Graduate School of Science and Engineering, Mathematics, Physics, and Earth Sciences, Graduate School of Science and Engineering Mathematics,Physics, and Earth Sciences

Journal of Differential Equations 257 2430-2463 Oct 2014

We consider the semilinear elliptic equation δu+K(|x|)up=0 in RN for N>2 and p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, a...

Global attractivity and convergence rate in the weighted norm for a supercritical semilinear heat equation

NAITO Yuki

Differential Integral Equations 28 777-800 2015 [Refereed]

International Journal of Differential Equations 2013 Nov 2013

We study a new kind of asymptotic behaviour near t = 0 for the nonautonomous system of two linear differential equations: x ' (t) = A (t) x (t), t ε (0, t0], where the matrix-valued function A = A (t) has a kind of singularity at t = 0. It is call...

Communications on Pure and Applied Analysis 12 1861-1880 Sep 2013

In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simplified version of a chemotaxis model, and is also a model of self-interacting particles. The behavior of soluti...

Journal of Differential Equations 253 3029-3060 Dec 2012

We consider the Cauchy problem, where N>2, p>1, and u 0 is a bounded continuous non-negative function in R N. We study the case where u 0(x) decays at the rate |x| -2/(p-1) as |x|→∞, and investigate the stability and instability properties of forw...

Discrete and Continuous Dynamical Systems 32 3691-3713 Jul 2012

We consider a parabolic-elliptic system of equations that arises in modelling the chemotaxis in bacteria and the evolution of self-attracting clusters. In the case space dimension 3 ≤ N ≤ 9, we will derive criteria of the blow-up rate of solutions...

Bulletin of the London Mathematical Society 44 545-562 Jun 2012

We consider conditionally oscillatory second-order linear differential equations with a parameter, and investigate the asymptotic behaviour and number of zeros of solutions to the equations. In particular, we find criteria for the equations to be ...

Annali di Matematica Pura ed Applicata 191 25-51 Jan 2012

We consider the non-homogeneous critical semilinear elliptic problems where Ω is a bounded smooth domain in R N, f ε H -1 (ω), f ≥ 0 in Ω, ε R is a fixed constant, and λ > 0 is a parameter. We investigate the multiplicity of positive solutions to ...

Discrete and Continuous Dynamical Systems- Series A 1111-1118 Sep 2011

In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simplified version of a chemotaxis model, and is also a model of self-interacting particles. The behavior of soluti...

We consider the boundary value problem involving the one dimensional p- Laplacian, and establish the precise intervals of the parameter for the existence and nonexistence of solutions with prescribed numbers of zeros. Our argument is based on the ...

Nonlinear Analysis, Theory, Methods and Applications 69(9) 3070-3083 Nov 2008

We consider the boundary value problem involving the one-dimensional p-Laplacian (| u′ |p - 2 u′)′ + a (x) f (u) = 0, 0 < x < 1, u (0) = u (1) = 0, where p > 1. We establish sharp conditions for the existence of solutions with prescribed numbers o...

Indiana University Mathematics Journal 57(3) 1283-1315 Aug 2008

The Cauchy problem for a semilinear heat equation with singular initial data (Equation Presented) is studied, where N>2,p = (N + 2) / (N - 2), λ > O is a parameter, and a ≥ 0, a ≄ 0. We show that there exists a constant λ* > 0 such that the proble...

Journal of Differential Equations 235(2) 439-483 Apr 2007

We consider the existence of solutions to the semilinear elliptic problem(*)κ{(- Δ u + u = up + κ ∑i = 1 m ci δai in D′ (RN),; u > 0 a.e. in RN and u (x) → 0 as | x | → ∞,). with prescribed given finite points {ai}i = 1 m in RN and positive numbe...

Journal of Differential Equations 232(1) 176-211 Jan 2007

We consider the blowup rate of solutions for a semilinear heat equationut = Δ u + | u |p - 1 u, x ∈ Ω ⊂ RN, t > 0, with critical power nonlinearity p = (N + 2) / (N - 2) and N ≥ 3. First we investigate the profiles of backward self-similar solutio...

Royal Society of Edinburgh - Proceedings A 136(4) 807-835 Sep 2006

The Cauchy problem for semi-linear heat equations with singular initial data wt = wp + wp in ℝN × (0, ∞) and w(x, 0) = ℓ|x|-2/(p-1) in ℝ N \ {0}, is studied, where N >2, p > (N + 2)/N, and ℓ > 0 is a parameter. We establish the existence and multi...

A variational approach to self-similar solutions for semilinear heat equations

Advanced Studies in Pure Mathematics, Asymptotic Analysis and Singuralities 47(2) 675-688 2007

The Cauchy problem for semilinear heat equations with singular initial data wt = Δw + wp in RN × (0, ∞) and w(x, 0) = λa (x/|x|) |x|-2/(p-1) in RN \ {0} is studied, where N ≥ 2, λ > 0 is a parameter, and a > 0, a ≢ 0. We show that when p > (N + 2)...

Taiwanese Journal of Mathematics 8(1) 43-55 Dec 2004

We study the forward self-similar solutions to a parabolic-elliptic system ut = Δu - ∇ · (u∇v), 0 = Δv + u in the whole space R2. First it is proved that self-similar solutions (u, v) must be radially symmetric about the origin. Then the structure...

Asymptotically self-similar solutions for the parabolic systems modelling chemotaxis

Banach Center Publ.Self-similar solutions of nonlinear PDE, 74 149-160 2006

Nonlinear Analysis, Theory, Methods and Applications 56(6) 919-935 Mar 2004

We consider the boundary value problem for nonlinear second-order differential equations of the form u″ + a(x)f(u) = 0, 0 < x < 1, u(0) = u(1) = 0. We establish the precise condition concerning the behavior of the ratio f(s)/s at infinity and zero...

Journal of Differential Equations 184(2) 386-421 Sep 2002

We study the forward self-similar solutions to a parabolic system modeling chemotaxis ut = ∇ · (∇u - u∇v), τvt = Δv + u in the whole space ℝ2, where τ is a positive constant. Using the Liouville-type result and the method of moving planes, it is p...

Nonlinear Analysis, Theory, Methods and Applications 47(6) 3661-3670 Aug 2001

The radial symmetry of classical solutions for semilinear elliptic equations is studied. An approach based on the maximum principle in unbounded domains together with the method of moving plates is presented. This approach helps to develop argumen...

Japan Journal of Industrial and Applied Mathematics 17(3) 427-451 Oct 2000

We investigate a semilinear elliptic equation (SE) -△v - ε/2x · ▽v = λe-1/4|x|2 ev in R2 with a parameter λ > 0 and a constant 0 < ε < 2, and obtain a structure of the pair (λ, v) of a parameter and a solution which decays at infinity. This equati...

Journal of the Mathematical Society of Japan 52(3) 637-644 Jul 2000

We consider the global properties of nonnegative solutions of the semilinear elliptic equations in the entire space. By employing Pohozaev identity in the entire space and the results concerning the asymptotic behavior of nonnegative solutions, we...

Journal of Differential Equations 163(2) 407-428 May 2000

The symmetry properties of positive solutions of the equation Δu+1/x · ∇u+ 1/p-1 u + up = 0 in Rn, where n ≥ 2, p > (n + 2)/n, was studied. It was proved that u must be radially symmetric about the origin provided u(x) = o(|x|-2/(p-1)) as \x\ → ∞,...

Pacific Journal of Mathematics 189(1) 107-115 May 1999

We treat the Dirichlet problem for elliptic equations on annular regions, and show the monotonicity and symmetry properties of positive solutions with respect to the sphere. We generalize the argument of the method of moving spheres to more genera...

Radial symmetry of positive solutions for semilinear elliptic equations in R^n

Journal of the Korean Mathematical Society 37(5) 751-761 2000

Canadian Mathematical Bulletin 40 244-253 Jun 1997

This paper treats the quasilinear elliptic inequality div(|Du|m-2 Du) ≥ p(x)uσ, x ∈ ℝN, where N ≥ 2, m > 1, σ > m - 1, and p:ℝN → (0, ∞) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When ...