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Blow-up for a degenerate and singular parabolic equation with a nonlocal source

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Abstract

This article studies the blow-up phenomenon for a degenerate and singular parabolic problem. Conditions for the local and global existence of solutions for the problem are given. In the case that blow-up occurs, the blow-up set for the problem is investigated. Finally, the asymptotic behaviour of the solution when time converges to the blow-up time is studied.

Introduction

In this paper, we consider the blow-up phenomenon of the following degenerate and singular parabolic equation with a nonlocal source:

$$ \left . \textstyle\begin{array}{@{}l} u_{t}=(x^{\beta }(u^{m})_{x})_{x}+\int _{0}^{a}u^{p}(x,t)\,dx,\quad (x,t) \in (0,a) \times (0,\infty ), \\ u(0,t)=u(a,t)=0,\quad t>0, \\ u(x,0)=g(x),\quad x \in {}[ 0,a], \end{array}\displaystyle \right \} $$
(1)

where \(\beta \in {}[ 0,1)\) and \(p>m>1\) and g satisfies the following hypotheses:

  1. (H1)

    \(g\in C^{2+\alpha }(0,a)\cap C[0,a]\) with \(0<\alpha <1\),

  2. (H2)

    \(g>0\) on \((0,a)\), \(g(0)=g(a)=0\), and \(g^{\prime }(0)>0\) and \(g^{\prime }(a)<0\),

  3. (H3)

    \((x^{\beta }(g^{m})^{\prime })^{\prime }+\int _{0}^{a}g ^{p}(x)\,dx>0 \) for \(x\in (0,a)\),

  4. (H4)

    \(\lim_{x\rightarrow 0^{+}}(x^{\beta }(g^{m})^{\prime })^{ \prime }=-\int _{0}^{a}g^{p}(x)\,dx\) and \(\lim_{x\rightarrow a^{-}}(x ^{\beta }(g^{m})^{\prime })^{\prime }=-\int _{0}^{a}g^{p}(x)\,dx\),

  5. (H5)

    \((x^{\beta }(g^{m})^{\prime })^{\prime }\leq 0\) for \(x\in (0,a)\).

We note that the idea for constructing the function g satisfying the assumptions (H1)–(H5) is in the appendix of [14]. Since \(\beta \in {}[ 0,1)\), coefficients of terms \(u_{x}\), \(u_{xx}\) may tend to 0 or ∞ as x converges to 0+. We thus can regard (1) as degenerate and singular. Let us introduce the definition of blow-up in a finite time.

Definition 1.1

The solution u of (1) is said to show blow-up at the point \(x_{b}\) in a finite time \(T_{b}\) (>0) if there exists a sequence \(\{(x_{n},t_{n})\}\) in \((0,a) \times (0,\infty )\) such that \((x_{n},t_{n})\rightarrow (x_{b},T_{b})\) as \(n\rightarrow \infty \) and \(\lim_{n\rightarrow \infty }u(x_{n},t_{n})=\infty \). The point \(x_{b}\) and the time \(T_{b}\) are called a blow-up point and blow-up time, respectively. Furthermore, we call the set of all blow-up points to be the blow-up set, which is denoted by S. If \(S=[0,a]\), we say that the solution u of (1) shows global blow-up.

The first paper concerning the blow-up problem for the reaction-diffusion equation was written by Fujita [9]. He studied the Cauchy problem: \(u_{t}- \Delta u=u^{1+\alpha }\), \(\alpha >0\) and shown that if \(0< N\alpha <2\) (N is the space dimension), then the initial value problem had no non-trivial global solutions while if \(N\alpha >2\), there were non-trivial global solutions. In this second case, it was essential that the initial values were sufficiently small. After the publication of Fujita’s paper, the blow-up phenomenon for the reaction-diffusion equations has been the object of intensive research. Degenerate parabolic equations with/without nonlocal source have been studied by under various types of initial and boundary conditions since the early 1970s by many researchers ([1, 3, 5, 10,11,12] and [15]).

In 1997, Aderson and Deng [2] studied the following problem:

$$ \left . \textstyle\begin{array}{@{}l} u_{t}=((u^{m})_{x}+\varepsilon u^{n})_{x}+au \Vert u \Vert _{q}^{p-1}, \quad (x,t)\in (0,1)\times (0,\infty ), \\ u(0,t)=u(1,t)=0,\quad t>0, \\ u(x,0)=u_{0}(x), \quad x \in {}[ 0,1]. \end{array}\displaystyle \right \} $$
(2)

They showed that the solution of (2) blows up in a finite time for a sufficiently large data \(u_{0}\) if \(p>\max \{1, \max \{m,n\}\}\). They, however, did not consider the blow-up profile of the blow-up solution.

In 2001, Deng et al. [6] considered the following problem:

$$ \left . \textstyle\begin{array}{@{}l} u_{t}=(u^{m})_{xx}+a\int _{-l}^{l}u^{q}\,dx, \quad (x,t)\in (-l,l)\times (0,\infty ), \\ u(-l,t)=u(l,t)=0, \quad t>0, \\ u(x,0)=u_{0}(x), \quad x \in {}[ -l,l], \end{array}\displaystyle \right \} $$
(3)

with \(l>0\), \(a>0\) and \(q>m>1\). They established that, under certain conditions, the solution of (3) either exists globally or blows up completely in a finite time. Moreover, they obtained

$$ C_{1}(T_{b}-t)^{-1/(q-1)}\leq \max _{x\in {}[ -l,l]}u(x,t)\leq C _{2}(T_{b}-t)^{-1/(q-1)}. $$

In 2003, Liu et al. [14] studied the following problem:

$$ \left . \textstyle\begin{array}{@{}l} u_{t}=x^{\alpha }(u^{m})_{xx}+\int _{0}^{a}u^{p}\,dx-ku^{q},\quad (x,t) \in (0,a)\times (0,\infty ), \\ u(0,t)=u(a,t)=0, \quad t>0, \\ u(x,0)=u_{0}(x),\quad x\in {}[ 0,a], \end{array}\displaystyle \right \} $$
(4)

and, under some assumptions, they proved the local existence and uniqueness of a classical solution of (4) and obtained some sufficient conditions for blow-up in a finite time of a solution of (4). Furthermore, they showed that the blow-up set of the solution is the whole domain.

In 2003, Li et al. [13] considered the following problem:

$$ \left . \textstyle\begin{array}{@{}l} u_{t}=\Delta (u^{m})+au^{p}\int _{\varOmega }u^{q}\,dx, \quad (x,t)\in \varOmega \times (0,\infty ), \\ u(x,t)=0, \quad x\in \partial \varOmega , t>0, \\ u(x,0)=u_{0}(x),\quad x\in \overline{\varOmega }, \end{array}\displaystyle \right \} $$
(5)

where \(\varOmega \subset R^{N}\) is a bounded domain with sufficiently smooth boundary ∂Ω. They showed that the solution of (5) either exists globally or blows up in a finite time. Moreover, if \(p+q>m\), then they showed

$$ C_{1}(T_{b}-t)^{-1/(p+q-1)}\leq \max _{x\in {}[ -l,l]}u(x,t)\leq C _{2}(T_{b}-t)^{-1/(p+q-1)}. $$

This paper is organized as follows. In the next section, we establish local existence and uniqueness of the solution of (1). We give some criteria for the solution of (1) to exist globally or blow up in a finite time in Sect. 3. The blow-up set and blow-up profile of the solution are presented in Sect. 4.

Local existence

Since (1) is degenerate and singular, the standard theory of parabolic type cannot be applied directly to obtain the existence and uniqueness of its classical solution. To investigate the local existence of the solution of (1), we need some transformation. Let \(v=u^{m}\), \(t=\frac{\tau }{ma^{\beta -2}}\) and \(x=a\xi \) in (1). Then (1) becomes

$$ \left . \textstyle\begin{array}{@{}l} v_{\tau }=v^{r} [ (\xi ^{\beta }v_{\xi })_{\xi }+a^{3-\beta }\int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi ] , \quad (\xi ,\tau )\in (0,1) \times (0,\infty ), \\ v(0,\tau )=v(1,\tau )=0, \quad \tau >0, \\ v(\xi ,0)=k(\xi ), \quad \xi \in {}[ 0,1], \end{array}\displaystyle \right \} $$
(6)

where \(0< r=\frac{m-1}{m}<1\), \(q=\frac{p}{m}>1\), \(k=g^{m}\) and k satisfies the following:

(H1′):

\(k\in C^{2+\alpha }(0,1)\cap C[0,1]\) with \(0<\alpha <1\),

(H2′):

\(k>0\) on \((0,1)\), \(k(0)=k(1)=0\), and \(k^{\prime }(0)\geq 0\) and \(k^{\prime }(1)\leq 0\),

(H3′):

\((\xi ^{\beta }k^{\prime })^{\prime }+a^{3- \beta }\int _{0}^{1}k^{q}(\xi )\,d\xi >0\) for \(\xi \in (0,1)\),

(H4′):

\(\lim_{\xi \rightarrow 0^{+}}(\xi ^{\beta }k ^{\prime }(\xi ))^{\prime }=-a^{3-\beta }\int _{0}^{1}k^{q}(\xi )\,d\xi \) and \(\lim_{\xi \rightarrow 1^{-}}(\xi ^{\beta }k^{\prime }(\xi ))^{ \prime }=-a^{3-\beta }\int _{0}^{1}k^{q}(\xi )\,d\xi \),

(H5′):

\((\xi ^{\beta }k^{\prime })^{\prime }\leq 0\) for \(\xi \in (0,1)\).

In the part of showing the local existence of problem (6), we need the following lemma.

Lemma 2.1

Let \(b_{i}\) is bounded and continuous and \(b_{i}(\xi ,\tau )\geq 0\) on \([0,1]\times {}[ 0, T]\) for \(i=1,2,3,4\) and \(d(\xi ,\tau ) \geq 0\) on \([0,1]\times {}[ 0, T]\) with \(0< T\leq \infty \). Suppose that \(w\in C^{2,1} ( (0,1)\times (0,T) ) \cap C([0,1]\times {}[ 0,T])\) satisfies

$$\begin{aligned}& w_{\tau }-d(\xi ,\tau ) \bigl(\xi ^{\beta }w_{\xi } \bigr)_{\xi }\geq b_{1}w_{ \xi }+b_{2}w+b_{3} \int _{0}^{1}b_{4}w(\xi ,\tau )\,d\xi , \quad (\xi ,\tau ) \in (0,1)\times (0,T], \\& w(0,\tau )\geq 0,\qquad w(1,\tau )\geq 0,\quad \tau \in (0,T], \\& w(\xi ,0)\geq 0,\quad \xi \in {}[ 0,1]. \end{aligned}$$

Then \(w\geq 0\) on \([0,1]\times {}[ 0, T]\).

Proof

Suppose that there exists a point \((\xi _{0},\tau _{0})\) in \((0,1) \times (0,T]\) such that \(w(\xi _{0},\tau _{0})<0\). Let \(B_{i}= \max_{(\xi ,\tau )\in {}[ 0,1]\times {}[ 0,T]}b_{i}(\xi , \tau )\) for \(i=1,2,3,4\) and let \(w(\xi ,\tau )=e^{c\tau }v(\xi , \tau )\) for \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,T]\) where c is a positive constant and \(c>B_{2}+B_{3}B_{4}\). Then we have, for any \((\xi ,\tau )\in (0,1)\times (0,T]\),

$$ v_{\tau }-d(\xi ,\tau ) \bigl(\xi ^{\beta }v_{\xi } \bigr)_{\xi }-b_{1}v_{\xi }+(c-b _{2})v-b_{3} \int _{0}^{1}b_{4}v(\xi ,\tau )\,d\xi \geq 0. $$

It follows from \(w(\xi _{0},\tau _{0})<0\) and \(w=e^{c\tau }v\) that \(v(\xi _{0},\tau _{0})<0\). Since v is non-negative on the parabolic boundary and \(v\in C^{2,1} ( (0,1)\times (0,T) ) \cap C([0,1] \times {}[ 0,T])\), there exists a point \((\xi _{1},\tau _{1})\) in \((0,1)\times (0,T]\) such that v attains its negative minimum at the point \((\xi _{1},\tau _{1})\). This yields \(v(\xi _{1},\tau _{1})<0\), \(v_{\tau }(\xi _{1},\tau _{1}) = 0\), \(v_{\xi }(\xi _{1},\tau _{1})=0\) and \(v_{\xi \xi }(\xi _{1},\tau _{1})\geq 0\). By the second mean value theorem for integrals, we find that there exists a \(\xi _{2}\in (0,1)\) such that

$$ \int _{0}^{1}b_{4}(\xi ,\tau )v(\xi , \tau )\,d\xi =v(\xi _{2},\tau ) \int _{0}^{1}b_{4}(\xi ,\tau )\,d\xi \quad \text{for any }\tau \in (0,T]. $$

It is clear that \(v(\xi _{2},\tau _{1})\geq v(\xi _{1},\tau _{1})\). Let us consider that

$$\begin{aligned}& v_{\tau }(\xi _{1},\tau _{1})-d(\xi _{1},\tau _{1}) \bigl[ \xi _{1}^{ \beta }v_{\xi \xi }( \xi _{1},\tau _{1})+\beta \xi _{1}^{\beta -1}v_{ \xi }( \xi _{1},\tau _{1}) \bigr] -b_{1}(\xi _{1},\tau _{1})v_{\xi }(\xi _{1},\tau _{1}) \\& \qquad {}+\bigl(c-b_{2}(\xi _{1},\tau _{1}) \bigr)v(\xi _{1},\tau _{1})-b_{3}(\xi _{1},\tau _{1})v( \xi _{2},\tau _{1}) \int _{0}^{1}b_{4}(\xi ,\tau _{1})\,d\xi \\& \quad \leq (c-B_{2})v(\xi _{1},\tau _{1})-B_{3}B_{4}v( \xi _{1},\tau _{1}) \\& \quad = -v(\xi _{1},\tau _{1}) \bigl[ -(c-B_{2})+B_{3}B_{4} \bigr] \\& \quad < 0. \end{aligned}$$

This contradiction implies that \(w(\xi ,\tau )\geq 0\) for any \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,T]\). Hence, the proof of Lemma 2.1 is completed. □

Since (6) is also degenerate, we will prove the local existence of the solution of (6) by considering the following problem:

$$ \left . \textstyle\begin{array}{@{}l} ( v_{1} ) _{\tau }=(v_{1}+\delta )^{r} [ ( \xi ^{\beta } ( v_{1} ) _{\xi } ) _{\xi }+a^{3-\beta }\int _{0}^{1} ( v_{1} ) ^{q}(\xi ,\tau )\,d\xi ] , \quad (\xi ,\tau )\in (0,1)\times (0,\infty ), \\ v_{1}(0,\tau ;\delta )=v_{1}(1,\tau ;\delta )=0, \quad \tau >0, \\ v_{1}(\xi ,0;\delta )=k(\xi ), \quad \xi \in {}[ 0,1], \end{array}\displaystyle \right \} $$
(7)

where δ is a positive constant and \(\delta <1\). We note that the function \(v_{1}=v_{1}(x,t;\delta )\) depends on x, t and δ. Let ε be a positive constant and \(\varepsilon <\delta \). To show the existence of the classical solution of (7), we have to use the function given by Dunford and Schwartz [7]. There exists a non-decreasing and continuously differentiable function ρ such that \(\rho =0\) if \(\xi \leq 0\) and \(\rho =1\) if \(\xi \geq 1\). Let

$$ \rho (\xi ;\varepsilon )= \textstyle\begin{cases} 0, & \xi \leq \varepsilon , \\ \rho ( \frac{\xi }{\varepsilon }-1 ) , & \varepsilon < \xi < 2\varepsilon , \\ 1, & \xi \geq 2\varepsilon , \end{cases} $$

and let \(k(\xi ;\varepsilon )=\rho (\xi ;\varepsilon )k(\xi )\). We note that

$$ \frac{\partial }{\partial \varepsilon }k(\xi ;\varepsilon )= \textstyle\begin{cases} 0, & \xi \leq \varepsilon , \\ -\frac{\xi }{\varepsilon ^{2}}\rho ^{\prime } ( \frac{\xi }{\varepsilon }-1 ) k(\xi ), & \varepsilon < \xi < 2\varepsilon , \\ 0, & \xi \geq 2\varepsilon . \end{cases} $$

By the non-decreasing property of ρ, we have \(\frac{\partial }{ \partial \varepsilon }k(\xi ;\varepsilon )\leq 0\). It follows from \(0\leq \rho \leq 1\) that \(k(\xi )\geq k(\xi ;\varepsilon )\) and \(\lim_{\varepsilon \rightarrow 0}k(\xi ;\varepsilon )=k(\xi )\).

We see that (7) is degenerate and singular. By the regularization technique again, we consider the problem:

$$ \left . \textstyle\begin{array}{@{}l} ( v_{2} ) _{\tau }=(v_{2}+\delta )^{r} [ ( \xi ^{\beta } ( v_{2} ) _{\xi } ) _{\xi }+a^{3-\beta }\int _{\varepsilon }^{1} ( v_{2} ) ^{q}\,d\xi ] , \quad (\xi ,\tau )\in (\varepsilon ,1)\times (0,\infty ), \\ v_{2}(0,\tau ;\delta ,\varepsilon )=v_{2}(1,\tau ;\delta ,\varepsilon )=0,\quad \tau >0, \\ v_{2}(\xi ,0;\delta ,\varepsilon )=k(\xi ;\varepsilon ), \quad \xi \in {}[ \varepsilon ,1]. \end{array}\displaystyle \right \} $$
(8)

Now, the function \(v_{2}=v_{2}(\xi ,\tau ;\delta ,\varepsilon )\) depends on ξ, τ, δ and ε. It is clear that, since the zero function is a lower solution of (8), that is, \(v_{2}\geq 0\). The next lemma show that the solution \(v_{2}\) of (8) is non-decreasing in τ.

Lemma 2.2

Let ε and δ be any positive real number with \(\varepsilon <\delta <1\). If \(\frac{\partial }{\partial \xi } ( \xi ^{\beta }\frac{\partial }{\partial \xi }k(\xi ;\varepsilon ) ) +a^{3-\beta }\int _{\varepsilon }^{1}k^{q}(\xi ;\varepsilon )\,d\xi >0\) for any \(\xi \in (\varepsilon ,1)\), then \(( v_{2} ) _{\tau } \geq 0\) for any \((\xi ,\tau )\in {}[ \varepsilon ,1]\times {}[ 0,\infty )\).

Proof

Let \(z= ( v_{2} ) _{\tau }\). Then we have, for any \((\xi ,\tau )\in (\varepsilon ,1)\times (0,\infty )\),

$$\begin{aligned} z_{\tau } =&r(v_{2}+\delta )^{-1}\bigl( ( v_{2} ) _{\tau }\bigr)^{2}+(v _{2}+\delta )^{r}\bigl(\xi ^{\beta }z_{\xi }\bigr)_{\xi } \\ &{}+q(v_{2}+ \delta )^{r}a ^{3-\beta } \int _{\varepsilon }^{1}v_{2}^{q-1}(\xi , \tau ;\delta , \varepsilon )z(\xi ,\tau )\,d\xi . \end{aligned}$$

Thus, the function z satisfies the following:

$$\begin{aligned}& z_{\tau }-(v_{2}+\delta )^{r}\bigl(\xi ^{\beta }z_{\xi }\bigr)_{\xi }\geq q(v _{2}+ \delta )^{r}a^{3-\beta } \int _{\varepsilon }^{1}v_{2}^{q-1}z( \xi ,\tau )\,d\xi ,\quad (\xi ,\tau )\in (\varepsilon ,1)\times (0,\infty ), \\& z(\varepsilon ,\tau )= ( v_{2} ) _{\tau }(\varepsilon , \tau ; \delta ,\varepsilon )=0, \qquad z(1,\tau )= ( v_{2} ) _{ \tau }(1,\tau ;\delta ,\varepsilon )=0, \quad \tau >0, \\& z(\xi ,0)=\bigl(k(\xi ;\varepsilon )+\delta \bigr)^{r} \biggl[ \frac{d}{d\xi } \biggl( \xi ^{\beta }\frac{d}{d\xi }k(\xi ;\varepsilon ) \biggr) +a ^{3-\beta } \int _{\varepsilon }^{1}k^{q}(\xi ;\varepsilon )\,d \xi \biggr] >0, \quad \xi \in {}[ \varepsilon ,1]. \end{aligned}$$

Lemma 2.1 implies that \(( v_{2} ) _{\tau }\geq 0\) for any \((\xi ,\tau )\in {}[ \varepsilon ,1]\times {}[ 0,\infty )\). □

The boundedness and monotonicity properties of \(v_{2}\) are shown in Lemma 2.3 and Lemma 2.4, respectively.

Lemma 2.3

There exist a time \(\tau _{1}\) and a function \(f\in C^{1}[0,\tau _{1}]\) such that, for all \(\varepsilon ,\delta >0\) with \(\varepsilon <\delta <1\), (8) has a unique classical solution \(v_{2}\) and \(0\leq v_{2}(\xi ,\tau ;\delta ,\varepsilon )\leq f( \tau )\) for any \((\xi ,\tau )\in [\varepsilon ,1]\times {}[ 0, \tau _{1}]\).

Proof

Let us consider the following ordinary differential equation:

$$ \left . \textstyle\begin{array}{@{}l} f^{\prime }(\tau )=a^{3-\beta }f^{q}(\tau )(f(\tau )+1)^{r},\quad \tau >0, \\ f(0)=\max_{\xi \in {}[ 0,1]}k(\xi ). \end{array}\displaystyle \right \} $$
(9)

Then there exists a positive constant \(\tau _{1}\) such that (9) has a unique positive solution f on \([0,\tau _{1}]\). We next show that, for all \(\varepsilon ,\delta >0\) with \(\varepsilon <\delta <1\), \(f(\tau )\geq v_{2}(\xi ,\tau ;\delta , \varepsilon )\) for any \((\xi ,\tau )\in {}[ \varepsilon ,1]\times {}[ 0,\tau _{1}]\). Let \(z(\xi ,\tau )=f(\tau )-v_{2}(\xi ,\tau ; \delta ,\varepsilon )\) for \((\xi ,\tau )\in {}[ \varepsilon ,1] \times {}[ 0,\tau _{1}]\). We then consider that, for any \((\xi ,\tau )\in (\varepsilon ,1)\times (0,\tau _{1}]\),

$$\begin{aligned} z_{\tau } \geq &\bigl(f(\tau )+\delta \bigr)^{r}a^{3-\beta } \int _{\varepsilon }^{1}f^{q}(\tau )\,d\xi -(v_{2}+\delta )^{r} \biggl[ \bigl( \xi ^{\beta } ( v_{2} ) _{\xi } \bigr) _{\xi }+a^{3-\beta } \int _{\varepsilon }^{1}v_{2}^{q}\,d\xi \biggr] \\ =&\bigl(f(\tau )+\delta \bigr)^{r}\bigl(\xi ^{\beta }z_{\xi } \bigr)_{\xi }+r\eta _{1}^{r-1}(v _{2}+ \delta )^{-r} ( v_{2} ) _{\tau }z \\ &{}+qa^{3-\beta }\bigl(f(\tau )+\delta \bigr)^{r} \int _{\varepsilon }^{1}\eta _{2} ^{q-1}z( \xi ,\tau )\,d\xi, \end{aligned}$$

where \(\eta _{1}\) and \(\eta _{2}\) are some intermediate values between h and \(v_{2}\). Thus, the function z satisfies \(z_{\tau }-(f( \tau )+\delta )^{r}(\xi ^{\beta }z_{\xi })_{\xi }\geq \frac{r\eta _{1} ^{r-1} ( v_{2} ) _{\tau }}{(v_{2}+\delta )^{r}}z+qa^{3- \beta }(f(\tau )+\delta )^{r}\int _{\varepsilon }^{1}\eta _{2}^{q-1}z( \xi ,\tau )\,d\xi \) for any \((\xi ,\tau )\in (\varepsilon ,1)\times (0, \tau _{1}]\) and on the parabolic boundary:

$$\begin{aligned}& z(\varepsilon ,\tau )=f(\tau )>0,\qquad z(1,\tau )=f(\tau )>0, \quad \tau \in (0,\tau _{1}], \\& z(\xi ,0)=f(0)-v_{2}(\xi ,0;\delta ,\varepsilon )=\max _{s \in {}[ 0,1]}k(s )-k(\xi ;\varepsilon )\geq 0,\quad \xi \in {}[ \varepsilon ,1]. \end{aligned}$$

Lemma 2.1 ensures that \(z\geq 0\), that is, \(v_{2}\leq f\) for any \((\xi ,\tau )\in {}[ \varepsilon ,1]\times {}[ 0,\tau _{1}]\). By modifying the proof of Theorem A.1. in [6], we see that there exists a unique classical positive solution \(v_{2}\) of (8) and \(0\leq v_{2}\leq f\) for all ε and δ. The proof of this lemma is completed. □

Lemma 2.4

Let \(0<\varepsilon _{1}<\varepsilon _{2}<\delta <1\) and assume that \(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})\) and \(v_{2}(\xi ,\tau ; \delta ,\varepsilon _{2})\) are solutions of (8). Then \(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})>v_{2}(\xi ,\tau ;\delta , \varepsilon _{2})\) for any \((\xi ,\tau )\in [\varepsilon _{2},1] \times {}[ 0, \tau _{1}]\).

Proof

Let \(z(\xi ,\tau )=v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})-v_{2}( \xi ,\tau ;\delta ,\varepsilon _{2})\) on \([\varepsilon _{2},1]\times {}[ 0, \tau _{1}]\). We have, for any \((\xi ,\tau )\in (\varepsilon _{2},1) \times (0,\tau _{1}]\),

$$\begin{aligned} z_{\tau } \geq &\bigl(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})+\delta \bigr)^{r} \biggl[ \bigl( \xi ^{\beta } \bigl( v_{2}(\xi ,\tau ;\delta , \varepsilon _{1}) \bigr) _{\xi } \bigr) _{\xi }+a^{3-\beta } \int _{\varepsilon _{2}}^{1}v_{2}^{q}(\xi , \tau ;\delta ,\varepsilon _{1})\,d\xi \biggr] \\ &{}-\bigl(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{2})+ \delta \bigr)^{r} \biggl[ \bigl( \xi ^{\beta } \bigl( v_{2}(\xi ,\tau ;\delta ,\varepsilon _{2}) \bigr) _{\xi } \bigr) _{\xi }+a^{3-\beta } \int _{\varepsilon _{2}}^{1}v_{2}^{q}(\xi , \tau ; \delta ,\varepsilon _{2})\,d\xi \biggr] \\ =&\bigl(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})+\delta \bigr)^{r}\bigl(\xi ^{ \beta }z_{\xi } \bigr)_{\xi }+r\eta _{3}^{r-1}\bigl(v_{2}( \xi ,\tau ;\delta , \varepsilon _{2})+\delta \bigr)^{-r} \bigl( v_{2}(\xi ,\tau ;\delta , \varepsilon _{2}) \bigr) _{\tau }z \\ &{}+qa^{3-\beta }\bigl(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})+\delta \bigr)^{r} \int _{\varepsilon _{2}}^{1}\eta _{4}^{q-1}z( \xi ,\tau )\,d\xi, \end{aligned}$$

where \(\eta _{3}\) and \(\eta _{4}\) are some intermediate values between \(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})\) and \(v_{2}(\xi ,\tau ; \delta ,\varepsilon _{2})\). Then it follows from \(\frac{\partial }{ \partial \varepsilon }k(\xi ;\varepsilon )\leq 0\) that the function z satisfies

$$\begin{aligned}& z_{\tau }-\bigl(v_{2}(\xi ,\tau ;\delta ,\varepsilon _{1})+\delta \bigr)^{r}\bigl( \xi ^{\beta }z_{\xi } \bigr)_{\xi } \\& \quad \geq \frac{r\eta _{3}^{r-1} ( v_{2}( \xi ,\tau ;\delta ,\varepsilon _{2}) ) _{\tau }}{(v_{2}(\xi , \tau ;\delta ,\varepsilon _{2})+\delta )^{r}}z +qa^{3-\beta }\bigl(v_{2}(\xi , \tau ;\delta ,\varepsilon _{1})+\delta \bigr)^{r} \int _{\varepsilon _{2}}^{1}\eta _{4}^{q-1}z( \xi ,\tau )\,d\xi , \\& \qquad (\xi ,\tau )\in (\varepsilon _{2},1)\times (0,\tau _{1}], \\& z(\varepsilon _{2},\tau )=v_{2}(\varepsilon _{2},\tau ;\delta , \varepsilon _{1})\geq 0,\qquad z(1,\tau )=0,\quad \tau \in (0,\tau _{1}], \\& z(\xi ,0)=k(\xi ;\varepsilon _{1})-k(\xi ;\varepsilon _{2})\geq 0,\quad \xi \in {}[ \varepsilon ,1]. \end{aligned}$$

By Lemma 2.1, we can conclude that \(v_{2}(\xi ,\tau ;\delta , \varepsilon _{1})>v_{2}(\xi ,\tau ;\delta ,\varepsilon _{2})\) for any \((\xi ,\tau )\in [\varepsilon _{2},1]\times {}[ 0, \tau _{1}]\). The proof is completed. □

From Lemma 2.3 and Lemma 2.4, we can construct the function \(v_{1}\) which is a good candidate for the solution for (7), by

$$ v_{1}(\xi ,\tau ;\delta )= \textstyle\begin{cases} \lim_{\varepsilon \rightarrow 0^{+}}v_{2}(\xi ,\tau ;\delta , \varepsilon ), & (\xi ,\tau )\in (\varepsilon ,1]\times {}[ 0,\tau _{1}], \\ 0, & (\xi ,\tau )\in \{ 0 \} \times {}[ 0,\tau _{1}], \end{cases} $$
(10)

for all \(\delta >0\). By modifying the proofs of Theorem 2.3 in [8] and Lemma 10 and Theorem 12 in [4], we obtain the existence result.

Theorem 2.5

Assume that (H1′)(H3′) hold. The function \(v_{1}(\xi ,\tau ;\delta )\) given by (10) is a unique classical solution of (7) for any \((\xi ,\tau )\in [0,1]\times {}[ 0, \tau _{1}]\) and \(\delta >0\).

In the next step, we show the existence of solutions of (6). By using the same technique as in Lemma 2.2 and Lemma 2.3, we can show that the solution \(v_{1}\) of (7) satisfies \(\frac{\partial }{\partial \tau }v _{1}(x,t;\delta )\geq 0\) for all δ and \(k(\xi )\leq v_{1}( \xi ,\tau ;\delta )\leq f(\tau )\) for any \((\xi ,\tau )\in {}[ 0,1] \times {}[ 0,\tau _{1}]\) where the function f is given in Lemma 2.3. We give an additional property of \(v_{1}\) which is the monotonicity property with respect to δ.

Lemma 2.6

Let \(0<\delta _{1}<\delta _{2}<1\) and suppose that \(v_{1}(\xi ,\tau ; \delta _{1})\) and \(v_{1}(\xi ,\tau ;\delta _{2})\) are solutions of (7). Then \(v_{1}(\xi ,\tau ;\delta _{2})>v_{1}( \xi ,\tau ;\delta _{1})\) for any \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,\tau _{1}]\).

Proof

Let \(z=v_{1}(\xi ,\tau ;\delta _{2})-v_{1}(\xi ,\tau ;\delta _{1})\) for any \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,\tau _{1}]\). By (7), we obtain, for any \((\xi ,\tau )\in (0,1) \times (0,\tau _{1}]\),

$$\begin{aligned} z_{\tau } \geq &\bigl(v_{1}(\xi ,\tau ;\delta _{2})+\delta _{2}\bigr)^{r} \biggl[ \bigl( \xi ^{\beta } \bigl( v_{1}(\xi ,\tau ;\delta _{2}) \bigr) _{ \xi } \bigr) _{\xi }+a^{3-\beta } \int _{0}^{1}v_{1}^{q}(\xi , \tau ; \delta _{2})\,d\xi \biggr] \\ &{}-\bigl(v_{1}(\xi ,\tau ;\delta _{1})+\delta _{2}\bigr)^{r} \biggl[ \bigl( \xi ^{\beta } \bigl( v_{1}(\xi ,\tau ;\delta _{1}) \bigr) _{\xi } \bigr) _{\xi }+a^{3-\beta } \int _{0}^{1}v_{1}^{q}(\xi , \tau ;\delta _{1})\,d\xi \biggr] \\ =&\bigl(v_{1}(\xi ,\tau ;\delta _{2})+\delta _{2}\bigr)^{r}\bigl(\xi ^{\beta }z_{ \xi } \bigr)_{\xi }+r\eta _{5}^{r-1}\bigl(v_{1}( \xi ,\tau ;\delta _{1})+\delta _{1}\bigr)^{-r} \bigl( v_{1}(\xi ,\tau ;\delta _{1}) \bigr) _{\tau }z \\ &{}+qa^{3-\beta }\bigl(v_{1}(\xi ,\tau ;\delta _{2})+ \delta _{2}\bigr)^{r} \int _{0} ^{1}\eta _{6}^{q-1}z( \xi ,\tau )\,d\xi , \end{aligned}$$

where \(\eta _{5}\) and \(\eta _{6}\) are some intermediate values between \(v_{1}(\xi ,\tau ;\delta _{1})\) and \(v_{1}(\xi ,\tau ;\delta _{2})\). Then the function z satisfies \(z_{\tau }-(v_{1}(\xi ,\tau ;\delta _{2})+ \delta _{2})^{r}(\xi ^{\beta }z_{\xi })_{\xi }\geq \frac{r\eta _{5}^{r-1} ( v_{1}(\xi ,\tau ;\delta _{1}) ) _{\tau }}{(v_{1}(\xi , \tau ;\delta _{1})+\delta _{1})^{r}}z+qa^{3-\beta }(v_{1}(\xi ,\tau ; \delta _{2})+\delta _{2})^{r}\int _{0}^{1}\eta _{6}^{q-1}z(\xi ,\tau )\,d\xi \) for \((\xi ,\tau )\in (0,1)\times (0,\tau _{1}]\) and on the parabolic boundary: \(z(0,\tau )=0\), \(z(1,\tau )=0\) for \(\tau \in (0, \tau _{1}]\) and \(z(\xi ,0)=0\) for \(\xi \in {}[ 0,1]\). By Lemma 2.1, we have \(v_{1}(\xi ,\tau ;\delta _{2})>v_{1}(\xi ,\tau ;\delta _{1})\) for any \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,\tau _{1}]\). □

By Lemma 2.6 and \(k(\xi )\leq v_{1}(\xi ,\tau ;\delta )\leq f(\tau )\) for any \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,\tau _{1}]\) and for all δ, we define the function v by

$$ v(\xi ,\tau )=\lim_{\delta \rightarrow 0}v_{1}(\xi ,\tau ;\delta ), \quad ( \xi ,\tau )\in (0,1)\times (0,\tau _{1}]. $$
(11)

Based on Lemma 2.7 in [6], and Lemma 10 and Theorem 12 in [4], we get the following theorem.

Theorem 2.7

Assume that (H1′)(H3′) hold. The function v given by (11) is a unique classical solution of (6) on \([0,1]\times {}[ 0, \tau _{1}]\) for some positive constant \(\tau _{1}\).

Note that by the transformations \(v=u^{m}\), \(t=\frac{\tau }{ma^{ \beta -2}}\) and \(x=a\xi \) and Theorem 2.7, we find the following.

Corollary 2.8

Assume that (H1)(H3) hold. Then there exists a time \(\widetilde{\tau }_{1}>0\) such that (1) admits a unique non-negative classical solution on \([0,a]\times {}[ 0, \widetilde{\tau }_{1}]\).

Blow-up in a finite time

The sufficient condition for the occurrence of blow-up in a finite time of (1) is given in this section. Let us consider the following problem:

$$ \left . \textstyle\begin{array}{@{}l} -(\xi ^{\beta }\varphi ^{\prime }(\xi ))^{\prime }=\lambda \varphi ( \xi ),\quad \xi \in (0,1), \\ \varphi (0)=\varphi (1)=0. \end{array}\displaystyle \right \} $$
(12)

From [4], the eigenvalue problem (12) is solvable. Denote the first eigenvalue of (12) by \(\lambda _{1}>0\) and the corresponding eigenfunction by \(\varphi _{1}\), with the normalization \(\varphi _{1}>0\) in \((0,1)\) and \(\max_{\xi \in [0,1]}\varphi _{1}(\xi )=1\). The next theorem deals with the condition that guarantee for the occurrence of blow-up in a finite time depending on the value of the constant a.

Theorem 3.1

Suppose that the function k satisfies (H1′)(H3′). If the constant a satisfies

$$ a>\max \biggl\{ \biggl( \frac{\lambda _{1}}{\int _{0}^{1}\varphi _{1}( \xi )\,d\xi } \biggr) ^{\frac{q}{3-\beta }}, \biggl( \frac{1}{\int _{0} ^{1}k^{q}(\xi )\,d\xi } \biggr) ^{\frac{1}{3-\beta }} \biggr\} , $$

then the solution v of (6) blows up in a finite time.

Proof

Let \(H(\tau )=\int _{0}^{1}v^{1-r}(\xi ,\tau )\varphi _{1}(\xi )\,d\xi \). We then have

$$\begin{aligned} \frac{1}{1-r}H^{\prime }(\tau ) =& \int _{0}^{1}\bigl(\xi ^{\beta }v_{\xi } \bigr)_{ \xi }\varphi _{1}(\xi )\,d\xi +a^{3-\beta } \int _{0}^{1}v^{q}\,d\xi \int _{0}^{1}\varphi _{1}(\xi )\,d\xi \\ =&-\lambda _{1} \int _{0}^{1}v(\xi ,\tau )\varphi _{1}( \xi )\,d\xi +a^{3- \beta } \int _{0}^{1}v^{q}\,d\xi \int _{0}^{1}\varphi _{1}(\xi )\,d\xi . \end{aligned}$$

From

$$\begin{aligned} \int _{0}^{1}v(\xi ,\tau )\varphi _{1}( \xi )\,d\xi \leq &\frac{1}{a^{(3- \beta )/q}} \biggl( \int _{0}^{1}a^{3-\beta }v^{q}\,d\xi \biggr) ^{ \frac{1}{q}} \biggl( \int _{0}^{1}\varphi _{1}^{\frac{q}{q-1}}( \xi )\,d\xi \biggr) ^{1-\frac{1}{q}} \\ \leq &\frac{1}{a^{(3-\beta )/q}} \biggl( \int _{0}^{1}a^{3-\beta }v ^{q}\,d \xi \biggr) ^{\frac{1}{q}}, \end{aligned}$$

we obtain

$$ \frac{1}{1-r}H^{\prime }(\tau )\geq - \frac{\lambda _{1}}{a^{(3-\beta )/q}} \biggl( \int _{0}^{1}a^{3-\beta }v ^{q}\,dx \biggr) ^{\frac{1}{q}}+a^{3-\beta } \int _{0}^{1}v^{q}\,dx \int _{0} ^{1}\varphi _{1}(\xi )\,d\xi . $$

From \(v_{\tau }\geq 0\) and \(a^{3-\beta }\int _{0}^{1}k^{q}(\xi )\,d\xi \geq 1\), we obtain \(a^{3-\beta }\int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi \geq 1\). It follows that \(( \int _{0}^{1}a^{3-\beta }v^{q}\,d\xi ) ^{\frac{1}{q}}\leq a^{3-\beta }\int _{0}^{1}v^{q}(\xi , \tau )\,d\xi\) with \(q>1\). Then

$$\begin{aligned} \frac{1}{1-r}H^{\prime }(\tau ) \geq &-\frac{\lambda _{1}a^{3-\beta }}{a ^{(3-\beta )/q}} \int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi +a^{3-\beta } \int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi \int _{0}^{1}\varphi _{1}(\xi )\,d\xi \\ =&a^{3-\beta } \int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi \biggl[ -\frac{ \lambda _{1}}{a^{(3-\beta )/q}}+ \int _{0}^{1}\varphi _{1}(\xi )\,d\xi \biggr] . \end{aligned}$$

By the assumption that \(\lambda _{1}< a^{(3-\beta )/q}\int _{0}^{1} \varphi _{1}(\xi )\,d\xi \), we have \(- \frac{\lambda _{1}}{a^{(3-\beta )/q}}+\int _{0}^{1}\varphi _{1}(\xi )\,d\xi \geq \eta _{9}\) where \(\eta _{9}\) is a positive constant. Thus,

$$ \frac{1}{1-r}H^{\prime }(\tau )\geq \eta _{9}a^{3-\beta } \int _{0}^{1}v ^{q}(\xi ,\tau )\,d\xi . $$

Since

$$ \int _{0}^{1}v^{1-r}(\xi ,\tau )\varphi _{1}(\xi )\,d\xi \leq \biggl( \int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi \biggr) ^{\frac{1-r}{q}} \biggl( \int _{0}^{1}\varphi _{1}^{\frac{q}{q+r-1}}( \xi )\,d\xi \biggr) ^{ \frac{q+r-1}{q}}, $$

we get

$$ \int _{0}^{1}v^{q}(\xi ,\tau )\,d\xi \geq \biggl( \int _{0}^{1}v ^{1-r}(\xi ,\tau )\varphi _{1}(\xi )\,d\xi \biggr) ^{\frac{q}{1-r}}\Big/ \biggl( \int _{0}^{1}\varphi _{1}^{ \frac{q}{q+r-1}}( \xi )\,d\xi \biggr) ^{\frac{q+r-1}{1-r}}. $$

We then obtain

$$\begin{aligned} \frac{1}{1-r}H^{\prime }(\tau ) \geq & \eta _{9}a^{3-\beta } \biggl( \int _{0}^{1}v^{1-r}(\xi ,\tau )\varphi _{1}(\xi )\,d\xi \biggr) ^{\frac{q}{1-r}}\Big/ \biggl( \int _{0}^{1}\varphi _{1}^{ \frac{q}{q+r-1}}( \xi )\,d\xi \biggr) ^{\frac{q+r-1}{1-r}} \\ \geq &\eta _{9}a^{3-\beta }H^{\frac{q}{1-r}}(\tau ), \end{aligned}$$

that is,

$$ \bigl( H^{1-q/(1-r)}(\tau ) \bigr) ^{\prime }\leq \eta _{9}a^{3- \beta }(1-r-q). $$
(13)

Integrating (13) over \((0,\tau )\), we get

$$ H^{1-q/(1-r)}(\tau )-H^{1-q/(1-r)}(0)\leq \eta _{9}a^{3-\beta }(1-r-q) \tau $$

or

$$ H^{\frac{q}{1-r}-1}(\tau )\geq \frac{H^{\frac{q}{1-r}-1}(0)}{1-\eta _{9}a ^{3-\beta }(q+r-1)H^{\frac{q}{1-r}-1}(0)\tau }. $$

We can see that \(H^{\frac{q}{1-r}-1}(\tau )\) exists for \(\tau \in {}[ 0,T_{b})\) but \(H^{\frac{q}{1-r}-1}(\tau )\) is unbounded as τ converges to \(T_{b}\) where

$$ T_{b}=\frac{H^{1-\frac{q}{1-r}}(0)}{\eta _{9}a^{3-\beta }(q+r-1)}=\frac{1}{ \eta _{9}a^{3-\beta }(q+r-1)} \biggl( \int _{0}^{1}k^{1-r}(\xi )\varphi _{1}( \xi )\,d\xi \biggr) ^{\frac{-(q+r-1)}{1-r}}. $$

Therefore, H blows up in a finite time. This implies that v blows up in a finite time. Then the proof of this theorem is completed. □

By the transformation technique and Theorem 3.1, we obtain the following.

Corollary 3.2

Suppose that g satisfies (H1)(H3). Then the solution u of (1) blows up in a finite time if the constant a is sufficiently large.

In the following, we show that under some conditions, the solution v of (6) can exist globally. To obtain the desired results, we need the following comparison theorem.

Lemma 3.3

Let v be the solution of (6) and suppose that a non-negative function \(w\in C^{2,1}((0,1)\times (0,T))\cap C([0,1] \times {}[ 0,T])\) satisfies

$$\begin{aligned}& w_{\tau }\geq (\leq )\ w^{r} \biggl[ \bigl(\xi ^{\beta }w_{\xi }\bigr)_{\xi }+a^{3- \beta } \int _{0}^{1}w^{q}(\xi ,\tau )\,d\xi \biggr] , \quad (\xi ,\tau ) \in (0,1)\times (0,T], \\& w(0,\tau )\geq (=)\ 0,\qquad w(1,\tau )\geq (=)\ 0, \quad \tau \in (0,T], \\& w(\xi ,0)\geq (\leq )\ k(\xi ),\quad \xi \in {}[ 0,1]. \end{aligned}$$

Then \(w\geq (\leq )\ v\) on \([0,1]\times {}[ 0, T]\).

Proof

We first consider in the case “≥”. Let \(z(\xi ,\tau )=w(\xi , \tau )-v(\xi ,\tau )\) on \([0,1]\times {}[ 0, T]\). It is clear that, from Lemma 2.1 and (H2′), \(v>0\) in \((0,1)\times (0,T]\). We then have, for any \((0,1)\times (0,T]\),

$$ z_{\tau }=w^{r}\bigl(\xi ^{\beta }z_{\xi } \bigr)_{\xi }+r\eta _{7}^{r-1}v^{-r}v _{\tau }z+qa^{3-\beta }w^{r} \int _{0}^{1}\eta _{8}^{q-1}z( \xi ,\tau )\,d\xi, $$

where \(\eta _{7}\) and \(\eta _{8}\) are some intermediate values between w and v. Then the function z satisfies

$$\begin{aligned}& z_{\tau }-w^{r}\bigl(\xi ^{\beta }z_{\xi } \bigr)_{\xi }=r\eta _{7}^{r-1}v^{-r}v _{\tau }z+qa^{3-\beta }w^{r} \int _{0}^{1}\eta _{8}^{q-1}z( \xi ,\tau )\,d\xi ,\quad (\xi ,\tau )\in (0,1)\times (0,T], \\& z(0,\tau )\geq 0,\qquad z(1,\tau )\geq 0,\quad \tau \in (0,T], \\& z(\xi ,0)\geq 0, \quad \xi \in {}[ 0,1]. \end{aligned}$$

Lemma 2.1 implies that \(w(\xi ,\tau )\geq v(\xi ,\tau )\) for any \((\xi ,\tau )\in {}[ 0,1]\times {}[ 0,T]\). By using the above technique, we can get the result in the case “≤”. The proof of this lemma is completed. □

Let us consider the following boundary value problem:

$$ -\bigl(\xi ^{\beta }\psi ^{\prime }(\xi )\bigr)^{\prime }=1, \quad \xi \in (0,1)\quad \text{and}\quad \psi (0)=\psi (1)=0. $$

The solution ψ is given by \(\psi (\xi )=\frac{1}{2-\beta } \xi ^{1-\beta }(1-\xi )\) for \(\xi \in (0,1)\). By direct computation, we obtain \(\int _{0}^{1}\psi ^{q}(\xi )\,d\xi =\frac{B(q(1-\beta )+1,q+1)}{(2- \beta )^{q}}\) where \(B(l,m)\) is the Beta function which is defined by \(B(l,m)=\int _{0}^{1}\xi ^{l-1}(1-\xi )^{m-1}\,d\xi \). The following theorem deals with the global existence result.

Theorem 3.4

Suppose that k satisfies (H1′)(H3′). Then the solution v of (6) exists globally if a is small enough.

Proof

Let \(z(\xi ,\tau )=M_{1}\psi (\xi )\) on \([0,1]\times {}[ 0, \infty )\) where \(M_{1}\) is a positive constant and \(M_{1}\psi (\xi ) \geq k\). We choose \(a\leq ( \frac{(2-\beta )^{q}}{M^{q-1}B(q(1-\beta )+1,q+1)} ) ^{\frac{1}{3- \beta }}\) and then the function z satisfies

$$\begin{aligned}& z_{\tau }-z^{r} \biggl[ \bigl(\xi ^{\beta }z_{\xi } \bigr)_{\xi }+a^{3-\beta } \int _{0}^{1}z^{q}(\xi ,\tau )\,d\xi \biggr] \\& \quad = M_{1}^{r}\psi ^{r}(\xi ) \biggl[ M_{1}-a^{3-\beta }M_{1}^{q} \frac{B(q(1- \beta )+1,q+1)}{(2-\beta )^{q}} \biggr] \end{aligned}$$

for any \((\xi ,\tau )\in (0,1)\times (0,\infty )\). Thus, \(z_{\tau }-z ^{r} [ (\xi ^{\beta }z_{\xi })_{\xi }+a^{3-\beta }\int _{0}^{1}z ^{q}(\xi ,\tau )\,d\xi ] \geq 0\) for \((\xi ,\tau )\in (0,1) \times (0,\infty )\). Furthermore, \(z(0,\tau )=z(1,\tau )=0\) for \(\tau >0\) and \(z(\xi ,0)=M_{1}\psi (\xi )\geq k(\xi )\) for \(\xi \in {}[ 0,1]\). Lemma 3.3 implies that \(z\geq v\) on \([0,1]\times {}[ 0, \infty )\). We can conclude that the solution v of (6) exists globally. □

It follows from the transformation technique and Theorem 3.4 that we have the following.

Corollary 3.5

Suppose that g satisfies (H1)(H3). Then the solution u of (1) exists globally if a is small enough.

Blow-up set and uniform blow-up profile

In this section, we assume that the solution u of (1) blows up at the blow-up time \(T_{b.}\) Then we discuss the set of blow-up points and the blow-up profile for the solution u of (1). From the assumptions (H1)–(H5), we know that there are a sufficiently small positive constant \(\varepsilon _{1}\) and a non-negative function \(h(\xi ;\varepsilon )\) for \(0<\varepsilon \leq \varepsilon _{1}\) such that:

(H1):

\(h(\xi ;\varepsilon )\in C^{2+\alpha }(\varepsilon ,1-\varepsilon )\cap C[\varepsilon ,1-\varepsilon ]\) with \(\alpha \in (0,1)\),

(H2):

\(h(\varepsilon ;\varepsilon )=0\) and \(h(1- \varepsilon ;\varepsilon )=0\),

(H3):

\(h(\xi ;\varepsilon )< k(\xi )\) for \(\xi \in ( \varepsilon ,2\varepsilon )\cup (1-2\varepsilon ,1-\varepsilon )\) and \(h(\xi ;\varepsilon )=k(\xi )\) for \(\xi \in (2\varepsilon ,1-2\varepsilon )\),

(H4):

\(( \xi ^{\beta }h_{\xi }(\xi ;\varepsilon ) ) _{\xi }\leq 0\) for \(\xi \in (\varepsilon ,1-\varepsilon )\),

(H5):

\(h(\xi ;\varepsilon )\) is non-increasing with respect to \(\varepsilon \in (0,\varepsilon _{1}]\), \(\lim_{\xi \rightarrow \varepsilon } ( \xi ^{\beta }h_{\xi }( \xi ;\varepsilon ) ) _{\xi }=-a^{3-\beta } \int _{\varepsilon } ^{1-\varepsilon }h^{q}(\xi ;\varepsilon )\,d\xi \) and \(\lim_{\xi \rightarrow 1-\varepsilon } ( \xi ^{\beta }h_{\xi }( \xi ;\varepsilon ) ) _{\xi }=-a^{3-\beta }\int _{\varepsilon } ^{1-\varepsilon }h^{q}(\xi ;\varepsilon )\,d\xi \),

(H6):

\(( \xi ^{\beta }h_{\xi }(\xi ;\varepsilon ) ) _{\xi }+a^{3-\beta }\int _{\varepsilon }^{1-\varepsilon }h ^{q}(\xi ;\varepsilon )\,d\xi \geq 0\) for \(\varepsilon \in (0, \varepsilon _{1}]\) and \(\xi \in (\varepsilon ,1-\varepsilon )\).

It is obvious that \(\lim_{\varepsilon \rightarrow 0}h(\xi ;\varepsilon )=k(\xi )\). We next consider the following regularized problem:

$$ \left . \textstyle\begin{array}{@{}l} w_{\tau }=(w+\delta )^{r} [ (\xi ^{\beta }w_{\xi })_{\xi }+a^{3- \beta }\int _{\varepsilon }^{1-\varepsilon }w^{q}(\xi ,\tau ;\delta , \varepsilon )\,d\xi ] , \\ \quad (\xi ,\tau )\in (\varepsilon ,1-\varepsilon )\times (0,\infty ), \\ w(\varepsilon ,\tau ;\delta ,\varepsilon )=w(1-\varepsilon ,\tau ; \delta ,\varepsilon )=0, \quad \tau >0, \\ w(\xi ,0;\delta ,\varepsilon )=h(\xi ;\varepsilon ),\quad \xi \in {}[ \varepsilon ,1-\varepsilon ]. \end{array}\displaystyle \right \} $$
(14)

In the same way as before, it is not difficult to show that the regularized problem (14) has a unique positive solution w and

$$ \lim_{\delta \rightarrow 0,\varepsilon \rightarrow 0}w(\xi ,\tau ; \delta ,\varepsilon )=v(\xi ,\tau ), $$

where v is the solution of (6). To find the blow-up set and blow-up profile of the blow-up solution u of (1), we need the following lemma.

Lemma 4.1

Assume that k satisfies (H1′)(H5′). Before blow-up occurs, \((\xi ^{\beta }v_{\xi })_{\xi }\leq 0\) for \((\xi , \tau )\in (0,1)\times {}[ 0,T_{b})\).

Proof

Let ε and δ be positive constants with \(\varepsilon <\delta <1\). From \((\xi ^{\beta }h_{\xi })_{\xi }+a^{3- \beta }\int _{\varepsilon }^{1-\varepsilon }h^{q}(\xi ; \varepsilon )\,d\xi \geq 0\) for \(\xi \in (\varepsilon ,1-\varepsilon )\), we have \(w_{\tau }\geq 0\) for \((\xi ,\tau )\in (\varepsilon ,1-\varepsilon ) \times {}[ 0,T_{b})\). Let \(z(\xi ,\tau )=(\xi ^{\beta }w_{\xi })_{ \xi }\) for \((\xi ,\tau )\in (\varepsilon ,1-\varepsilon )\times {}[ 0,T_{b})\). We consider that, for \((\xi ,\tau )\in (\varepsilon ,1-\varepsilon )\times (0,T_{b})\),

$$\begin{aligned}& z_{\tau }-(w+\delta )^{r}\bigl(\xi ^{\beta }z_{\xi } \bigr)_{\xi }-2r(w+\delta )^{r-1} \xi ^{\beta }w_{\xi }z_{\xi }-r(w+ \delta )^{-1}w_{\tau }z \\& \quad = r(r-1) (w+\delta )^{-2}\xi ^{\beta }w_{\tau }(w_{\xi })^{2}. \end{aligned}$$

This means that \(z_{\tau }-(w+\delta )^{r}(\xi ^{\beta }z_{\xi })_{ \xi }-2r(w+\delta )^{r-1}\xi ^{\beta }w_{\xi }z_{\xi }-r(w+\delta )^{-1}w _{\tau }z \leq 0\). for \((\xi ,\tau )\in (\varepsilon ,1-\varepsilon ) \times (0,T_{b})\). Furthermore, we have

$$ z(\varepsilon ,\tau )=\bigl(\xi ^{\beta }w_{\xi } \bigr)_{\xi }\vert _{\xi =\varepsilon }=-a^{3-\beta } \int _{\varepsilon }^{1-\varepsilon }w^{q}(\xi ,\tau ;\delta , \varepsilon )\,d\xi < 0 $$

and

$$ z(1-\varepsilon ,\tau )=\bigl(\xi ^{\beta }w_{\xi } \bigr)_{\xi }\vert _{\xi =1-\varepsilon }=-a^{3-\beta } \int _{\varepsilon }^{1-\varepsilon }w^{q}(\xi ,\tau ;\delta , \varepsilon )\,d\xi < 0. $$

It follows from (H4) that \(z(\xi ,0)\leq 0\) for \(\xi \in {}[ \varepsilon ,1-\varepsilon ]\). By applying Lemma 2.1, we obtain \(z\leq 0\) on \([\varepsilon ,1-\varepsilon ]\times {}[ 0, T_{b})\). Since ε and δ are arbitrary, we have \((\xi ^{ \beta }v_{\xi })_{\xi }\leq 0\) for \((\xi ,\tau )\in (0,1)\times {}[ 0,T_{b})\). Hence, the proof of this lemma is completed. □

From Lemma 4.1, we obtain the following corollary.

Corollary 4.2

Assume that g satisfies (H1)(H5). Before blow-up occurs, \(( x^{\beta }u_{x}^{m} ) _{x}\leq 0\) for \((x,t)\in (0,a) \times {}[ 0,T_{b})\).

The next theorem states about the set of blow-up point of the solution u of (1). By modifying techniques in [16], we obtain this result.

Theorem 4.3

Assume that the solution u of (1) blows up in a finite time \(T_{b}\). Then \(S=[0,a]\).

Proof

Let ε be any positive constant. We construct functions ϕ and Φ by \(\phi (t)= \int _{0}^{a}u^{p}(x,t)\,dx\) and \(\varPhi (t)=\int _{0}^{t}\phi (s)\,ds\). We set \(M_{2}= \inf_{x\in (\varepsilon ,a-\varepsilon )}\mu (x)\) where μ is the unique positive solution of the following problem

$$\begin{aligned}& -\frac{d}{dx} \biggl( x^{\beta }\frac{d}{dx}\mu ^{m}(x) \biggr) =1,\quad x \in (0,a), \\& \mu (0)=\mu (a)=0. \end{aligned}$$

Corollary 4.2 yields, for \(t\in (0,T_{b})\),

$$ \int _{0}^{a}u^{m}(x,t)\,dx=- \int _{0}^{a}u^{m}(x,t)\frac{d}{dx} \biggl( x ^{\beta }\frac{d}{dx}\mu ^{m}(x) \biggr) \,dx\geq -M_{2}^{m} \int _{\varepsilon }^{a-\varepsilon }\bigl(x^{\beta }u_{x}^{m} \bigr)_{x}\,dx. $$

We then obtain

$$ 0\leq \lim_{t\rightarrow T_{b}}\frac{-M_{2}^{m}\int _{\varepsilon } ^{a-\varepsilon }(x^{\beta }u_{x}^{m})_{x}\,dx}{\phi (t)}\leq \lim _{t\rightarrow T_{b}}\frac{\int _{0}^{a}u^{m}(x,t)\,dx}{\int _{0}^{a}u ^{p}(x,t)\,dx}=0 $$

and this implies that \(\lim_{t\rightarrow T_{b}}\frac{\int _{\varepsilon }^{a-\varepsilon }(x^{\beta }u_{x}^{m})_{x}\,dx}{\phi (t)}=0\). As \(\varepsilon \rightarrow 0\), we obtain

$$ \lim_{t\rightarrow T_{b}}\frac{(x^{\beta }u_{x}^{m})_{x}}{\phi (t)}=0\quad \text{for }x\in (0,a). $$
(15)

Integrating the first equation in (1) with respect to t from 0 to t, we have

$$ u(x,t)-g(x)= \int _{0}^{t}\bigl(x^{\beta }u_{x}^{m}(x,s) \bigr)_{x}\,ds+\varPhi (t). $$
(16)

Since u blows up at the finite time \(T_{b}\), \(\lim_{t\rightarrow T _{b}}u(x_{b},t)=\infty \) for some \(x_{b}\in (0,a)\), and then we obtain

$$ \lim_{t\rightarrow T_{b}}u(x_{b},t)-\lim_{t\rightarrow T_{b}}g(x_{b})= \lim_{t\rightarrow T_{b}} \int _{0}^{t}\bigl(x_{b}^{\beta }u_{x}^{m}(x_{b},s) \bigr)_{x}\,ds+ \lim_{t\rightarrow T_{b}}\varPhi (t) $$

or

$$ \lim_{t\rightarrow T_{b}}\varPhi (t)=\infty . $$
(17)

It follows from (15) and (17) that

$$ \lim_{t\rightarrow T_{b}}\frac{\int _{0}^{t}(x^{\beta }u_{x}^{m}(x,s))_{x}\,ds}{ \varPhi (t)}=0 \quad \text{for }x\in (0,a). $$
(18)

Let be a fixed point in \((0,a)\). We have, by (16),

$$ \lim_{t\rightarrow T_{b}}\frac{u(\widetilde{x},t)}{\varPhi (t)}= \lim_{t\rightarrow T_{b}} \frac{g(\widetilde{x})}{\varPhi (t)}+ \lim_{t\rightarrow T_{b}}\frac{\int _{0}^{t}(\widetilde{x}^{\beta }u _{x}^{m}(\widetilde{x},s))_{x}\,ds}{\varPhi (t)}+1. $$

Equations (17) and (18) imply

$$ \lim_{t\rightarrow T_{b}}\frac{u(\widetilde{x},t)}{\varPhi (t)}=1, $$
(19)

which means that the solution u of (1) blows up at the point . Since is arbitrary in \((0,a)\), we can conclude that the solution u of (1) blows up everywhere in \((0,a)\). For \(\widetilde{x}\in \{0,a\}\), we can always find a sequence \(\{(x_{n},t_{n})\}\) in \((0,a)\times (0,T_{b})\) such that \((x_{n},t_{n})\rightarrow \{\widetilde{x},T_{b}\}\) and \(\lim_{n\rightarrow \infty }u(x_{n},t_{n})=\infty \). Hence, the blow-up set is \([0,a]\). The proof of Theorem 4.3 is completed. □

Finally, we consider the uniform blow-up profile of the solution u of (1).

Theorem 4.4

Assume that g satisfies (H1)(H5).

Then \(u(x,t)\sim [ a(p-1)(T_{b}-t) ] ^{-\frac{1}{p-1}}\) for any \(x\in (0,a)\) as \(t\rightarrow T_{b}\).

Proof

Equation (19) tells us that, for any \(x \in (0,a)\),

$$ u(x,t)\sim \varPhi (t) \quad \text{as } t \rightarrow T_{b}. $$
(20)

Then we get

$$ \varPhi ^{\prime }(t)= \int _{0}^{a}u^{p}(x,t)\,dx\sim a\varPhi ^{p}(t)\quad \text{as } t\rightarrow T_{b}. $$
(21)

Integrating (21) over \((t,T_{b})\), we have, by (17),

$$ \varPhi (t)\sim \bigl[ a(p-1) (T_{b}-t) \bigr] ^{-\frac{1}{p-1}}\quad \text{as }t\rightarrow T_{b}. $$
(22)

It follows from (20) and (22) that, as t approaches the blow-up time \(T_{b}\), \(u(x,t)\sim [ a(p-1)(T_{b}-t) ] ^{-\frac{1}{p-1}}\) for any \(x \in (0,a)\). □

Conclusion

In this paper, we study a degenerate and singular parabolic problem with a nonlocal term. We show that such a problem has a local classical solution. Furthermore, the conditions that the solution exists globally or blows up in finite time are given. Finally, we demonstrate the uniform blow-up profile of the blow-up solution.

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Author information

NS and PS formulated the research problem and wrote the paper. NS,PS and WS participated in the derivation of the mathematical results. All authors read and approved the final manuscript.

Correspondence to Panumart Sawangtong.

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MSC

  • 35B44
  • 35K67
  • 36K65

Keywords

  • Degenerate and singular parabolic equations
  • Nonlocal source
  • Blow-up set
  • Uniform blow-up profile