Robert Stańczy (2006)
Banach Center Publications
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A priori estimates for solutions of a system describing the interaction of gravitationally attracting particles with a self-similar pressure term are proved. The presented theory covers the case of the model with diffusions that obey either Fermi-Dirac statistics or a polytropic one.
Piotr Biler, Tadeusz Nadzieja (1995)
Applicationes Mathematicae
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Radially symmetric solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles in a bounded container are studied. Conditions ensuring either global-in-time existence of solutions or their finite time blow up are given.
Piotr Biler, Lorenzo Brandolese (2006)
Colloquium Mathematicae
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We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and Debye-Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method due to S. Montgomery-Smith.
Ignacio Guerra (2004)
Banach Center Publications
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In this article we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see e.g. M. P. Brenner et al. 1999, Nonlinearity 12, 1071-1098); one type is self-similar, and may be viewed as a trade-off between...
Andrzej Raczyński (2004)
Banach Center Publications
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We study the existence of solutions to a nonlinear parabolic equation describing the temporal evolution of a cloud of self-gravitating particles with a given external potential. The initial data are in spaces of (generalized) pseudomeasures. We prove existence of local and global-in-time solutions, and also a kind of stability of global solutions.
Marras, M. (2011)
Serdica Mathematical Journal
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2000 Mathematics Subject Classification: 35K55, 35K60. We investigate the blow-up of the solutions to a nonlinear parabolic system with Robin boundary conditions and time dependent coefficients. We derive sufficient conditions on the nonlinearities and the initial data in order to obtain explicit lower and upper bounds for the blow up time t*.
Zhou, Jun (2007)
Surveys in Mathematics and its Applications
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Andrzej Raczyński (2005)
Applicationes Mathematicae
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We study existence and nonexistence of solutions (both stationary and evolution) for a parabolic-elliptic system describing the electrodiffusion of ions. In this model the evolution of temperature is also taken into account. For stationary states the existence of an external potential is also assumed.
Grzegorz Karch, Kanako Suzuki (2011)
Applicationes Mathematicae
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A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.
Piotr Biler, Danielle Hilhorst, Tadeusz Nadzieja (1994)
Colloquium Mathematicae
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We study the existence and nonexistence in the large of radial solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles. The blow-up of solutions defined in the n-dimensional ball with large initial data is connected with the nonexistence of radial stationary solutions with a large mass.
Jens A. Griepentrog (2004)
Banach Center Publications
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A nonlocal model of phase separation in multicomponent systems is presented. It is derived from conservation principles and minimization of free energy containing a nonlocal part due to particle interaction. In contrast to the classical Cahn-Hilliard theory with higher order terms this leads to an evolution system of second order parabolic equations for the particle densities, coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction...
L.A. Peletier, B. Kawohl (1989)
Mathematische Zeitschrift
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Zhou, Jun, Mu, Chunlai (2010)
Electronic Journal of Qualitative Theory of Differential Equations [electronic only]
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Rossi, J.D. (1998)
Acta Mathematica Universitatis Comenianae. New Series
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Youpeng Chen, Baozhu Zheng (2015)
Annales Polonici Mathematici
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This paper deals with the blow-up properties of positive solutions to a localized singular parabolic equation with weighted nonlocal nonlinear boundary conditions. Under certain conditions, criteria of global existence and finite time blow-up are established. Furthermore, when q=1, the global blow-up behavior and the uniform blow-up profile of the blow-up solution are described; we find that the blow-up set is the whole domain [0,a], including the boundary, in contrast to the case of...
Piotr Biler (1995)
Studia Mathematica
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The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.
Fila, M., Filo, J.
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Carole Rosier, Lionel Rosier (2004)
Banach Center Publications
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Results on the global existence and uniqueness of variational solutions to an elliptic-parabolic problem occurring in statistical mechanics are provided.
Pierre-Henri Chavanis (2004)
Banach Center Publications
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We introduce a new class of nonlocal kinetic equations and nonlocal Fokker-Planck equations associated with an effective generalized thermodynamical formalism. These equations have a rich physical and mathematical structure that can describe phase transitions and blow-up phenomena. On general grounds, our formalism can have applications in different domains of physics, astrophysics, hydrodynamics and biology. We find an aesthetic connexion between topics (stars, vortices, bacteries,...)...