CPAA
Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation
Shota Sato
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.
keywords: Semilinear parabolic equation blow-up at space infinity moving singularity critical exponent.
CPAA
Appearance of anomalous singularities in a semilinear parabolic equation
Shota Sato Eiji Yanagida
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.
keywords: backward self-similar solution singular solution Semilinear parabolic equation critical exponent.
DCDS-S
Singular backward self-similar solutions of a semilinear parabolic equation
Shota Sato Eiji Yanagida
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.
keywords: critical exponent. Semilinear parabolic equation backward self-similar solution singular solution
DCDS
Forward self-similar solution with a moving singularity for a semilinear parabolic equation
Shota Sato Eiji Yanagida
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.
keywords: critical exponent. Semilinear parabolic equation moving singularity forward self-similar
DCDS
Asymptotic behavior of singular solutions for a semilinear parabolic equation
Shota Sato Eiji Yanagida
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.
keywords: critical exponent. Semilinear parabolic equation singular solution convergence to a steady state

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