arXiv subjects
Publications and source records attributed to Tetsutaro Shibata.
The one-dimensional nonlocal Kirchhoff type bifurcation problems which are derived from logistic equation of population dynamics are studied. We obtain the precise asymptotic shapes of $L^2$ bifurcation curves $λ= λ(α)$ as $α\to \infty$, where $α:= \Vert u_λ\Vert_2$.
We study the one-dimensional nonlocal Kirchhoff type bifurcation problem related to logistic equation of population dynamics. We establish the precise asymptotic formulas for bifurcation curve $λ= λ(α)$ as $α\to \infty$ in $L^2$-framework, where $α:= \Vert u_λ\Vert_2$.
We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with convolutional Kirchhoff functions. We establish the exact solutions $u_λ$ and bifurcation curves $λ(α)$, where $α:= \Vert u_λ\Vert_\infty$.
We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with oscillatory nonlinear term. We establish the precise asymptotic formulas for the bifurcation curves $λ(α)$ as $α\to \infty$ and $α\to 0$, where $α:= \Vert u_λ\Vert_\infty$ and $u_λ$ is the solution associated with $λ$. We show that the second term of $λ(α)$ is oscillatory as $α\to \infty$.
We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with logarithmic Kirchhoff function. We establish the precise asymptotic formulas for the solution $u_λ(x)$ as $λ\to \infty$. Here, $λ> 0$ is the bifurcation parameter.
We study the one-dimensional nonlocal elliptic equation \begin{eqnarray*} -\left(\int_0^1 \vert u(x)\vert^p dx + b\right)^q u''(x) &=& λu(x)^p, \quad x \in I:= (0,1), \ u(x) > 0, \ x\in I, \\ u(0) &=& u(1) = 0, \end{eqnarray*} where $b \ge 0, p \ge 1, q > 1 - \frac{1}{p}$ are given constants and $λ> 0$ is a bifurcation parameter. We establish the global behavior of bifurcation diagrams and precise asymptotic formulas for $u_λ(x)$ as $λ\to \infty$.
We study the one-dimensional Kirchhoff type equation $$ -(b + a\Vert u'\Vert^{2}) u''(x) = λu(x)^p, x \in I:= (-1,1), \enskip u(x) > 0, \enskip x\in I, \enskip u(\pm 1) = 0, $$ where $\Vert u'\Vert = \left(\int_I u'(x)^2 dx\right)^{1/2}$, $a > 0, b > 0, p> 0$ are given constants and $λ> 0$ is a bifurcation parameter. We establish the exact solution $u_λ(x)$ and complete shape of the bifurcation curves $λ= λ(ξ)$, where $ξ:= \Vert u_λ\Vert_\infty$. We also study the nonlinear eigenvalue problem $$ -\Vert u'\Vert^{p-1} u''(x) = μu(x)^p, x \in I, \enskip u(x) > 0, x\in I, \enskip u(\pm 1) = 0, $$ where $p > 1$ is a given constant and $μ> 0$ is an eigenvalue parameter. We establish the first eigenvalue and eigenfunction of this problem by using a simple time map method.
We consider the nonlinear eigenvalue problem $[D(u(t))u(t)']' + λg(u(t)) = 0$, $u(t) > 0$, $t \in I := (0,1)$, $u(0) = u(1) = 0$, which comes from the porous media type equation. Here, $D(u) = pu^{2n} + \sin u$ ($n \in \mathbb{N}$, $p > 0$: given constants), $g(u) = u$ or $g(u) = u + \sin u$. $λ> 0$ is a bifurcation parameter which is a continuous function of $α= \Vert u_λ\Vert_\infty$ of the solution $u_λ$ corresponding to $λ$, and is expressed as $λ= λ(α)$. Since our equation contains oscillatory term in diffusion term, it seems significant to study how this oscillatory term gives effect to the structure of bifurcation curves $λ(α)$. We prove that the simplest case $D(u) = u^{2n} + \sin u$ and $g(u) = u$ gives us the most significant phenomena to the global behavior of $λ(α)$.