Refined estimates for the propagation speed of the transition solution to a free boundary problem with a nonlinearity of combustion type

TL;DR

This paper refines the estimates for the propagation speed of transition solutions in a combustion-type free boundary problem, showing how nonlinearity influences the speed with more precise bounds.

Contribution

It introduces two approaches to establish sharper upper and lower bounds on the free boundary's propagation speed, highlighting the impact of nonlinearity.

Findings

More accurate bounds on $h(t)$ for transition solutions.

Nonlinearity $f$ significantly influences propagation speed.

Transition solutions converge to ignition temperature $ heta$.

Abstract

We are concerned with the nonlinear problem $u_t=u_{xx}+f(u)$, where $f$ is of combustion type, coupled with the Stefan-type free boundary $h(t)$. According to [4,5], for some critical initial data, the transition solution $u$ locally uniformly converges to $\theta$, which is the ignition temperature of $f$, and the free boundary satisfies $h(t)=C\sqrt{t}+o(1)\sqrt{t}$ for some positive constant $C$ and all large time $t$. In this paper, making use of two different approaches, we establish more accurate upper and lower bound estimates on $h(t)$ for the transition solution, which suggest that the nonlinearity $f$ can essentially influence the propagation speed.

Equations

$$\displaystyle\left\{\begin{array}[]{ll}u_{t}=u_{xx}+f(u),&t>0,\ g(t)<x<h(t),\\ u(t,g(t))=u(t,h(t))=0,&t>0,\\ g^{\prime}(t)=-\mu u_{x}(t,g(t)),&t>0,\\ h^{\prime}(t)=-\mu u_{x}(t,h(t)),&t>0,\\ -g(0)=h(0)=h_{0},u(0,x)=u_{0}(x),&-h_{0}\leq x\leq h_{0},\end{array}\right.$$

$$f(0)=0,f(u)<0\mathrm{for}u>1.$$

$$\begin{array}{rl}& f(u)=0\mathrm{in}[0,\theta ],f(u)>0\mathrm{in}(\theta ,1),\\ & f(u)<0\mathrm{in}(1,\infty )\end{array}$$

$$\displaystyle u_{0}\in\mathscr{X}(h_{0}):=\left\{\phi\in C^{2}[-h_{0},h_{0}]:\begin{tabular}[]{ll}$\phi(-h_{0})=\phi(h_{0})=0,\ \phi^{\prime}(-h_{0})>0,$\\ $\phi^{\prime}(h_{0})<0,\ \phi(x)>0\ {\rm in}\ (-h_{0},h_{0})$.\end{tabular}\right\}$$

$$g_{\infty }:=\underset{t\to \infty }{\lim }g(t)\in [-\infty ,-h_0),h_{\infty }:=\underset{t\to \infty }{\lim }h(t)\in (h_0,+\infty ]$$

$$\underset{t\to \infty }{\lim }u(t,x)=1\text{mbox}locallyuniformlyin\mathbb{R};$$

$$\underset{t\to \infty }{\lim }\underset{x\in [g(t),h(t)]}{\max }u(t,x)=0;$$

$$\underset{t\to \infty }{\lim }u(t,x)=\theta \text{mbox}locallyuniformlyin\mathbb{R}.$$

$$\underset{t\to \infty }{\lim }(h(t)-c^{\ast }t-H_{\infty })=0,\underset{t\to \infty }{\lim }{h}^{\prime }(t)=c^{\ast },$$

$$\underset{t\to \infty }{\lim }(g(t)+c^{\ast }t-G_{\infty })=0,\underset{t\to \infty }{\lim }{g}^{\prime }(t)=-c^{\ast },$$

$$\displaystyle\left\{\begin{array}[]{l}q_{zz}-c^{*}q_{z}+f(q)=0\ \ {\rm for}\ z\in(0,\infty),\\ q(0)=0,\ \mu q(0)=c^{*},\ q(\infty)=1,\ q(z)>0\ \ {\rm for}\ z\in(0,\infty).\end{array}\right.$$

$$h(t),-g(t)=2{\xi }_0\sqrt{t}\{1+o(1)\},\mathrm{as}t\to \infty ,$$

$$2{\xi }_0{e}^{{\xi }_0^2}{\int }_0^{{\xi }_0}{e}^{-s^2}ds=\mu \theta .$$

$$f(u)=(u-\theta {)}^p\mathrm{in}[\theta ,\theta +\sigma ),$$

$$\displaystyle 2\xi_{0}\sqrt{t}+mt^{\frac{1}{2}-\frac{1}{p-1}}\leq-g(t),\ \ h(t)\leq 2\xi_{0}\sqrt{t}+Mt^{\frac{1}{2}-\frac{1}{p+1}},\ \ \mbox{as $t\to\infty$},$$

$$\displaystyle 2\xi_{0}\sqrt{t}+O(1)\leq-g(t),\ \ h(t)\leq 2\xi_{0}\sqrt{t}+Mt^{\frac{1}{2}-\frac{1}{p+1}},\ \ \mbox{as $t\to\infty$},$$

$$\displaystyle\left\{\begin{array}[]{ll}\overline{u}_{t}\geq\overline{u}_{xx}+f(\overline{u}),&0<t\leq T,\ \xi(t)<x<\overline{h}(t),\\ \overline{u}(t,\xi(t))\geq u(t,\xi(t)),&0<t\leq T,\\ \overline{u}(t,\overline{h}(t))=0,\ \overline{h}^{\prime}(t)\geq-\mu\overline{u}_{x}(t,\overline{h}(t)),&0<t\leq T.\\ \end{array}\right.$$

$$\xi (t)\ge g(t)in[0,T],h_0\le \overline{h}(0),u_0(x)\le \overline{u}(0,x)in[\xi (0),h_0],$$

$$h(t)\le \overline{h}(t)in(0,T],$$

$$u(t,x)\le \overline{u}(t,x)fort\in (0,T]and\xi (t)<x<\overline{h}(t).$$

$$\displaystyle\left\{\begin{array}[]{ll}\overline{u}_{t}\geq\overline{u}_{xx}+f(\overline{u}),&0<t\leq T,\ \xi(t)<x<\overline{h}(t),\\ \left\{\begin{array}[]{ll}\overline{u}(t,\xi(t))\geq 0,&{\rm if}\ h(t)\leq\xi(t),\\ \overline{u}(t,\xi(t))\geq u(t,\xi(t)),&{\rm if}\ \xi(t)<h(t),\end{array}\right.&0<t\leq T,\\ \overline{u}(t,\overline{h}(t))=0,&0<t\leq T,\\ \overline{h}^{\prime}(t)\geq-\mu\overline{u}_{x}(t,\overline{h}(t)),&0<t\leq T.\end{array}\right.$$

$$\xi (t)\ge 0,0\le t\le T,$$

$$h_0\le \xi (0)<\overline{h}(0),$$

$$\overline{u}(0,x)\ge 0,\xi (0)\le x\le \overline{h}(0),$$

$$\overline{u}(0,x)>0,\xi (0)<x<\overline{h}(0),$$

$$u(t,x)\le \overline{u}(t,x),0<t\le T,x\in (0,h(t))\cap (\xi (t),\overline{h}(t)),$$

$$h(t)<\overline{h}(t),0<t\le T.$$

$$h(t_{\ast })=\xi (t_{\ast }),$$

$$\xi (t)<h(t)<\overline{h}(t),t\in (t_{\ast },t^{\ast }),$$

$$h(t^{\ast })=\overline{h}(t^{\ast }).$$

$$h^{\prime }(t^{\ast })\ge {\overline{h}}^{\prime }(t^{\ast }).$$

$$u(t,x)\le \overline{u}(t,x)\mathrm{in}{\Omega }_{\ast }.$$

$$u(t,x)<\overline{u}(t,x)\mathrm{in}{\Omega }_{\ast }.$$

$$w_t-w_{xx}-c(t,x)w\ge 0\mathrm{in}{\Omega }_{\ast }$$

Full text

2010 Mathematics Subject Classification. 35K20, 35K55, 35R35

Key words and phrases. free boundary problem, nonlinear diffusion equation, combustion, propagation speed

*†*School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China. (Email: leichengxia001@163.com)

§National Institute of Technology, Numazu College, 3600 Ooka, Numazu City, Shizuoka 410-8501, Japan. (Email: hmatsu@numazu-ct.ac.jp)

*‡*School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China. (Email: pengrui $\b{}$ seu@163.com)

♮School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia. (Email: mzhou6@une.edu.au)

H. Matsuzawa was partially supported by Grant-in-Aid for Scientific Research (C) 17K05340, and R. Peng was partially supported by NSF of China (No. 11671175, 11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (No. PPZY2015A013) and Qing Lan Project of Jiangsu Province.

1. Introduction and Main Results

In this paper, we consider the following free boundary problem of nonlinear diffusion equations:

[TABLE]

where $x=g(t)$ and $x=h(t)$ are the expanding fronts, $\mu$, $h_{0}$ are given positive constants. The nonlinear reaction term $f$ is a locally Lipschitz continuous function satisfying

[TABLE]

Throughout the paper, unless otherwise specified, we assume that $f$ is of combustion type:

[TABLE]

for some constant $\theta\in(0,1)$.

For any given $h_{0}>0$, the initial function $u_{0}(x)$ satisfies

[TABLE]

It was shown by [4] that under condition (1.7), (1.6) has a unique globally defined classical solution $(u(t,x),h(t),g(t))$. In addition, $g^{\prime}(t)<0$ and $h^{\prime}(t)>0$ for $t>0$ and therefore

[TABLE]

always exist.

Denote $\mathbb{R}=(-\infty,\infty)$. In [4], the authors obtained the following trichotomy result and the sharp threshold dynamics when $f$ satisfies (1.8).

Theorem A** ([4, Theorem 1.4]).**

Assume that $f$ is of combustion type, and $h_{0}>0$, $u_{0}\in\mathscr{X}(h_{0})$. Then one of the following situations occurs:

1. (i)

Spreading:* $(g_{\infty},h_{\infty})=\mathbb{R}$ and*

[TABLE] 2. (ii)

Vanishing:* $(g_{\infty},h_{\infty})$ is a finite interval and*

[TABLE] 3. (iii)

Transition:* $(g_{\infty},h_{\infty})=\mathbb{R}$ and*

[TABLE]

Moreover, if $u_{0}=\nu\phi$ ($\nu>0$) for some $\phi\in\mathscr{X}(h_{0})$, then there exists $\nu^{*}=\nu^{*}(h_{0},\phi)\in(0,\infty]$ such that vanishing happens when $0<\nu<\nu^{*}$, spreading happens when $\nu>\nu^{*}$, and transition happens when $\nu=\nu^{*}$.

Remark 1**.**

In [4], it was assumed that $f$ is $C^{1}$; in view of the argument of [6], one can easily see that $f$ can be relaxed to be locally Lipschitz and Theorem A remains true.

In the paper [7], the authors derived the following propagation speed when spreading happens.

Theorem B** ([7, Theorem 1.2]).**

Assume that $f$ is of combustion type with $f\in C^{1}$ and $f^{\prime}(1)<0$. If spreading happens, then there exist $H_{\infty}$, $G_{\infty}\in\mathbb{R}$ such that

[TABLE]

where $c^{*}$ is uniquely determined by the following equation

[TABLE]

The more recent paper [5] established some estimates on $h(t)$ in the transition case.

Theorem C** ([5, Theorem 1.2]).**

Assume that $f$ is of combustion type and transition happens. Then we have

[TABLE]

where $\xi_{0}>0$ is uniquely determined by

[TABLE]

Regarding related studies on the free boundary problem, one may refer to [1, 2, 3, 8] and the references therein.

The purpose of the current paper is to obtain more delicate estimates on the small order $o(1)\sqrt{t}$ in Theorem C for a special type of nonlinearity $f$. Precisely, by assuming that there exist constants $p\geq 1$ and small $\sigma>0$ such that

[TABLE]

we aim to investigate how the power $p$ in (1.13) influences the asymptotic behavior $h(t)$ for large time $t$ in the transition case. Our main result reads as follows.

Theorem 1.1**.**

Assume that $f$ is of combustion type satisfying (1.13) for some $p\geq 1$ and $\sigma>0$, and that the transition happens. Then the following assertions hold.

1. (1)

If $p>3$, we have

[TABLE]

where $m$ and $M$ are some positive constants and $\xi_{0}>0$ is the constant uniquely given by (1.12). 2. (2)

If $1\leq p\leq 3$, we have

[TABLE]

where $\xi_{0}>0$ is the constant uniquely given by (1.12).

Remark 2**.**

Theorem 1.1 shows that when $p>3$, $h(t)-2\xi_{0}\sqrt{t}$ and $g(t)+2\xi_{0}\sqrt{t}$ are unbounded for large time $t$ while $h(t)-2\xi_{0}\sqrt{t}$ and $g(t)+2\xi_{0}\sqrt{t}$ become bounded when $p=1$. Therefore, our results reveal that the nonlinear reaction term $f$ has a qualitative impact on the propagation speed for the transition solution.

Remark 3**.**

In light of Theorem 1.1, it is unclear to us that $h(t)-2\xi_{0}\sqrt{t}$ and $g(t)+2\xi_{0}\sqrt{t}$ are unbounded or bounded for large time $t$ when $1<p\leq 3$. It would be of interest to investigate the refined propagation speed of the transition solution for the nonlinearity $f$ satisfying (1.13) with $0<p<1$ or $1<p\leq 3$.

2. Preliminary results

In this section, we give some basic facts, which will be used frequently in the forthcoming sections. We shall begin with some comparison principles associated with problem (1.6). The first one, which will be used later to estimate the lower bound of $h(t)$, is due to [4].

Lemma 2.1** ([4, Lemma 2.2]).**

Suppose that $f\in C^{0,1}_{\rm loc}$ and $f(0)=0$, $T\in(0,\infty)$, $\xi$, $\overline{h}\in C^{1}[0,T]$, $\overline{u}\in C(\overline{D}_{T})\cap C^{1,2}(D_{T})$ with $D_{T}=\{(t,x)\in\mathbb{R}^{2}:0<t\leq T,\ \xi(t)<x<\overline{h}(t)\}$, and

[TABLE]

If

[TABLE]

where $(u,g,h)$ is a solution to (1.6), then

[TABLE]

The function $\overline{u}$ or the triple $(\overline{u},\xi,\overline{h})$ in Lemma 2.1 is usually called an upper solution of problem (1.6). We can define a lower solution by reversing all the inequalities in the suitable places above.

The following version of comparison principle will be used to estimate the upper bound of $h(t)$.

Lemma 2.2**.**

Suppose that $f\in C^{0,1}_{\rm loc}$ and $f(0)=0$, $T\in(0,\infty)$, $\xi$, $\overline{h}\in C^{1}[0,T]$, $\overline{u}\in C(\overline{D}_{T})\cap C^{1,2}(D_{T})$ with $D_{T}=\{(t,x)\in\mathbb{R}^{2}:0<t\leq T,\ \xi(t)<x<\overline{h}(t)\}$, and

[TABLE]

If

[TABLE]

and $u$ is the unique solution to problem (1.6), then

[TABLE]

Proof.

If $h(t)\leq\xi(t)$ for $t\in(0,T]$, nothing is needed to prove. Suppose that there exist $t_{*}$, $t^{*}\in(0,T]$ with $t_{*}<t^{*}$ such that

[TABLE]

Then we have

[TABLE]

Let $\Omega_{*}=\{(t,x)\in\mathbb{R}^{2}|t_{*}<t\leq t^{*},\ \xi(t)<x<h(t)\}$. It follows from the maximum principle for parabolic equations that

[TABLE]

Since $\overline{u}(t,h(t))>0=u(t,h(t))$ for $t\in(t_{*},t^{*})$, the strong maximum principle for parabolic equations infers that

[TABLE]

By setting $w(t,x)=\overline{u}(t,x)-u(t,x)$, then $w$ satisfies

[TABLE]

for some bounded function $c(t,x)$. We also have

[TABLE]

An application of the Hopf lemma allows one to conclude that $w_{x}(t^{*},h(t^{*}))<0$, that is,

[TABLE]

which is a contradiction against (2.1). ∎

The following lemma also plays a key role in our later analysis. To stress the dependence of the unique solution $(u(t,x),g(t),h(t))$ of (1.6) on the initial function $u_{0}$, we will sometimes use notation $u(t,x)=u(t,x;u_{0})$, $h(t)=h(t;u_{0})$ and $g(t)=g(t;u_{0})$.

Lemma 2.3** ([5, Lemma 4.1]).**

Suppose that $f$ is of combustion type and that for any compactly supported functions $u_{0}^{1}$ and $u_{0}^{2}$ satisfying (1.11) such that the transition happens for $u(t,x;u_{0}^{i})$, $i=1,2$. Then there exists a constant $C>0$ such that

[TABLE]

In light of the above lemma, to prove Theorem 1.1, it suffices to deal with some special initial function. Without loss of generality, we fix an initial datum $\tilde{u}_{0}$ to be symmetrically decreasing and

[TABLE]

so that the transition case happens. Consequently, the solution $(u,g,h)$ with such initial datum $\tilde{u}_{0}$ satisfies

[TABLE]

Remark 4**.**

We note that the above chosen initial datum $\tilde{u}_{0}$ may vary with respect to the nonlinearity $f$. Thus in what follows whenever such $\tilde{u}_{0}$ is used, it should be understood that the function $f$ is fixed first.

With the above chosen initial datum $\tilde{u}_{0}$, we now need to show that when $p>3$, it holds

[TABLE]

for some positive constants $m$ and $M$ and when $1\leq p\leq 3$, it holds

[TABLE]

3. Proof of the main theorem

Throughout this section, we assume that $f$ is of combustion type and that (1.13) holds for $p\geq 1$ and $\sigma>0$, and that $(u,g,h)$ is the solution of (1.6) with the initial datum $\tilde{u}_{0}$.

3.1. Upper bound estimate

We first observe that $u(t,0)>\theta$ for all $t\geq 0$. Otherwise there exists $t_{0}>0$ such that $u(t_{0},0)\leq\theta$ and hence $u(t_{0},x)<\theta$ for $x\in[-h(t_{0}),h(t_{0})]\backslash\{0\}$. By the strong maximum principle we easily deduce $u(t,x)<\theta$ for $t>t_{0}$ and $x\in[-h(t),h(t)]$, which in turn implies $u(t,x)\to 0$ as $t\to\infty$, contradicting the assumption that the transition happens. Combining this and (2.3), we can see that for each $t\geq 0$, there exists a unique $\theta(t)\in(0,h(t))$ such that

[TABLE]

To obtain an upper bound estimate of $h(t)$, as a first step, we will derive an estimate for $\theta(t)$. For this purpose, given a constant $b\in(0,1-\theta)$, let us recall the property of the initial value problem:

[TABLE]

Through a phase plane analysis, (3.2) has a unique solution $V_{b}$ and numbers $l(b)$, $L(b)\in(0,\infty)$ with $l(b)<L(b)$ such that

[TABLE]

Furthermore, we have

Lemma 3.1**.**

The following assertions hold.

1. (i)

If $p>1$, then $l(b)$ is strictly decreasing in $b\in(0,\sigma)$. 2. (ii)

If $p=1$, then $l(b)=\frac{\pi}{2}$ for $b\in(0,\sigma)$. 3. (iii)

If $p\geq 1$, then $L(b)$ is strictly decreasing in $b\in(0,\sigma)$.

Proof.

By virtue of the equation for $V_{b}$, direct calculation gives

[TABLE]

where

[TABLE]

By integrating we then obtain

[TABLE]

and so

[TABLE]

that is,

[TABLE]

Integrating the above identity over $[0,l(b)]$, we have

[TABLE]

where

[TABLE]

For $0<u<\sigma$ and $p>1$, it follows that

[TABLE]

and

[TABLE]

where

[TABLE]

This implies that $l(b)$ is strictly decreasing in $b\in(0,\sigma)$ if $p>1$.

When $p=1$, $l(b)=\frac{\pi}{2}$ is an obvious fact.

We next investigate $L(b)$. As $V_{b}(x)$ is a linear function over $[l(b),L(b)]$, we have

[TABLE]

This gives

[TABLE]

By (3.4) we know $V_{b}^{\prime}(l(b))=-\sqrt{G(b)}$. Thus, from (3.7) and (3.5) it follows

[TABLE]

and

[TABLE]

Hence, $L(b)$ is strictly decreasing in $b\in(0,\sigma)$. ∎

Lemma 3.2** ([5, Lemma 4.3]).**

If $0<b<\frac{1-\theta}{2}$, then $L(b)\to+\infty$ if and only if $b\to 0$.

As one will see below, for any fixed $t\geq 1$, the sign-changing pattern of the function $u(t,x)-V_{b}(x)$ becomes crucial in obtaining a refined estimate for $\theta(t)$. Indeed, such kind of properties were studied in Lemmas 4.6–4.8 in [5].

By Lemma 4.5 of [5], there exists $\delta_{0}>0$ such that for each $b\in(0,\delta_{0})$,

[TABLE]

satisfies

[TABLE]

and has a unique zero in $[0,h(1)]$, and the zero is nondegenerate.

As $t\mapsto w_{b}(t,x)$, $t\mapsto(w_{b})_{x}(t,x)$, and $t\mapsto h(t)$ are all continuous uniformly in $x$, we see that for each $b\in(0,\delta_{0})$ there exists $\epsilon_{0}>0$ small such that for each fixed $t\in[1-\epsilon_{0},1+\epsilon_{0}]$, $w_{b}(t,x)$ satisfies

[TABLE]

and has a unique zero in $[0,h(t)]$ which is nondegenedete.

We define

[TABLE]

Clearly $T_{1}^{b},T_{2}^{b}\geq 1+\epsilon_{0}$. Due to $h(t)\to+\infty$ and $w_{b}(t,0)\to-b<0$ as $t\to\infty$, both $T_{1}^{b}$ and $T_{2}^{b}$ are finite.

Now we are ready to present the estimate of $\theta(t)$ in the following lemma.

Lemma 3.3**.**

Let $(u,g,h)$ be the solution of (1.6) with the initial datum $\tilde{u}_{0}$. Then there exists $M>0$ such that

[TABLE]

Proof.

Let $\sigma_{1}\in(0,\sigma)$ be arbitrarily given. By Lemma 3.1, for any $h\in(L(\sigma_{1}),+\infty)$, there exists a unique $b\in(0,\sigma_{1})$ such that $L(b)=h$. By the fact $h(t)\to\infty$ as $t\to\infty$, there exists $T_{0}>0$ such that

[TABLE]

Hence for any $t\geq T_{0}$, there exists a unique $b(t)\in(0,\sigma_{1})$ such that

[TABLE]

Claim 1. $\theta(t)\leq l(b(t))$ for all large $t$.

Since $\lim_{t\to\infty}u(t,x)=\theta$ locally uniformly in $\mathbb{R}^{1}$, there exists $T_{0}^{\prime}>T_{0}$ such that

[TABLE]

It is sufficient to prove that

[TABLE]

We proceed indirectly and suppose that there exists $t_{0}>T_{0}^{\prime}$ such that $\theta(t_{0})>l(b(t_{0}))=:l(b_{0})$. In view of $h(t_{0})=L(b_{0})$ and Lemmas 4.6–4.8 of [5], we know that

[TABLE]

Recalling that $\theta(t_{0})>l(b_{0})$, it is easily seen that

[TABLE]

Hence $u(t_{0},x)-V_{b_{0}}(x)$ has a zero in $(l(b_{0}),L(b_{0}))$. This case can only happen in (ii) of Lemma 4.7 in [5]. Then $u(t_{0},x)-V_{b_{0}}(x)$ has the sign-changing pattern of $[+0-0]$. This and (3.10) imply that

[TABLE]

Define

[TABLE]

First we show that set $A$ is nonempty. We already have

[TABLE]

Since $l(b)$ is nonincreasing in $b$ (due to Lemma 3.1) and $V_{b}$ is continuous with respect to $b$, we obtain

[TABLE]

for $b_{2}>b_{0}$ but close to $b_{0}$. This implies that $A$ is nonempty. In addition, from (3.9) for all $b\in A$, it follows

[TABLE]

Then $b_{*}\leq u(t_{0},0)-\theta\leq\sigma_{1}$. Therefore $b_{*}$ is well-defined. In the sequel, we further claim that

[TABLE]

Suppose that $b_{*}<u(t_{0},0)-\theta$. Then $V_{b_{*}}(0)<u(t_{0},0)$, that is, $w_{b_{*}}(t_{0},0)>0$. Note that $b^{*}>b_{0}$ and $L(b)$ is decreasing in $b$. So

[TABLE]

Then Lemmas 4.6–4.8 of [5] imply that

[TABLE]

By Lemma 4.7 (iii) in [5], $u(t_{0},x)-V_{b_{*}}(x)$ has the sign-changing pattern $[+0-0+]$. We then conclude that

[TABLE]

For any $b\in(b_{0},b_{*})$, it follows that

[TABLE]

Letting $b\nearrow b_{*}$ gives

[TABLE]

Noting that

[TABLE]

we apply (3.10) and Lemma 3.1 to assert that

[TABLE]

Since $u(t_{0},x)-V_{b_{*}}(x)$ has the sign-changing pattern $[+0-0+]$, there holds

[TABLE]

By continuity of $V_{b}$ with respect to $b$, one can find $\varepsilon>0$ such that

[TABLE]

This contradicts the definition of $b_{*}$. So we have $b_{*}=u(t_{0},0)-\theta$, that is, $u(t_{0},0)=V_{b_{*}}(0)$ and

[TABLE]

However, by Lemmas 4.6–4.8 of [5], if $u(t_{0},x)-V_{b_{*}}(x)$ has a tangency at $x=0$, then $u(t_{0},x)<V_{b_{*}}(x)$ in $(0,\delta)$ for some small $\delta>0$. This is a contradiction. Thus we have shown that $\theta(t)\leq l(b(t))$ for all large $t$.

Claim 2. $\theta(t)\leq Mt^{\frac{1}{2}-\frac{1}{p+1}}$ for all large $t$.

In the case of $p=1$, we have $l(b)=\frac{\pi}{2}$ by (3.6).

Now we handle the situation of $p>1$. By Theorem 1.2 in [5](see Theorem C) and (3.8)

[TABLE]

By Lemma 3.2,

[TABLE]

Since $h(t)\to\infty$ as $t\to\infty$, one further gets

[TABLE]

From (3.11) and (3.12), we have

[TABLE]

and in turn

[TABLE]

from which we obtain

[TABLE]

This and (3.6) yield

[TABLE]

Therefore there exists $M>0$ such that

[TABLE]

∎

Consider the Stefan problem

[TABLE]

It is easily seen that the solution to (3.16) is given by

[TABLE]

where

[TABLE]

By utilizing the explicit solution of the Stefan problem (3.16) to construct a suitable upper solution of (1.6), we can obtain the upper estimate of $h(t)$. Specifically, we can state

Lemma 3.4**.**

If $p>1$ and let $(u,g,h)$ be the solution of (1.6) with the initial datum $\tilde{u}_{0}$, we have

[TABLE]

Proof.

In view of Lemma 3.3, we can find some $K>0$ and $T_{0}>1$ such that

[TABLE]

Fix $T_{0}$ and choose $M>0$ so that

[TABLE]

Define

[TABLE]

where $\rho$, $r$ is given by (3.17).

We will check that $(\overline{u},\xi,\overline{h})$ satisfies the condition of Lemma 2.2, that is,

[TABLE]

From the choice of $M$, it follows that

[TABLE]

and so (3.18) holds.

By the definition of $\overline{u}$, (3.21) is trivially true.

When $t\geq T_{0}$ and $\xi(t)>h(t)$, clearly $\overline{u}(t,\xi(t))\geq 0$. As

[TABLE]

we get

[TABLE]

for $t\geq T_{0}$, $\xi(t)<h(t)$. Thus (3.24) holds.

By the definition of $\overline{u}$ again, we have $\overline{u}(t,\overline{h}(t))=\rho(t,r(t))=0$, verifying (3.25).

We now show that (3.26) holds. In fact, for $t\geq T_{0}$,

[TABLE]

Therefore (3.26) is fulfilled.

Finally we verify (3.27). Direct calculation gives

[TABLE]

Then we have, for $t\geq T_{1}$ and $\xi(t)<x<\overline{h}(t)$,

[TABLE]

Since

[TABLE]

we find

[TABLE]

Here we used the facts that $0\leq\overline{u}\leq\theta$ and $f(\overline{u})=0$. Consequently, (3.27) is satisfied.

With the help of Lemma 2.2, we can assert

[TABLE]

The proof is thus complete. ∎

Lemma 3.5**.**

Assume that $p=1$ and let $(u,g,h)$ be the solution of (1.6) with the initial datum $\tilde{u}_{0}$. Then we have

[TABLE]

Proof.

When $p=1$, it is clear that $l(b)=\pi/2$ for $b\in(0,\delta)$. On the other hand, from the proof of Claim 1 in the lemma above, we have $\theta(t)\leq l(b(t))$ for large $t$. By choosing a sufficiently large constant $M$, one can easily check that $(\bar{u},\bar{h}):=(\rho(t,x-M),r(t)+M)$ is an upper solution of $(u,g,h)$ by a similar argument as in Lemma 3.4. Therefore, $h(t)\leq 2\xi_{0}\sqrt{t}+M$ follows. ∎

3.2. Lower bound estimate for $p>3$

Our proof of the upper bound of $h(t)$ heavily relies on the upper bound estimate for $\theta(t)$; it seems that such an idea ceases to produce a refined lower bound of $h(t)$. To the aim, in the following we will develop a different approach by firstly deriving a lower bound of $u(t,0)$.

Lemma 3.6**.**

Assume that $p>3$ and let $(u,g,h)$ be the solution of (1.6) with the initial datum $\tilde{u}_{0}$. Then there exist constants $a>0$ and $T>0$ such that

[TABLE]

Proof.

We first recall a well known result of the self-similar solution to the following nonlinear heat equation

[TABLE]

Consider an associated problem

[TABLE]

By the result of Haraux and Weissler [9], for

[TABLE]

the above problem has a unique solution $\varphi$, which is defined for all $y>0$, and $\varphi(y)>0,\ \forall y\geq 0$. If we define

[TABLE]

then $v$ satisfies (3.28).

Set

[TABLE]

Then $w$ solves

[TABLE]

In view of $v(t,\cdot)\to 0$ as $t\to\infty$ locally uniformly in $\mathbb{R}$ (actually in $C^{1}(\mathbb{R})$), it follows that $w(t,\cdot)\to\theta$ locally uniformly in $\mathbb{R}$ as $t\to\infty$.

Recall that $u(t,0)>\theta$ for all $t\geq 0$. Thus there exists $T>0$ such that

[TABLE]

Furthermore we may assume that $u(0,x)$ and $w(T,x)$ have exact one intersection point $y\in(0,h_{0})$.

Denote

[TABLE]

Then we claim

[TABLE]

Otherwise there exists $t_{1}>0$ such that

[TABLE]

Since $\eta(t,0)>0>\eta(t,h(t))$ for $t\in(0,t_{1})$ and the unique existence of $y(t)$ (zero of $\eta$) for $t\in(0,\varepsilon_{1})$ ($\varepsilon_{1}>0$ is a small constant), we deduce that $\eta(t,x)$ has a unique nondegenerate zero over $[0,h(t)]$ for $t\in(0,t_{1})$ due to Lemma 2.2 in [5]. Hence $y(t)$ can be extended to all $t\in(0,t_{1})$.

Let us look at the limit of $y(t)$ as $t$ increases to $t_{1}$. If the limit does not exist, then as the proof of Lemma 2.4 in [5] we have $\eta(t_{1},x)\equiv 0$ over $[0,h(t_{1})]$ which contradicts with $\eta(t_{1},h(t_{1}))<0$. Hence the limit exists and we denote it by $y(t_{1})$. As $\eta(t_{1},y(t_{1}))=0$, clearly $y(t_{1})<h(t_{1})$. If $y(t_{1})=0$, the maximum principle, as applied to $\eta$ over $\{(t,x)|0<t<t_{1},y(t)<x<h(t)\}$, gives

[TABLE]

It is easily seen that

[TABLE]

By the strong maximum principle,

[TABLE]

For any $t_{2}>t_{1}$, we choose $\varepsilon_{0}>0$ small enough such that

[TABLE]

We can apply the comparison principle to conclude that

[TABLE]

By taking $(1+\varepsilon_{0})u(t_{2},\cdot)$ as a new initial value, Theorem A concludes that

[TABLE]

which is a contradiction.

If $y(t_{1})\in(0,h(t_{1}))$, then by the maximum principle applied to $\eta$ over $\{(t,x)|0\leq t\leq t_{1},\ 0<x<y(t)\}$, one can see that $\eta(t_{1},x)>0$ for $0<x<y(t_{1})$. In light of the Hopf lemma, we further have

[TABLE]

which implies

[TABLE]

This is a contradiction to $u_{x}(t_{1},0)=0$. Thus, our previous claim holds.

As a result, we derive

[TABLE]

Denoting $a=\gamma$, we obtain the desired assertion. ∎

Inspired by the Stefan problem (3.17), we are now able to deduce the lower bound of $h(t)$ using Lemma 3.6. Indeed, we have

Lemma 3.7**.**

Assume that $p>3$ and Let $(u,g,h)$ be the solution of (1.6) with the initial datum $\tilde{u}_{0}$. Then there exists a constant $m>0$ such that

[TABLE]

where $\xi_{0}>0$ is given by (1.12).

Proof.

Let

[TABLE]

where $T$ is the constant determined by Lemma 3.6 and $0<b<a$ is a constant to be chosen later. We first observe that there exists $\beta\in C^{1}$ such that $\beta(t)>0$ for $t>0$ and

[TABLE]

where $E$ is defined in (3.17). In fact, by setting

[TABLE]

we have

[TABLE]

Since $\Phi(0)=0$ and $\lim_{x\to\infty}\Phi(x)=\infty$, there exists a unique $\beta=\beta(t)$ such that $\Phi(\beta(t))=\mu(\theta+\xi(t))$. The implicit function theorem guarantees that $\beta\in C^{1}$. One can further claim that $\beta^{\prime}(t)<0$. Indeed, by (3.29),

[TABLE]

which indicates $\beta^{\prime}(t)<0$ for $t>0$.

Now we define a lower solution as follows:

[TABLE]

Fix $T_{0}>0$ and choose $b$ so that

[TABLE]

Then we have

[TABLE]

We next show that for sufficiently small $\varepsilon>0$, $(\underline{u},\xi,\underline{h})$ is a lower solution to (1.6), that is,

[TABLE]

Basic calculation yields

[TABLE]

As $\beta^{\prime}(t)<0$, $\underline{u}_{t}-\underline{u}_{xx}-f(\underline{u})\leq 0$, and so (3.31) holds.

By the definition, clearly (3.32) is true.

We then check (3.33). By Lemma 3.6 and (3.30) we have

[TABLE]

for any sufficiently small $\varepsilon>0$ and $0\leq x\leq\underline{h}(\varepsilon)$. By the continuity of $u(T_{0},x)$, one can find a small $\delta>0$ such that

[TABLE]

By choosing $\varepsilon>0$ to be small enough so that

[TABLE]

we deduce

[TABLE]

and

[TABLE]

for $x\in[0,\underline{h}(\varepsilon)]=[0,2\beta(\varepsilon)\sqrt{\varepsilon}]$. This indicates (3.33).

In view of (3.30), we have

[TABLE]

which verifies (3.34).

Finally we check (3.35). From the definition of $\beta(t)$ and (3.29), it follows

[TABLE]

Therefore,

[TABLE]

Now, applying the comparison principle (Lemma 2.1), we can conclude

[TABLE]

In view of

[TABLE]

we have

[TABLE]

As a consequence, we obtain

[TABLE]

for sufficiently large $t$, and then there exists $m>0$ such that

[TABLE]

∎

3.3. Lower bound estimate for $1\leq p\leq 3$

It turns out that the lower bound estimate of $h(t)$ in the case $1\leq p\leq 3$ is easier to establish.

Lemma 3.8**.**

Assume that $1\leq p\leq 3$ and let $(u,g,h)$ be the solution of (1.6) with the initial datum $\tilde{u}_{0}$. Then we have

[TABLE]

where $\xi_{0}>0$ is given by (1.12).

Proof.

We construct a lower solution in the following manner:

[TABLE]

Then we have $\underline{u}(t,x)\leq\theta$ and $f(\underline{u})\geq 0$.

We will show that for sufficiently small $\varepsilon>0$ $(\underline{u},\xi,\underline{h})$ is a lower solution to (1.6) for $t>\varepsilon$, that is,

[TABLE]

From (3.16) and the fact that $f(\underline{u})\geq 0$, (3.36) follows.

By the definition, it is easily seen that (3.37) holds.

Next we check (3.38). As $u(t,0)>\theta$ for $t\geq 0$, clearly

[TABLE]

for any sufficiently small $\varepsilon>0$ and $0\leq x\leq\underline{h}(\varepsilon)$.

By the continuity of $u(0,x)$, we can find a small $\delta>0$ such that

[TABLE]

Taking $\varepsilon>0$ to be sufficiently small so that

[TABLE]

we have

[TABLE]

and

[TABLE]

for $x\in[0,\underline{h}(\varepsilon)]=[0,2\xi_{0}\sqrt{\varepsilon}]$.

In view of $\underline{u}(t,0)\leq\theta<u(t-\varepsilon,0)$ for $t\geq\varepsilon$, it is also clear that (3.39) holds.

Finally (3.16) ensures that (3.40) is satisfied.

Hence, Lemma 2.1 yields that

[TABLE]

and in turn,

[TABLE]

∎

Combining Lemmas 3.4–3.8, we obtain Theorem 1.1.