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### Open Access Journals

CPAA

The Cauchy problem for a parabolic partial differential
equation with a power nonlinearity is studied.
It is known that in some parameter
range, there exists a time-local solution whose singularity has the same
asymptotics as that of a singular steady state. In this paper, a sufficient condition
for initial data is given for the existence of a solution with a moving singularity
that becomes anomalous in finite time.

DCDS-S

We consider a parabolic partial differential equation with power nonlinearity.
Our concern is the existence of a singular solution whose
singularity becomes anomalous in finite time.
First we study the structure of singular radial
solutions for an equation derived by backward self-similar variables.
Using this, we obtain a singular backward self-similar solution
whose singularity becomes stronger or weaker than that of
a singular steady state.

DCDS

We study the Cauchy problem for a parabolic partial differential
equation with a power nonlinearity.
It was shown in our previous paper that in some parameter range, the problem
has a time-local solution with prescribed moving singularities.
Our concern in this paper is the existence of a time-global solution.
By using a perturbed Haraux-Weissler equation, it is shown that there exists
a forward self-similar solution with a moving singularity.
Using this result, we also obtain a sufficient condition
for the global existence of solutions with a moving singularity.

DCDS

The paper is concerned with stable subharmonic solutions of timeperiodic
spatially inhomogeneous reaction-diffusion equations. We show that such
solutions exist on any spatial domain, provided the nonlinearity is chosen suitably.
This contrasts with our previous results on spatially homogeneous equations that
admit stable subharmonic solutions on some, but not on arbitrary domains.

CPAA

We consider solutions of the heat equation with
time-dependent singularities.
It is shown that a singularity is removable if it is
weaker than the order of
the fundamental solution
of the Laplace equation.
Some examples of non-removable singularities are also given, which
show the optimality of the condition for removability.

DCDS

We consider the Cauchy problem for a parabolic partial differential
equation with a power nonlinearity. It is known that in some range of
parameters, this equation has a family of singular steady states with
ordered structure. Our concern in this paper is the existence of
time-dependent singular solutions and their asymptotic behavior.
In particular, we prove the convergence of solutions to singular
steady states. The method of proofs is based on the analysis of a
related linear parabolic equation with a singular coefficient and the
comparison principle.

CPAA

Our aim in this paper is to prove the instability of multi-spot
patterns in a shadow system,
which is obtained as a limiting system of a reaction-diffusion model as one of the
diffusion coefficients goes to infinity.
Instead of investigating each eigenfunction for a linearized operator,
we characterize the eigenspace spanned by unstable eigenfunctions.

MBE

This paper is concerned with an indefinite weight linear eigenvalue
problem in cylindrical domains. We investigate the minimization of the positive
principal eigenvalue under the constraint that the weight is bounded by
a positive and a negative constant and the total weight is a fixed negative
constant. Biologically, this minimization problem is motivated by the question
of determining the optimal spatial arrangement of favorable and unfavorable
regions for a species to survive. Both our analysis and numerical simulations
for rectangular domains indicate that there exists a threshold value such that
if the total weight is below this threshold value, then the optimal favorable
region is a circular-type domain at one of the four corners, and a strip at the
one end with shorter edge otherwise.

CPAA

We propose a method to investigate the structure of positive radial
solutions to semilinear elliptic problems with various boundary conditions.
It is already shown that the boundary value problems can be reduced to a
canonical form by a suitable change of variables. We show structure theorems
to canonical forms to equations with power nonlinearities and various boundary
conditions. By using these theorems, it is possible to study the properties of
radial solutions of semilinear elliptic equations in a systematic way, and make
clear unknown structure of various equations.

CPAA

We consider positive solutions of Matukuma's
equation, which is described by a nonlinear elliptic equation with a weight.
Any radially symmetric solution of this equation is said to be regular or singular according
to its behavior near the origin and infinity. We investigate the structure of
positive radial regular and singular solutions.

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