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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.
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.
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.
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.
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.
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.
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.
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.
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.
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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