195 research outputs found
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)
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
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
Global existence for a hydrogen storage model with full energy balance
A thermo-mechanical model describing hydrogen storage by use of metal
hydrides has been recently proposed in a paper by Bonetti, Fr\'emond and
Lexcellent. It describes the formation of hydrides using the phase transition
approach. By virtue of the laws of continuum thermo-mechanics, the model leads
to a phase transition problem in terms of three state variables: the
temperature, the phase parameter representing the fraction of one solid phase,
and the pressure, and is derived within a generalization of the principle of
virtual powers proposed by Fr\'emond accounting for micro-forces, responsible
for the phase transition, in the whole energy balance of the system. Three
coupled nonlinear partial differential equations combined with initial and
boundary conditions have to be solved. The main difficulty in investigating the
resulting system of partial differential equations relies on the presence of
the squared time derivative of the order parameter in the energy balance
equation. Here, the global existence of a solution to the full problem is
proved by exploiting known and sharp estimates on parabolic equations with
right hand side in L^1. Some complementary results on stability and steady
state solutions are also given.Comment: Key-words: phase transition model; hydrogen storage; nonlinear
parabolic system; existenc
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