Publications of Peter Polacik

with D. Valdebenito,  Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity.
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with D. Valdebenito, Existence of quasiperiodic solutions of elliptic equations on  R^{N+1} via center manifold and KAM theorems, J. Differential Equations 262 (2017),  6109–6164.
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Convergence and quasiconvergence properties of solutions of parabolic equations on the real line: an overview.  Proceedings of the conference "Patterns of Dynamics" (Berlin, July 25-29, 2016), to appear
 Abstract    Preprint.pdf
Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations, SIAM J. Math. Analysis, to appear
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with H. Matano, An entire solution of a bistable parabolic equation on R with two colliding pulses, J. Funct. Anal. 272 (2017), 1956-1979
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Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R, Mem. Amer. Math. Soc., to appear
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Propagating terraces in a proof of the Gibbons conjecture and related results, J. Fixed Point Theory Appl. 19 (2017), 113-128.
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with H. Matano, Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences, Comm. Partial Differential Equations 41 (2016), 785–811
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Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations, J. Dynam. Differential Equations 28 (2016), 605-625
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Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations, In Contributions to nonlinear elliptic equations and systems. A tribute to Djairo Guedes de Figueiredo on the occasion of his 80th Birthday, A.N. Carvalho et al. (eds), Springer 2015, 404-423
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Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on R, J. Evol. Equ. 15 (2015), 281-307
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with J. Foldes, Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains, J. Differential Equations 258 (2015), 1859-1888
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with Y. Du, Locally uniform convergence to an equilibrium for nonlinear parabolic equations on R^N, Indiana Univ. Math. J. 64 (2015), 787-824
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with E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Analysis 46 (2014), 3481–3496
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with E. Yanagida, Global unbounded solutions of the Fujita equation in the intermediate range, Math. Ann. 360 (2014), 255–266
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On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains, Discrete Cont. Dynam. Syst. 34 (2014), 2657–2667
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with S. Terracini, Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains, Proc. Amer. Math. Soc. 142 (2014), 1249–1259
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A discussion of nonnegative solutions of elliptic equations on symmetric domains, Nonlinear Partial Di fferential Equations, Dynamical Systems and Their Applications (M. Taniguchi, ed.), RIMS Kokyuroku, vol. 1881, Kyoto University, 2014, pp. 92-106
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Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains, J. Funct. Anal. 262 (2012), 4458-4474
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with J. Foldes, On asymptotically symmetric parabolic equations, Networks and Heterogeneous Media 7 (2012), 673-689
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with Y. Du and R. Peng, The parabolic logistic equation with blow-up initial and boundary values, J. Anal. Math. 118 (2012), 297–316
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On symmetry of nonnegative solutions of elliptic equations. Ann. Inst. H. Poincare Anal. Non Lineaire 29 (2012), 1-19
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with J. Foldes, Convergence to a steady state for asymptotically autonomous semilinear heat equations on R^N. J. Differential Equations 251 (2011), 1903–1922
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Symmetry of nonnegative solutions of elliptic equations via a result of Serrin. Comm. Partial Differential Equations 36 (2011), 657-669.
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Threshold solutions and sharp transitions for nonautonomous parabolic equations on R^N. Arch. Rational Mech. Anal. 199 (2011), 69-97
Abstract    Preprint.pdf    Addendum.pdf
with T. Bartsch and P. Quittner, Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations. J. European Math. Soc. 13 (2011), 219-247
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with J. Foldes, On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete Cont. Dynamical Systems 21 (2009), 329-341
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Symmetry properties of positive solutions of parabolic equations: a survey. In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, Y. Du et al. (eds), World Scientific 2009, 170-208
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with E. Yanagida, Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation. J. Dynam. Differential Equations 21 (2009), 329-341
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with P. Quittner, Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation. Asymptot. Anal., 57 (2008), 125-141
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with V. Sverak, Zeros of complex caloric functions and singularities of complex viscous Burgers equation. J. reine angew. Math. 616 (2008), 205--217
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with J. Huska, Exponential separation and principal Floquet bundles for linear parabolic equations on R^N. Discrete Cont. Dynamical Systems 20 (2008), 81–113
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with J. Huska and M. V. Safonov, Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007), 711-739
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with P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations, Indiana Univ. Math. J. 56 (2007), 879-908.
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with P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems. Duke Math. J. 139 (2007), 555-579
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Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains. Arch. Rational Mech. Anal. 183 (2007), 59-91
Abstract    Preprint.pdf    Addendum.pdf
with Y. Lou and S. Martinez, Loops and branches of coexistence states in a Lotka-Volterra competition model. J. Differential Equations, J. Differential Equations 230 (2006), 720--742
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Symmetry properties of positive solutions of parabolic equations on $R^N$: II. Entire solutions. Comm. Partial Differential Equations 31 (2006): 1615-1638
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with P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation. Nonlinear Analysis, TMA 64 (2006), 1679-1689
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Symmetry properties of positive solutions of parabolic equations on $R^N$: I. Asymptotic symmetry for the Cauchy problem, Comm. Partial Differential Equations 30 (2005), 1567-1593
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with J. Huska and M.V. Safonov, Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations, Proceedings of the 5th International Conference on Dynamical Systems and Differential Equations, Pomona 2004. Disc. Cont. Dynamical Systems, Supplement 2005, pp. 427-435
Abstract   
with M. Fila and H. Matano, Immediate regularization after blow-up. SIAM J. Math. Analysis 37 (2005), 752-776
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with P. Quittner, Liouville-type theorems and complete blow-up for indefinite superlinear parabolic equations, In Nonlinear Elliptic and Parabolic Problems, M. Chipot and J. Escher (eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 64, Birkhauser, Basel, 2005, 391-402
Abstract   
Asymptotic symmetry of positive solutions of parabolic equations, Proceedings of Equadiff 2003, F. Dumortier et al. (eds), World Scientific 2005, 99-108
Abstract   
with M. Fila and H. Matano, Immediate regularization after blow-up. Proceedings of Equadiff 2003, F. Dumortier et al. (eds), World Scientific 2005, 284-289.
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with E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation. J. Differential Equations, 208 (2005), 194-214.
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On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete Contin. Dynamical Systems, 12 (2005), 13-26.
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with J. Huska, The principal Floquet bundle and exponential separation for linear parabolic equations. J. Dynam. Differential Equations, 16 (2004), 347-375.
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with E. Yanagida, Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation. Differential Integral Equations, 17 (2004), 535-548.
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with M.-A. Jendoubi, Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Proc. Royal Soc. Edinburgh 133A (2003), 1137-1153.
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with E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327 (2003), 745-771.
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with V. Hutson, Y. Lou and K. Mischaikow, Competing species near a degenerate limit. SIAM J. Math. Analysis 35 (2003), 453-491.
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On $L^1$-heteroclinic solutions of a semilinear parabolic equation, Proceedings of the Conference on Nonlinear Partial Differential Equations: Diffusion, Cross-diffusion and Related Topics", Ryukoku 2002; Y. Morita et al. (eds.), Ryukoku University, Ryukoku 2003, pp. 62-68.
Abstract   
with J. Busca and M.-A. Jendoubi, Convergence to equilibrium for semilinear parabolic problems in $\R^N$. Comm. Partial Differential Equations 27 (2002), 1793-1814.
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with F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Differential Equations 186 (2002), 586-610.
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Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations, Proc. Equadiff 10, Math. Bohemica 127 (2002), 301-310.
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with M. Fila and H. Matano, Existence of $L^1$-connections between equilibria of a semilinear parabolic equation. J. Dynam. Differential Equations 14 (2002), 463-491.
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Parabolic equations: asymptotic behavior and dynamics on invariant manifolds (survey). Handbook on Dynamical Systems vol. 2, B. Fiedler (ed.), Elsevier, Amsterdam 2002, pp. 835-883.
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with E. Yanagida, Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete Contin. Dynam. Systems 8 (2002), 209-218.
Abstract   
with V. Hutson and K. Mischaikow, The evolution of dispersal rates in a heterogeneous time-periodic environment. J. Math. Biology 43 (2001), 501-533.
Abstract   
Morse indices and bifurcations of positive solutions of $\Delta u+f(u)=0$ on $R^N$. Indiana Univ. Math. J. 50 (2001), 1407-1432.
Abstract   
with W.-M. Ni and E. Yanagida, Monotonicity of stable solutions in shadow systems. Trans. Amer. Math. Soc. 353 (2001), 5057-5069.
Abstract   
with E. Yanagida, Existence of stable subharmonic solutions for reaction-diffusion equations, J. Differential Equations 169 (2001), 255-280.
Abstract   
Stable subharmonic solutions of reaction-diffusion equations, Proc. of Equadiff 99 , B. Fiedler, K. Groger, J. Sprekels (eds), World Scientific, Singapore, 2000, pp. 722-725.
Abstract   
with E. Yanagida, Stable subharmonic solutions and asymptotic behavior in reaction-diffusion equations, E. J. Qualitative Theory Diff. Equations 2000: Proc. 6th Colloquium on Qualitative Properties of Differential Equations No. 22. (2000), pp. 1-10.
Abstract   
with E. Feireisl, Structure of periodic solutions and asymptotic behavior for time- periodic reaction-diffusion equations on R, Adv. Differential Equations 5 (2000), 583-622.
Abstract   
On the dynamics of scalar parabolic equations, in: New Trends in Nonlinear Partial Differential Equations, Y. Morita et al. (eds.), Ryukoku University, Ryukoku 2000, pp. 32-45.
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with M. Fila, Global nonexistence without blow-up for an evolution problem, Math. Zeitschrift 232 (1999), 531-545.
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Persistent saddle-connection in a class of  reaction-diffusion equations,  J. Differential  Equations 156 (1999), 182-210.
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with M. Fila,  Global solutions of a semilinear  parabolic equation,  Adv.   Differential  Equations 4 (1999), 163-196.
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with E. N. Dancer,  Realization of vector fields and dynamics of   spatially  homogeneous   parabolic equations,  Mem. Amer. Math. Soc.  140 (1999), viii+82.
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with P. Brunovsky,  Morse-Smale structure of a generic reaction diffusion equation in higher  space  dimension, J. Differential  Equations. 135 (1997), 129-181.
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Reaction-diffusion equations and realization of gradient vector  fields,  Proc. EQUADIFF 95, L. Magalhaes, C. Rocha  and L. Sanchez (eds.), World Scientific,  Singapore 1998, 197-206.
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with  P. Brunovsky, On the local structure of $\omega$-limit sets of maps, J. Appl. Math. Physics (ZAMP) 48 (1997), 976-986.
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with X.-Y. Chen,  Asymptotic periodicity of positive solutions of reaction-diffusion equations on a ball,  J. reine angew. Math. 472 (1996), 17-51.
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with V.  Sosovicka,  Stable periodic solutions of a spatially homogeneous nonlocal reaction-diffusion equation,  Proc. Royal Soc. Edinburgh 126A (1996), 867-884.
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with K. P. Rybakowski,  Nonconvergent bounded trajectories in semilinear heat equations,  J. Differential  Equations., 124 (1996), 472-494.
Abstract   
with K. P. Rybakowski,  Imbedding vector fields into Dirichlet BVPs,  Annali Scuola Norm. Sup. Pisa XXI (1995), 737-749.
Abstract   
Realization of dynamics of ODE's in scalar parabolic PDE's,  Equadiff 8, Bratislava 1993, Tatra Mountains Math. Publ. 4 (1994), pp. 179-185.
Abstract   
Transversal and nontransversal intersection of stable  and unstable manifolds of reaction-diffusion  equations on symmetric domains,  Differential Integral Equations 7 (1994), 1527-1545.
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with X.-Y. Chen,  Gradient-like structure and Morse decompositions  for time periodic one-dimensional parabolic equations, J. Dynam. Differential Equations. 7 (1995), 73-107.
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with P. Hess,  Symmetry and convergence properties for nonnegative solutions of nonautonomous   reaction- diffusion problems,  Proc. Royal Soc. Edinburgh 124A (1994), 573-587.
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with A. Haraux, Convergence to a positive equilibrium for some nonlinear evolution equations, Acta Math. Univ. Comenianae, LXI  (1992), 129-141 
Abstract   
High-dimensional $\omega$-limit sets and chaos in scalar parabolic equations,  J. Differential  Equations. 119 (1995), 24-53 
Abstract   
Dynamics of scalar parabolic equations,  in: Proc. Equadiff 1991, C. Perell\'o, C. Sim\'o, J. Sola-Morales    (eds.),  World Scientific, London 1993, pp 215-224 
with I. Terescak,  Exponential separation and invariant bundles for maps in ordered Banach spaces  with applications to parabolic equations,  J. Dynam. Differential Equations. 5 (1993), 279-303.
Abstract   
with P. Hess,  Boundedness of prime periods of stable cycles and convergence to fixed points in discrete  monotone dynamical systems, SIAM J. Math. Anal. 24 (1993), 1312-1330 
Abstract   
with P. Brunovsky, X. Mora and J. Sola-Morales, Asymptotic behavior of  semilinear elliptic equations on  an unbounded strip,  Acta. Math. Univ. Comenianae LX (1991), 163-183 
with. I. Terescak, Convergence to cycles as a typical asymptotic  behavior in smooth strongly  monotone  discrete-time dynamical systems,  Arch. Rational Mech. Anal. 116 (1991), 339-360.
Abstract   
Realization of any finite jet in a scalar semilinear parabolic equation  on the ball in $R^3$,  Ann. Scuola  Norm. Sup. Pisa, XVIII (1991), 83-102 
Imbedding of any vector field in a scalar semilinear parabolic equations,  Proc. Amer. Math. Soc. 115 (1992), 1001-1008. 
Abstract   
with P. Brunovsky and B. Sandstede, Convergence in general periodic parabolic equations in one space  dimension,  Nonlinear Analysis, TMA, 18 (1992),  209-215 
Abstract   
Complicated dynamics in scalar semilinear parabolic equations in higher   space dimension,  J.  Differential Equations 89 (1991), 244-271 
Existence of unstable sets for invariant sets in compact semiflows.  Applications in order-preserving  semiflows,  Comm. Math. Univ. Carolinae  31 (1990), 263-276 
with B. Fiedler,  Complicated dynamics of scalar reaction diffusion equations with a nonlocal term,  Proc. Royal Soc. Edinburgh, 115  (1990), 263-276.
Domains of attraction of equilibria and monotonicity properties of  convergent  trajectories  in   parabolic  systems admitting strong comparison principle,  J. reine angew. Math. 400 (1989), 32-56. 
Generic properties of strongly monotone semiflows defined by ordinary  and parabolic  differential equations,  in: Colloquia Mathematica Soc.  J. Bolyai 53, North-Holland, Budapest 1990, pp. 519-530. 
Convergence in smooth strongly monotone flows defined by semilinear  parabolic equations,  J. Differential Equations  79 (1989), 89-110. 
with J. Mierczynski,  Group actions on strongly monotone dynamical  systems, Math. Ann. 283 (1989), 1-11. 
with P. Brunovsk\'y, Generic hyperbolicity for    reaction-diffusion    equations on symmetric domains,  J. Applied Math. Physics (ZAMP) 38    (1987), 172-183. 
Generic hyperbolicity in one-dimensional   reaction-diffusion equation   with general boundary conditions, Nonlinear Analysis, TMA, 11 (1987), 593-597. 


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