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### Open Access Journals

CPAA

We consider 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 solution with a moving singularity
that becomes anomalous in finite time.
Our concern is a blow-up solution with a moving singularity.
In this paper, we show that there exists a solution with a moving singularity such that
it blows up at space infinity.

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

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.

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