prof. dr hab. Grzegorz Karch

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List of all publications

1. G. Karch, D. Pilarczyk, M.E. Schonbek 
   L^2-asymptotic stability of mild solutions to Navier-Stokes system (2013), preprint.
   See arXiv:1308.6667 [math.AP]
2. D. Iftimie, G. Karch, and Ch. Lacave 
   Asymptotics of solutions to the Navier-Stokes system in exterior domains, (2013), submitted.
   See arXiv:1307.7837 [math.AP]
3. M. Cannone and G. Karch 
   On self-similar solutions to the homogeneous Boltzmann equation, Kinetic and Related Models, Vol. 6, Issue 4 (2013), 801-808 
   Preprint in PDF format
4. P. Biler, C. Imbert, G. Karch 
   Nonlocal porous medium equation: Barenblatt profiles and other weak solutions,(2013), 28 pages, submitted. 
   See arXiv:1302.7219 [math.AP]
5. A. Marciniak-Czochra, S. Härting, G. Karch, and K. Suzuki 
   Dynamical spike solutions in a nonlocal model of pattern formation. (2013), 26 pages. submitted. 
   See arXiv:1307.6236 [math.AP]
6. A. Marciniak-Czochra, G. Karch, and K. Suzuki 
   Unstable patterns in autocatalytic reaction-diffusion-ODE systems, (2013), 30 pages. submitted. 
   See arXiv:1301.2002 [math.AP] 
7. A. Marciniak-Czochra, G. Karch, and K. Suzuki 
   Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl. (9) 99 (2013), no. 5, 509–543.
   The preprint of this work ( PDF file) is slightly longer comparing with the published version. Unpublished parts are denoted by blue color. 
   See also http://arxiv.org/abs/1104.3592
   DOI: http://dx.doi.org/10.1016/j.matpur.2012.09.011
8. D. Iftimie, G. Karch, and Ch. Lacave 
   Self-similar asymptotics of solutions to the Navier-Stokes system in two dimensional exterior domain, (2011) 1-13. 
   This paper will be never published and it exists in the preprint version, only. The result from this paper is a particular case of our more general work (see above) which is submitted for publication, now. "pdf". 
   See also http://arxiv.org/abs/1107.2054
9. (with Piotr Biler and Cyril Imbert) 
   Barenblatt profiles for a nonlocal porous medium equation, C. R. Acad. Sci. Paris, Ser. I 349 (2011) 641–645. "pdf". 
   See also http://arxiv.org/abs/1001.0910
   DOI: 10.1016/j.crma.2011.06.003
10. (with Dominika Pilarczyk) 
    Asymptotic Stability of Landau Solutions to Navier–Stokes System, Arch. Rational Mech. Anal. (2011) "pdf". 
    See also http://arxiv.org/abs/1104.3589
    DOI 10.1007/s00205-011-0409-z
11. (with Kanako Suzuki) 
    Spikes and diffusion waves in a one-dimensional model of chemotaxis, Nonlinearity 23 (2010), no. 12, 3119-3137 "pdf". 
    See also http://arxiv.org/abs/1008.0020
    DOI 10.1088/0951-7715/23/12/007
12. (with Kanako Suzuki) 
    Blow-up versus global existence of solutions to aggregation equations,APPLICATIONES MATHEMATICAE 38,3 (2011), 243–258. "pdf". 
    See also http://arxiv.org/abs/1004.4021
13. (with Marco Cannone) 
    Infinite energy solutions to the homogeneous Boltzmann equation, Comm. Pure Appl. Math. 63 (2010), no. 6, 747-778. "pdf". 
    See also http://arxiv.org/abs/0907.1676
14. Nonlinear evolution equations with anomalous diffusion, "pdf". 
    Lecture Notes presented at EVEQ 2008, International Summer School on Evolution Equations Prague; Lecture notes of the Neas Center for Mathematical Modeling MATFYZPRESS Publishing House of the Faculty of Mathematics and Physics Charles University Prague. "pdf". 
15. (with Cyril Imbert and Nathel Alibaud) 
    Asymptotic properties of entropy solutions to fractal Burgers equation, SIAM J. Math. Anal. 42 (2010), no. 1, 354-376. "pdf". 
    See also http://arxiv.org/abs/0903.3394
16. (with Piotr Biler and Philippe Laurecot)
    Blowup of solutions to a diffusive aggregation model, Nonlinearity 22 (2009), 1559-1568. "pdf".
    See also http://arxiv.org/abs/0903.2915
17. (with Piotr Biler)
    Blow up of solutions to generalized Keller-Segel model, J. Evol. Equ. 10 (2010), no. 2, 247-262. "pdf".
    See also http://arxiv.org/abs/0812.4982
18. (with Ahmad Fino)
    Decay of mass for nonlinear equation with fractional Laplacian, Monatshefteur Mathematik 160 (2010) , no. 4, 375-384. "pdf"
    See also http://arxiv.org/abs/0812.4977
    DOI 10.1007/s00605-009-0093-3
19. (with Piotr Biler and Regis Monneau)
    Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys. 294 (2010), no. 1, 145-168. "pdf". 
    See also http://arxiv.org/abs/0812.4979
20. (with Lorenzo Brandolese)
    Far field asymptotics of solutions to convection equation with anomalous diffusion,J. Evolution Equations 8 (2008), 307--326. "pdf". 
    See also http://arxiv.org/abs/0801.1884
21. (with Changxing Miao and Xiaojing Xu)
    On convergence of solutions of fractal Burgers equation toward rarefaction waves,SIAM J. Math. Anal. 39 (2008), 1536.1549. "pdf". 
    See also http://arxiv.org/abs/math/0702088
22. (with N. Prioux)
    Self-similarity in viscous Boussinesq equation, Proc. Amer. Math. Soc. 136 (2008) 879-888. "pdf".
23. (with J. Dolbeault)
    Large time behaviour of solutions to nonhomogeneous diffusion equations, 133--147; in: Self-Similar Solutions in Nonlinear PDE, Banach Center Publications 74, Polish Acad. Sci., Warsaw, 2006. "pdf".
24. (with W.A. Woyczynski)
    Fractal Hamilton-Jacobi-KPZ equations, Trans. Amer. Math. Soc. 360 (2008), 2423--2442. "dvi" , "ps", "pdf". 
    See also http://arxiv.org/abs/math/0601712
25. (with P. Biler, Ph. Laurencot, T. Nadzieja)
    The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane. Math. Methods in the Applied Sci. 29 (2006), 1563-1583. "pdf".
26. (with P. Biler, Ph. Laurencot, T. Nadzieja)
    The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 133--147. "dvi" , "ps", "pdf".
27. (with M. Cannone)
    On the validity of Picard algorithm for nonlinear parabolic equations , Proceedings of the Royal Society of Edinburgh 135A (2005), 947-958. "dvi" , "ps", "pdf".
28. (with P. Biler, M. Cannone)
    Asymptotic stability of Navier--Stokes flow past an obstacle, "Nonlocal elliptic and parabolic problem", Banach Center Publ. 66 (2004), 47-59. "dvi" , "ps", "pdf".
29. (with S. Benachour & Ph. Laurencot)
    Asymptotic profiles of solutions to convection-diffusion equations , C. R. Acad. Sci. Paris, Ser. I 338 (2004), 369--374. "dvi", "ps", "pdf".
30. (with S. Benachour & Ph. Laurencot)
    Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, , J. Math. Pures Appl. 83 (2004) 1275-1308. "dvi", "ps", "pdf".
31. (with M. Cannone)
    About the regularized Navier--Stokes equations, Journal of Mathematical Fluid Mechanics 7, No. 1 (2005), 1 - 28. "dvi", "ps", "pdf".
32. (Lecture notes in Polish): Rownanie ciepla
    "dvi" , "ps", "pdf".
33. (with P. Biler, M. Cannone, I. Guerra)
    Global regular and singular solutions for a model of gravitating particles, , Mathematische Annalen 330 No. 4 (2004), 693 - 708. "dvi" , "pdf", "ps".
34. (with M. Cannone)
    Smooth or singular solutions to the Navier--Stokes system ?, Journal of Differential Equations 197 No. 2 (2004), 247--274. "dvi" , "ps".
35. (in Polish) O szóstym problemie milenijnym: istnienie i regularność rozwiązań układu Naviera-Stokesa , (Sixth millennium problem: existence and regularity of solutions of the Navier-Stokes system). (Polish) Wiadom. Mat. 38 (2002), 121--130. "dvi", "ps".
36. (with P. Biler)
    Generalized Fokker--Planck equations and convergence to their equilibria, Evolution equations (Warsaw, 2001), 307--318, Banach Center Publ., 60, Polish Acad. Sci., Warsaw."dvi", "ps".
37. (with M. Cannone)
    Incompressible Navier-Stokes equations in abstract Banach spaces , (2001), in"Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics", Edited by H. Fujita, S. T. Kuroda, and H. Okamoto, Printed by UTP Production Center, distributed byYurinsha. "dvi" , "ps".
38. (with M.E. Schonbek)
    On zero mass solutions of viscous conservation laws, Comm. Partial Differential Equations 27 (2002), no. 9-10, 2071--2100. "dvi", "ps".
39. (with P. Biler and M. Guedda)
    Asymptotic properties of solutions of the viscous Hamilton - Jacobi equation , Journal of Evolution Equations 4 No. 1 (2004), 75 - 97. "dvi", "ps".
40. (in Polish) Autoreferat rozprawy habilitacyjnej: Automorficzna asymptotyka rozwiazań pewnych nieliniowych równań ewolucyjnych. , "pdf".
    Habilitation in Mathematical Institute, University of Wroclaw, October 27th, 2001. Title: Selfsimilar asymptotics of some nonlinear evolution equations.
41. Scaling in nonlinear parabolic equations: applications to Debye system , (2000) 1 - 6, Conference Proceedings: Disordered and Complex Systems, London, 10-14 July 2000, American Institute of Physics Publishing. "dvi", "ps".
42. (with P. Biler, J. Dolbeault, and M. Esteban)
    Stationary solutions, intermediate asymptotics and large time behaviour of type II Streater's models , Advances in Differential Equations 6, No. 4 (2001), 461-480. "dvi", "ps".
43. Selfsimilar profiles in large time asymptotics of solutions to damped wave equations , Studia Mathematica 143 (2000), 175-197. "dvi", "ps". MR 2001k:35209
44. (with P. Biler and W.A. Woyczynski)
    Asymptotics and High Dimensional Approximations for Nonlinear Pseudodifferential Equations Involving Levy Generators, , Demonstratio Math. 34 (2001), no. 2, 403--413. This is a survey on results from my papers with P. Biler and W. A. Woyczynski, "dvi", "ps".
45. (with P. Biler and W.A. Woyczynski)
    Critical nonlinearity exponent and self-similar asymptotics for Levy conservation laws , Ann. I.H. Poincare - Analyse non lineare, 18, No. 5, (2001), 613-637. "dvi", "ps".
46. (with P. Biler and W.A. Woyczynski)
    Multifractal and Levy conservation laws , C. R. Acad. Sci. Paris, t. 330, Serie I, (2000) 343-348, "dvi", "ps".
47. (with P. Biler and W.A. Woyczynski)
    Asymptotics for conservation laws involving Levy diffusion generators , Studia Math. 148 (2001), no. 2, 171--192. "dvi", "ps".
48. (with P. Biler)
    A Neumann problem for a convection-diffusion equation on the half-line, Annales Polonici Mathematici (the volume in memory of Bogdan Ziemian), 74 (2000), 79-95. "dvi", "ps"."fig1.ps". MR 2002b:35101
49. Long-time asymptotics of solutions to some nonlinear wave equations, Evolution Equations: Existence, Regularity and Singularities (Warsaw 1998), 147-152, Banach Center Publ., 52, Polish Acad. Sci., Warsaw, 2000. "dvi", "ps". MR 2001m:35272
50. Asymptotics of solutions to a convection-diffusion equation on the half-line, Proc. Royal Society of Edinburgh 130A (2000) 837-853. "dvi", "ps". MR 2001h:35102
51. (with P. Biler and W.A. Woyczynski)
    Asymptotics for multifractal conservation laws , Studia Mathematica 135 (3) (1999), 231-252. "dvi", "ps". MR 2000h:35098
52. Large-time behavior of solutions to nonlinear wave equations: higher-order asymptotics, Mathematical Methods in the Applied Sciences 22 Issue 18 (1999) 1671-1697."dvi", "ps". MR 2000k:35246
53. Scaling in nonlinear parabolic equations, Journal of Mathematical Analysis and Applications 234 (1999) 534-558. "dvi", "ps", MR 2000h:35068
54. Self-similar Large Time Behavior of Solutions to Korteweg-de Vries-Burgers Equation, Nonlinear Analysis 35 (1999) 199-219. "dvi" , "ps", MR 99h:35188
55. (with T. Ricciardi)
    Lorentz spaces and differentiability of weak solutions to elliptic equations, Bulletin of the Polish Academy of Sciences  45 no. 1 (1997) 111-116 . "dvi", "ps". MR 97m:35047
56. Asymptotic behaviour of solutions to some pseudoparabolic equations, Mathematical Methods in the Applied Sciences, 20 (1997), 271-289. "dvi", "ps". MR 97m:35154
57. $L^p$-decay of solutions to dissipative-dispersive perturbations of conservation laws, Annales Polonici Mathematici 67 no.1 (1997) 65--86. "dvi" , "ps" . MR 98b:35082
58. (with J. Dziubanski) 
    Nonlinear scattering for some dispersive equations generalizing Benjamin-Bona-Mahony equation, Monatshefte fur Mathematik 122 (1996) 35--43. "dvi", "ps". MR 97e:35158

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