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
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List of publications

• 1. M. Fila, A priori estimate of the derivative $u_x$ of solutions of quasilinear parabolic equations, Acta Math. Univ. Comen. 44-45 (1984), 251--255.(in Russian).
• 2. M. Fila, On some properties of solutions of the Cauchy problem for a quasilinear parabolic equation, Casopis pro pest. matematiky 109 (1984), 268--276.
• 3. M. Fila and J. Filo, Stabilization of solutions of certain one-dimensional degenerate diffusion equations, Math. Slovaca 37 (1987), 217--229.
• 4. M. Fila, Stabilization of solutions of a reaction diffusion equation, Banach Center Publications 19, PWN, Warszawa, 1987, 87--93.
• 5. M. Fila and J. Filo, Blow-up above stationary solutions of certain nonlinear parabolic equations, Comm. Math. Univ. Carol. 29 (1988), 179--193.
• 6. M. Fila and F. Marko, A remark on a flow with a homoclinic trajectory, Acta Math. Univ. Comen. 54-55 (1988), 15--19.
• 7. M. Fila and J. Filo, A blow-up result for nonlinear diffusion equations, Math. Slovaca 39 (1989), 331--346.
• 8. M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comm. Math. Univ. Carol. 30 (1989), 479--484.
• 9. M. Fila and J. Filo, Global behaviour of solutions to some nonlinear diffusion equations, Czechoslovak Math. J. 40 (1990), 226--238.
• 10. M. Fila and B. Kawohl, Is quenching in infinite time possible? Quarterly of Appl. Math. 48 (1990), 531--534.
• 11. M. Fila and J. Filo, Large time behaviour of solutions to a degenerate diffusion problem, Proc. of EQUADIFF 7 (Praha, 1989), Teubner, Leipzig, 1990, 163--166.
• 12. M. Fila, Remarks on blow-up for nonlinear parabolic equations with a gradient term, Proc. of Amer. Math. Soc. 111 (1991), 795--801.
• 13. M. Fila and J.Hulshof, A note on the quenching rate, Proc. of Amer. Math. Soc. 112 (1991), 473--477.
• 14. M. Fila and P. Quittner, The blow-up rate for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci. 14 (1991), 197--205.
• 15. M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for a parabolic equation with a nonlinear boundary condition, Acta Math. Univ. Comenianae 60 (1991), 35--103.
• 16. M. Fila, J. Hulshof and P. Quittner, The quenching problem on the N-dimensional ball, Nonlinear Diffusion Equations and Their Equilibrium States (Ni, Peletier and Serrin eds.), Birkhauser Boston, 1992, 183--196.
• 17. M. Fila, Boundedness of global solutions of nonlinear diffusion equations, Journal of Diff. Equations 98 (1992), 226--240.
• 18. M. Fila and B.Kawohl, Asymptotic analysis of quenching problems, Rocky Mountain J. of Math. 22 (1992), 563--577.
• 19. M. Fila, B. Kawohl and H.A. Levine, Quenching for quasilinear equations, Communications in PDE 17 (1992), 593--614.
• 20. M. Fila, H.A. Levine and J.L. Vazquez, Stabilization of solutions of weakly singular quenching problems, Proc. AMS 119 (1993), 555--559.
• 21. M. Fila and P. Quittner, Radial positive solutions for a semilinear elliptic equation with a gradient term, Advances in Math. Sci. Appl. 2 (1993), 39--45.
• 22. M. Fila and H.A. Levine, Quenching on the boundary, Nonlin. Anal. TMA 21 (1993), 795--802.
• 23. M. Fila, H.A. Levine and Y. Uda, A Fujita type global existence- global nonexistence theorem for a system of reaction diffusion equations with differing diffusivities, Math. Methods Appl. Sci. 17 (1994), 807--835.
• 24. K. Deng, M. Fila and H.A. Levine, On critical exponents for a system of heat equations coupled in the boundary conditions, Acta Math. Univ. Comenianae 63 (1994), 169--192.
• 25. M. Fila and G.M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Diff. and Int. Equations 7 (1994), 811--821.
• 26. M. Fila, Global existence of small solutions of a parabolic system via linearized stability, Pitman Series 314, Progress in PDE-s: the Metz surveys 3 (M. Chipot et al. eds.), Longman, Harlow, 1994, 215--220.
• 27. S.B. Angenent and M. Fila, Interior gradient blow-up in a semilinear parabolic equation, Diff. and Int. Equations 9 (1996), 865--877.
• 28. M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Advances in Diff. Eq. 1 (1996), 91--110.
• 29. M. Fila and H.A. Levine, On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition, J. Math. Anal. Appl. 204 (1996), 494--521.
• 30. M. Fila and J. Filo, Blow-up on the boundary: A survey, Singularities and Differential Equations, Banach Center Publications, Vol. 33, PWN, Warszawa, 1996, 67--78.
• 31. M. Fila and P.E. Sacks, The transition from decay to blow-up in some reaction-diffusion-convection equation, Proc. WCNA (Tampa 1992), V. Lakshmikantham ed., Walter de Gruyter, Berlin-New York, 1996, 455--463.
• 32. M. Fila, Boundedness of global solutions of nonlocal parabolic equations, Nonlin. Anal. TMA 30 (1997), 877--885.
• 33. M. Fila and P. Quittner, Global solutions of the Laplace equation with a nonlinear dynamical boundary condition, Math. Methods Appl. Sci. 20 (1997), 1325--1333.
• 34. M. Fila and H.A. Levine, On the boundedness of global solutions of abstract semi-linear parabolic equations, J. Math. Anal. Appl. 216 (1997), 654--666.
• 35. H. Amann and M. Fila, A Fujita-type theorem for the Laplace equation with a dynamical boundary condition, Acta Math. Univ. Comenianae 66 (1997), 321--328.
• 36. M. Fila and P. Polacik, Global solutions of a semilinear parabolic equation, Adv. Diff. Equations 4 (1999), 163--196.
• 37. M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $R^N_+$ with a nonlinear boundary condition, J. Math. Anal. Appl. 223 (1998), 429--471.
• 38. M. Fila and P. Polacik, Global nonexistence without blow-up for an evolution problem, Math. Zeitschrift 232 (1999), 511--525.
• 39. M. Fila and P. 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, Birkhauser Verlag, 1998, 252--272.
• 40. M. Fila, J. Filo and G.M. Lieberman, Blow-up on the boundary for the heat equation, Calculus of Variations and P.D.E. 10 (2000), 85--99.
• 41. M. Chlebik and M. Fila, From critical exponents to blow-up rates for parabolic problems, Rendiconti Mat. Appl. 19 (1999), 449--470.
• 42. M. Fila and P. Quittner, The blow-up rate for a semilinear parabolic system, J. Math. Anal. Appl. 238 (1999), 468--476.
• 43. M. Chlebik and M. Fila, Some recent results on blow-up on the boundary for the heat equation, Banach Center Publications 52 (2000), 61--71.
• 44. M. Fila and H. Matano, Connecting equilibria by blow-up solutions, Discrete and Continuous Dynamical Systems 6 (2000), 155--164.
• 45. M. Fila and H. Matano, Connecting equilibria by blow-up solutions, Int. Conf. on Differential Equations, Berlin 1999, World Scientific, Singapore, 2000, 741--743.
• 46. M. Chlebik and M. Fila, On the blow-up rate for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci. 23 (2000), 1323--1330.
• 47. M. Fila and J.-S. Guo, Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition, Nonlin. Anal. TMA 48 (2002), 995--1002.
• 48. M. Fila and B. Kawohl, Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition, Adv. Math. Sci. Appl. 11 (2001), 113--126.
• 49. M. Fila and H. Matano, Blow-up in nonlinear heat equations from the dynamical systems point of view, Handbook of Dynamical Systems II, Chapter 14, Edited by B. Fiedler, Elsevier Science B.V., 2002, 723--758.
• 50. M. Fila, Ph. Souplet and F.B. Weissler, Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions, Math. Annalen 320 (2001), 87--113.
• 51. M. Fila and Ph. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces and Free Boundaries 3 (2001), 337--344.
• 52. M. Chlebik, M. Fila and P. Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Cont. Discr. Imp. Systems (Series A: Math. Anal.) 10 (2003), 525--537.
• 53. M. Fila, Boundedness of global solutions of nonlinear parabolic equations, Proceedings of the 4th European Conf. on Elliptic and Parabolic Equations, Rolduc, World Scientific, 2002, pp. 88--102.
• 54. M. Chlebik, M. Fila and W. Reichel, Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition, NoDEA 10 (2003), 329--346.
• 55. M. Fila and Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains, NoDEA 8 (2001), 473--480.
• 56. M. Fila, Universal bounds for global solutions of nonlinear parabolic equations, Nonlinear Boundary Value Problems 12 (2002), 179--188.
• 57. M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Differ. Equations 205 (2004), 365--389.
• 58. M. Fila, H. Matano and P. Polacik, Immediate regularization after blow-up, SIAM J. Math. Anal. (2005), 752--776.
• 59. M. Fila, J.J.L. Velazquez and M. Winkler, Grow-up on the boundary for a semilinear parabolic problem, Progress in Nonlin. Differ. Equations and Their Applications 64 (2005), 137--150.
• 60. M. Fila, M. Winkler and E. Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity, J. Dyn. Diff. Equations 17 (2005), 53--73.
• 61. M. Fila, H. Ninomiya and J.L. Vazquez, Dirichlet boundary conditions can prevent blow-up in reaction diffusion equations and systems, Discr. Cont. Dyn. Systems 14 (2006), 63--74.
• 62. M. Fila, Blow-up of solutions of supercritical parabolic equations, Chapter II, Handbook of Evolution Equations, Elsevier, 2005, 105--158.
• 63. M. Fila, J.R. King, M. Winkler, and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Differ. Equations 228 (2006), 339-356.
• 64. M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up, Applied Mathematics Letters, 20 (2007), 578-582.

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