599 research outputs found
Large solutions of elliptic systems of second order and applications to the biharmonic equation
In this work we study the nonnegative solutions of the elliptic system \Delta
u=|x|^{a}v^{\delta}, \Delta v=|x|^{b}u^{\mu} in the superlinear case \mu
\delta>1, which blow up near the boundary of a domain of R^{N}, or at one
isolated point. In the radial case we give the precise behavior of the large
solutions near the boundary in any dimension N. We also show the existence of
infinitely many solutions blowing up at 0. Furthermore, we show that there
exists a global positive solution in R^{N}\{0}, large at 0, and we describe its
behavior. We apply the results to the sign changing solutions of the biharmonic
equation \Delta^2 u=|x|^{b}|u|^{\mu}. Our results are based on a new dynamical
approach of the radial system by means of a quadratic system of order 4,
combined with nonradial upper estimates
A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension
We study radial solutions in a ball of of a semilinear,
parabolic-elliptic Patlak-Keller-Segel system with a nonlinear sensitivity
involving a critical power. For , the latter reduces to the classical
linear model, well-known for its critical mass . We show that a critical
mass phenomenon also occurs for , but with a strongly different
qualitative behaviour. More precisely, if the total mass of cells is smaller or
equal to the critical mass M, then the cell density converges to a regular
steady state with support strictly inside the ball as time goes to infinity. In
the case of the critical mass, this result is nontrivial since there exists a
continuum of stationary solutions and is moreover in sharp contrast with the
case where infinite time blow-up occurs. If the total mass of cells is
larger than M, then all solutions blow up in finite time. This actually follows
from the existence (unlike for ) of a family of self-similar, blowing up
solutions with support strictly inside the ball.Comment: 35 page
Multiple positive solutions to elliptic boundary blow-up problems
We prove the existence of multiple positive radial solutions to the
sign-indefinite elliptic boundary blow-up problem where is a function superlinear at zero and at infinity,
and are the positive/negative part, respectively, of a sign-changing
function and is a large parameter. In particular, we show how the
number of solutions is affected by the nodal behavior of the weight function
. The proof is based on a careful shooting-type argument for the equivalent
singular ODE problem. As a further application of this technique, the existence
of multiple positive radial homoclinic solutions to is also considered
Blow-up of continuous and semidiscrete solutions to elliptic equations with semilinear dynamical boundary conditions of parabolic type
AbstractIn this paper we present existence of blow-up solutions for elliptic equations with semilinear boundary conditions that can be posed on all domain boundary as well as only on a part of the boundary. Systems of ordinary differential equations are obtained by semidiscretizations, using finite elements in the space variables. The necessary and sufficient conditions for blow-up in these systems are found. It is proved that the numerical blow-up times converge to the corresponding real blow-up times when the mesh size goes to zero
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