12,527 research outputs found
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity
We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian
Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions
We investigate blow-up phenomena for positive solutions of nonlinear
reaction-diffusion equations including a nonlinear convection term in a bounded domain of
under the dissipative dynamical boundary conditions . Some conditions on and are discussed
to state if the positive solutions blow up in finite time or not. Moreover, for
certain classes of nonlinearities, an upper-bound for the blow-up time can be
derived and the blow-up rate can be determinated.Comment: 20 page
Boundary fluxes for non-local diffusion
We study a nonlocal diffusion operator in a bounded smooth domain prescribing
the flux through the boundary. This problem may be seen as a generalization of
the usual Neumann problem for the heat equation. First, we prove existence,
uniqueness and a comparison principle. Next, we study the behavior of solutions
for some prescribed boundary data including blowing up ones. Finally, we look
at a nonlinear flux boundary condition
Nonlinear Schr\"odinger Equation with a White-Noise Potential: Phase-space Approach to Spread and Singularity
We propose a phase-space formulation for the nonlinear Schr\"odinger equation
with a white-noise potential in order to shed light on two issues: the rate of
spread and the singularity formation in the average sense. Our main tools are
the energy law and the variance identity. The method is completely elementary.
For the problem of wave spread, we show that the ensemble-averaged dispersion
in the critical or defocusing case follows the cubic-in-time law while in the
supercritical and subcritical focusing cases the cubic law becomes an upper and
lower bounds respectively.
We have also found that in the critical and supercritical focusing cases the
presence of a white-noise random potential results in different conditions for
singularity-with-positive-probability from the homogeneous case but does not
prevent singularity formation. We show that in the supercritical focusing case
the ensemble-averaged self-interaction energy and the momentum variance can
exceed any fixed level in a finite time with positive probability
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