199 research outputs found
Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system
Finite time blow-up is shown to occur for radially symmetric solutions to a
critical quasilinear Smoluchowski-Poisson system provided that the mass of the
initial condition exceeds an explicit threshold. In the supercritical case,
blow-up is shown to take place for any positive mass. The proof relies on a
novel identity of virial type
Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent
The large time behavior of non-negative solutions to the viscous
Hamilton-Jacobi equation in the whole space
is investigated for the critical exponent . Convergence
towards a rescaled self-similar solution of the linear heat equation is shown,
the rescaling factor being . The proof relies on the
construction of a one-dimensional invariant manifold for a suitable truncation
of the equation written in self-similar variables.Comment: 17 pages, no figur
The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in \RR^d,
It is known that, for the parabolic-elliptic Keller-Segel system with
critical porous-medium diffusion in dimension \RR^d, (also referred
to as the quasilinear Smoluchowski-Poisson equation), there is a critical value
of the chemotactic sensitivity (measuring in some sense the strength of the
drift term) above which there are solutions blowing up in finite time and below
which all solutions are global in time. This global existence result is shown
to remain true for the parabolic-parabolic Keller-Segel system with critical
porous-medium type diffusion in dimension \RR^d, , when the
chemotactic sensitivity is below the same critical value. The solution is
constructed by using a minimising scheme involving the Kantorovich-Wasserstein
metric for the first component and the -norm for the second component. The
cornerstone of the proof is the derivation of additional estimates which relies
on a generalisation to a non-monotone functional of a method due to Matthes,
McCann, & Savar\'e (2009)
On an age and spatially structured population model for Proteus Mirabilis swarm-colony development
Proteus mirabilis are bacteria that make strikingly regular spatial-temporal
patterns on agar surfaces. In this paper we investigate a mathematical model
that has been shown to display these structures when solved numerically. The
model consists of an ordinary differential equation coupled with a partial
differential equation involving a first-order hyperbolic aging term together
with nonlinear degenerate diffusion. The system is shown to admit global weak
solutions
Weak solutions to the continuous coagulation equation with multiple fragmentation
The existence of weak solutions to the continuous coagulation equation with
multiple fragmentation is shown for a class of unbounded coagulation and
fragmentation kernels, the fragmentation kernel having possibly a singularity
at the origin. This result extends previous ones where either boundedness of
the coagulation kernel or no singularity at the origin for the fragmentation
kernel were assumed
Singular behaviour of finite approximations to the addition model
Instantaneous gelation in the addition model with superlinear rate coefficients is investigated. The conjectured post-gelation solution is shown to arise naturally as the limit of solutions to some finite approximations as the number of equations grows to infinity. Non-existence of continuous solutions to the addition model is also established in that case
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