Publications of Peter Polacik

 with H. Matano, Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities Abstract    Preprint.pdf with P. Quittner, On the multiplicity of self-similar solutions of the semilinear heat equation Abstract    Preprint.pdf with D. Valdebenito, Existence of partially localized quasiperiodic solutions of homogeneous elliptic equations on R^(N+1), Annali Scuola Norm. Sup. Pisa,  to appear Abstract    Preprint.pdf Entire solutions and a Liouville theorem for a class of parabolic equations on the real line Abstract    Preprint.pdf with D. Valdebenito, Some generic properties of Schrodinger operators with radial potentials, Proc. Royal Soc. Edinburgh, to appear Abstract    Preprint.pdf with A. Pauthier, Large-time behavior of solutions of parabolic equations on the real line with convergent initial data, Nonlinearity 31 (2018), 4423-4441. Abstract    Preprint.pdf with D. Valdebenito,  Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity, Discrete Cont. Dynamical Syst., Ser. S, to appear Abstract    Preprint.pdf 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. Abstract    Preprint.pdf Convergence and quasiconvergence properties of solutions of parabolic equations on the real line: an overview.   In "Patterns of Dynamics,"  pp. 172–183, Springer Proc. Math. Stat. 205, Springer, Cham, 2017.  Abstract    Preprint.pdf Planar propagating terraces and the asymptotic one-dimensional symmetry of solutions of semilinear parabolic equations, SIAM J. Math. Anal. 49 (2017),  3716--3740. Abstract    Preprint.pdf with H. Matano, An entire solution of a bistable parabolic equation on R with two colliding pulses, J. Funct. Anal. 272 (2017), 1956-1979 Abstract    Preprint.pdf Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R, Mem. Amer. Math. Soc., to appear Abstract    Preprint.pdf Propagating terraces in a proof of the Gibbons conjecture and related results, J. Fixed Point Theory Appl. 19 (2017), 113-128. Abstract    Preprint.pdf 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 Abstract    Preprint.pdf Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations, J. Dynam. Differential Equations 28 (2016), 605-625 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on R, J. Evol. Equ. 15 (2015), 281-307 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf with Y. Du, Locally uniform convergence to an equilibrium for nonlinear parabolic equations on R^N, Indiana Univ. Math. J. 64 (2015), 787-824 Abstract    Preprint.pdf with E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Analysis 46 (2014), 3481–3496 Abstract    Preprint.pdf with E. Yanagida, Global unbounded solutions of the Fujita equation in the intermediate range, Math. Ann. 360 (2014), 255–266 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf 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 Abstract     Preprint.pdf 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 Abstract    Preprint.pdf Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains, J. Funct. Anal. 262 (2012), 4458-4474 Abstract    Preprint.pdf with J. Foldes, On asymptotically symmetric parabolic equations, Networks and Heterogeneous Media 7 (2012), 673-689 Abstract    Preprint.pdf with Y. Du and R. Peng, The parabolic logistic equation with blow-up initial and boundary values, J. Anal. Math. 118 (2012), 297–316 Abstract    Preprint.pdf On symmetry of nonnegative solutions of elliptic equations. Ann. Inst. H. Poincare Anal. Non Lineaire 29 (2012), 1-19 Abstract    Preprint.pdf with J. Foldes, Convergence to a steady state for asymptotically autonomous semilinear heat equations on R^N. J. Differential Equations 251 (2011), 1903–1922 Abstract    Preprint.pdf Symmetry of nonnegative solutions of elliptic equations via a result of Serrin. Comm. Partial Differential Equations 36 (2011), 657-669. Abstract    Preprint.pdf 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 Abstract    Preprint.pdf with J. Foldes, On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete Cont. Dynamical Systems 21 (2009), 329-341 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf with E. Yanagida, Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation. J. Dynam. Differential Equations 21 (2009), 329-341 Abstract    Preprint.pdf with P. Quittner, Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation. Asymptot. Anal., 57 (2008), 125-141 Abstract    Preprint.pdf with V. Sverak, Zeros of complex caloric functions and singularities of complex viscous Burgers equation. J. reine angew. Math. 616 (2008), 205--217 Abstract    Preprint.pdf with J. Huska, Exponential separation and principal Floquet bundles for linear parabolic equations on R^N. Discrete Cont. Dynamical Systems 20 (2008), 81–113 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf 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. Abstract    Preprint.pdf 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 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf Symmetry properties of positive solutions of parabolic equations on $R^N$: II. Entire solutions. Comm. Partial Differential Equations 31 (2006): 1615-1638 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf 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 Abstract    Preprint.pdf 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 Abstract 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. Abstract with E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation. J. Differential Equations, 208 (2005), 194-214. Abstract On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete Contin. Dynamical Systems, 12 (2005), 13-26. Abstract with J. Huska, The principal Floquet bundle and exponential separation for linear parabolic equations. J. Dynam. Differential Equations, 16 (2004), 347-375. Abstract with E. Yanagida, Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation. Differential Integral Equations, 17 (2004), 535-548. Abstract with M.-A. Jendoubi, Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Proc. Royal Soc. Edinburgh 133A (2003), 1137-1153. Abstract with E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327 (2003), 745-771. Abstract with V. Hutson, Y. Lou and K. Mischaikow, Competing species near a degenerate limit. SIAM J. Math. Analysis 35 (2003), 453-491. Abstract 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. Abstract with F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Differential Equations 186 (2002), 586-610. Abstract Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations, Proc. Equadiff 10, Math. Bohemica 127 (2002), 301-310. Abstract 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. Abstract 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. Abstract 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. Abstract with M. Fila, Global nonexistence without blow-up for an evolution problem, Math. Zeitschrift 232 (1999), 531-545. Abstract

 Persistent saddle-connection in a class of  reaction-diffusion equations,  J. Differential  Equations 156 (1999), 182-210. Abstract with M. Fila,  Global solutions of a semilinear  parabolic equation,  Adv.   Differential  Equations 4 (1999), 163-196. Abstract with E. N. Dancer,  Realization of vector fields and dynamics of   spatially  homogeneous   parabolic equations,  Mem. Amer. Math. Soc.  140 (1999), viii+82. Abstract with P. Brunovsky,  Morse-Smale structure of a generic reaction diffusion equation in higher  space  dimension, J. Differential  Equations. 135 (1997), 129-181. Abstract 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. Abstract with  P. Brunovsky, On the local structure of $\omega$-limit sets of maps, J. Appl. Math. Physics (ZAMP) 48 (1997), 976-986. Abstract    with X.-Y. Chen,  Asymptotic periodicity of positive solutions of reaction-diffusion equations on a ball,  J. reine angew. Math. 472 (1996), 17-51. Abstract with V.  Sosovicka,  Stable periodic solutions of a spatially homogeneous nonlocal reaction-diffusion equation,  Proc. Royal Soc. Edinburgh 126A (1996), 867-884. Abstract 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. Abstract 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. Abstract with P. Hess,  Symmetry and convergence properties for nonnegative solutions of nonautonomous   reaction- diffusion problems,  Proc. Royal Soc. Edinburgh 124A (1994), 573-587. Abstract 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.