Marek Fila, Blow-up of solutions of supercritical parabolic equations, Handbook of Differential Equations: Evolutionary Equations, Vol. 2, Amsterdam: Elsevier, 2005 S. 105–158.
  61. Marek Fila, Hirokazu Ninomiya, Reaction versus diffusion: blow-up induced and inhibited by diffusivity, Russian Mathematical Surveys, Vol. 60, No. 6 (2005), s. 1217–1235.
  62. Marek Fila, Michael Winkler, Eiji Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity, Journal of Dynamics and Diffierential Equations, Vol. 17, No. 2 (2005), s. 249–269.
  63. Marek Fila, Juan J. L. Velázquez, Michael Winkler, Grow-up on the boundary for a semilinear parabolic problem, Nonlinear Elliptic and Parabolic Problems, Basel: Birkhäuser Verlag, 2005 S. 137–150.
  64. Marek Fila, Hiroshi Matano, Peter Poláčik, Immediate regularization after blow-up, EQUADIFF 2003: Proceedings of the International Conference on Differential Equations, Singapore: World Scientific, 2005 S. 284–289.
  65. Marek Fila, Hiroshi Matano, Peter Poláčik, Immediate regularization after blow-up, SIAM Journal on Mathematical Analysis, Vol. 37, No. 3 (2005), s. 752–776.
  66. Marek Fila, Jonn R. King, Michael Winkler, Eiji Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, Journal of Differential Equations, Vol. 228, No. 1 (2006), s. 339–356.
  67. Marek Fila, Hirokazu Ninomiya, Juan Luis Vazquez, Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems, Discrete and Continuous Dynamical Systems, Vol. 14, No. 1 (2006), s. 63–74.
  68. Marek Fila, Jonn R. King, Michael Winkler, Eiji Yanagida, Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent, Advances in Differential Equations, Vol. 12, No. 1 (2007), s. 1–26.
  69. Marek Fila, Noriko Mizoguchi, Multiple continuation beyond blow-up, Differential and Integral Equations, Vol. 20, No. 6 (2007), s. 671–680.
  70. Marek Fila, Jari Taskinen, Michael Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up, Applied Mathematics Letters, Vol. 20, No. 5 (2007), s. 578–582.
  71. Marek Fila, Michael Winkler, Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed, Journal of the European Mathematical Society, Vol. 10, No. 1 (2008), s. 105–132.
  72. Marek Fila, Michal Winkler, Eiji Yanagida, Slow convergence to zero for a parabolic equation with a supercritical nonlinearity, Mathematische Annalen, Vol. 340, No. 3 (2008), s. 477–496.
  73. Marek Fila, Rate of convergence to a singular steady state of a supercritical parabolic equation, Journal of Evolution Equations, Vol. 8, No. 4 (2008), s. 673–692.
  74. Marek Fila, Michael Winkler, Eiji Yanagida, Convergence of solutions of a semilinear parabolic equation to selfsimilar solutions of the linear heat equation, Advances in Differential Equations, Vol. 13, No. 11–12 (2008), s. 1131–1149.
  75. Marek Fila, Jonn R. King, Michael Winkler, Eiji Yanagida, Linear behaviour of solutions of a superlinear heat equation, Journal of Mathematical Analysis and Applications, Vol. 340, No. 1 (2008), s. 401–409.
  76. Marek Fila, Michael Winkler, Eiji Yanagida, Convergence to self-similar solutions for a semilinear parabolic equation, Discrete and Continuous Dynamical Systems, Vol. 21, No. 3 (2008), s. 703–716.
  77. Marek Fila, Aappo Pulkkinen, Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation, Tohoku Mathematical Journal, Vol. 60, No. 3 (2008), s. 303–328.
  78. Xinfu Chen, Marek Fila, Jong-Shenq Guo, Boundedness of global solutions of a supercritical parabolic equation, Nonlinear Analysis—Theory, Methods & Applications, Vol. 68, No. 3 (2008), s. 621–628.
  79. Marek Fila, Aappo Pulkkinen, Backward selfsimilar solutions of supercritical parabolic equations, Applied Mathematics Letters, Vol. 22, No. 6 (2009), s. 897–901.
  80. Marek Fila, Michael Winkler, Stabilization of solutions of a semilinear parabolic equation modelling quorum sensing in a biofilm, Asymptotic Analysis, Vol. 62, No. 1–2 (2009), s. 89–106.

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