Yūki Naito

Nonexistence results of positive entire solutions for quasilinear elliptic inequalities

This paper treats the quasilinear elliptic inequality div(jDujm 2Du) 1⁄2 p(x)uõÒ x 2 RN Ò where N 1⁄2 2, m Ù 1, õ Ù m 1, and p:RN ! (0Ò1) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results. Received by the editors January 4, 1996. AMS subject classification: 35J70, 35B05. c Canadian Mathematical Society 1997. 244

An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations

The Cauchy problem for semi-linear heat equations with singular initial data is studied, where N > 2, p > ( N + 2)/ N , and l > 0 is a parameter. We establish the existence and multiplicity of positive self-similar solutions for the problem by applying the ordinary differential equation shooting method to the corresponding spatial profile problem.

Existence of self-similar solutions to a parabolic system modelling chemotaxis

We investigate a semilinear elliptic equation with a parameter λ > 0 and a constant 0 < ε < 2, and obtain a structure of the pair (λ, υ) of a parameter and a solution which decays at infinity. This equation arises in the study of self-similar solutions for the Keller-Segel system. Our main results are as follows: (i) There exists a λ* > 0 such that if 0 < λ < λ*, (SE) has two distinct solutionsυλ and guλ satisfyingυλ < guλ, and that if λ > λ*, (SE) has no solution, (ii) If λ = λ* and 0 < ε < 1, (SE) has the unique solution υ*; (iii) The solutionsυλ and υλ are connected through υ*.

Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at .

A VARIATIONAL APPROACH TO SELF-SIMILAR SOLUTIONS FOR SEMILINEAR HEAT EQUATIONS

Such self-similar solutions are global in time and often used to describe the large time behavior of global solutions to (1), see, e.g., [14, 15, 5, 21]. If w(x, t) is a self-similar solution of (1.1) and has an initial value A(x), then we easily see that A has the form A(x) = A(x/|x|)|x|−2/(p−1). Then the problem of existence of self-similar solutions is essentially depend on the solvablity of the Cauchy problem (1)-(2)λ. In this talk we consider the existence of self-similar solutions of the problem (1)-(2)λ. The idea of constructing self-similar solutions by solving the initial value problem for homogeneous initial data goes back to the study by Giga and Miyakawa [12] for the Navier-Stokes equation in vorticity form. It is well known by Fujita [9] that if 1 < p ≤ (N +2)/N then (1) has no time global solution w such that w ≥ 0 and w ≡ 0. (See also [25, 14].) Then the condition p > (N + 2)/N is necessary for the existence of positive self-similar solutions of (1).