Introduction

In this paper, we are concerned with the positive solutions of the initial value problem (P):

$$w''-\frac{y}{2}w'-\alpha w+w^{p}=0, \quad y > 0,$$ (1.1)

$$w(0) > 0, w'(0)=-w^{q}(0),$$ (1.2)

where $w=w(y), q=(p+1)/2$, and

$$\alpha=\frac{1}{p-1}.$$ (1.3)

Hereafter the prime denotes the differentiation with respect to $y$. We always assume that $p > 1$.

The problem (P) is closely related to the following semilinear parabolic problem

$$u_t =u_{xx}+u^{p}, \quad x\in(0,1), t>0,$$ (1.4)

$$u_x(0,t)=0, u_x(1,t)=u^q(1,t), \quad t>0,$$ (1.5)

$$u(x,0)=u_0(x), \quad x\in[0,1],$$ (1.6)

where $u_{0}(x)$ is a positive smooth function. We say that the solution $u$ of the problem (

1

1.1)-(

2

1.3) blows up if there is a finite time $T$ such that $\max_{x \in [0,1]}u(x,t) \to \infty$ as $t \uparrow T$. It has been shown by Lin and Wang [3] that the solution $u(x,t)$ of the problem (

3

1.1)-(

4

1.3) always blows up, since $p > 1$. Also, under some conditions (for example, $u'_{0}\ge 0$), $x=1$ is the only blow-up point. See [3] and [1].

In order to understand the time asymptotic behaviour of $u(x,t)$ as $t \uparrow T$, we make the following well-known Giga-Kohn transformation [2]

$$\begin{eqnarray*} &&y=\frac{1-x}{\sqrt{T-t}},\quad s=-\ln(T-t),\\ &&w(y,s)=(T-t)^\alpha u(x,t). \end{eqnarray*}$$

Then the function $w$ satisfies

$$\begin{eqnarray*} && w_{s}=w_{yy}-\frac{y}{2}w_{y}-\alpha w+w^{p}, \quad 0 < y <... ...n T)=T^{\alpha}u_{0}(1-y\sqrt{T}), \quad 0 \le y \le 1/\sqrt{T}. \end{eqnarray*}$$

It is nature to expect that, as $s\to\infty$ (or, $t \uparrow T$), $w(y,s)$ tends to a global positive solution of (P). Therefore, the existence and uniqueness of global positive (monotone decreasing) solution of (P) plays an important role in studying the time asymptotic behaviour of $u(x,t)$ as $t \uparrow T$. In fact, if $W(y)$ is the unique global positive monotone decreasing solution of (P), then we have

$$(T-t)^{\alpha}u(1-y\sqrt{T-t},t) \rightarrow W(y)$$

as $t \uparrow T$ uniformly for $y \in [0,C]$ for any $C > 0$. See [1] for more detail.

It has been proved in [4] (See also [1]) that there is a global positive monotone decreasing solution of (P) for any $p > 1$. Also, the uniqueness of global positive monotone decreasing solutions of (P) for $p \in (1,2]$ was proved in [1]. The main purpose of this paper is to show the following theorem on the structure of positive solutions of (P).

Theorem 1.1   Suppose that $1 < p \le 2$. Let $w(y;\eta)$ be the solution of (P) with the initial value $w(0;\eta)=\eta>0$. Then there exists a unique $\bar\eta>0$ such that

(i)

if $\eta>\bar\eta$, then $w(y;\eta)$ is decreasing to zero at some finite $R$;

(ii)

if $\eta=\bar\eta$, then $w(y;\eta)$ is a global positive monotone decreasing solution;

(iii)

if $\eta<\bar\eta$, then there exist $y_{0}, y_{1}, y_{2} >0$ such that $w'(y) < 0$ for $y \in [0,y_{0})$, $w'(y) > 0$ for $y \in (y_{0},y_{1})$, $w'(y) < 0$ for $y \in (y_{1},y_{2})$, and $w(y_{2})=0$.

We emphasize here that it follows from Theorem 

5

th1 that any solution of (P) must vanish at some finite $R$ except the solution starting with $\bar\eta$.

This paper is organized as follows. We first recall some facts from [1] in §2 and derive the assertions (i) and (ii) of Theorem 1.1. Then in §3 we prove the assertion (iii) of Theorem 1.1. Hence the only global positive solution of (P) is the monotone decreasing solution $w(y;\bar\eta )$.