Qualitative properties of solutions of elliptic and parabolic equations

Qualitative theory of elliptic and parabolic partial differential equations

Members of the research team from the DAMS

  • prof. RNDr. Marek Fila, DrSc.
  • prof. RNDr. Pavol Quittner, DrSc.
  • Mgr. Petra Macková

The research is focused on the study of properties of solutions of nonlinear diffusion equations and systems, as well as problems with nonlinear and dynamical boundary conditions. Besides questions related to the existence and uniqueness, we investigate mainly asymptotic properties of solutions.

Selected papers
  1. M. Fila, J. L. Vázquez, M. Winkler, E. Yanagida: Rate of convergence to Barenblatt profiles for the fast diffusion equation, Archive Rational Mech. Anal. 204 (2012), 599-625
  2. M. Fila, M. Winkler: Rate of convergence to separable solutions of the fast diffusion equation, Israel J. Math. 213 (2016).
  3. M. Fila, M. Winkler: Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation, Proc. Royal Soc. Edinburgh A 146 (2016), 309-324.
  4. M. Fila, K. Ishige, T. Kawakami: Minimal solutions of a semilinear elliptic equation with a dynamical boundary condition, J. Math. Pures Appl. 105 (2016), 788-809.
  5. M. Fila, M. Winkler: Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc. 95 (2017), 659-683.
  6. M. Fila, M. Winkler: A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, Adv. Math. 357 (2019), Art. No. 106823.
  7. T. Bartsch, P. Poláčik, P. Quittner: Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations, J. European Math. Soc. 13 (2011), 219-247.
  8. P. Quittner, Ph. Souplet: Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal. 44 (2012), 2545-2559.
  9. P. Quittner: Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure, Math. Ann. 364 (2016), 269-292.
  10. P. Quittner, Ph. Souplet: Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Cham 2019, Second (revised and expanded) edition.