1 Introduction

Payne et al. [1] studied the blow-up phenomenon for solutions to the following family of mixed problems:

$$\begin{aligned}& \frac{\partial u}{\partial t}=\bigl(\rho\bigl(|\nabla u|^{2}\bigr)u_{,i} \bigr)_{,i}+f(u) \quad \mbox{in } \Omega\times\bigl(0,t^{*} \bigr), \end{aligned}$$
(1.1)
$$\begin{aligned}& u(x,0)=g(x)\geq0 \quad\mbox{in } \Omega, \end{aligned}$$
(1.2)
$$\begin{aligned}& u(x,t)=0 \quad\mbox{in } \partial\Omega\times\bigl(0,t^{*}\bigr). \end{aligned}$$
(1.3)

They obtained a lower bound for the blow-up time \(t^{*}\) if the blow-up does really occur together with a criterion for getting a blow-up. Moreover, they proposed conditions that ensure that a blow-up cannot occur. In this paper, we continue the work of Payne, Philippin, and Schaefer. In [1], they obtained the lower bound for the blow-up time of solutions in a bounded domain \(\Omega\in\mathbb{R}^{N}\) for \(N=3\). If one is interested in generalizations to the case \(N>3\), then one important tool, which is important for proving the results obtained in [1], namely, the Sobolev inequality is no longer applicable. There are only a few papers dealing with a lower bound for the blow-up time when \(N>3\) (see [2, 3]). Our goal is to get a lower bound for the blow-up time of the solutions to (1.1)-(1.3) in \(\Omega\in\mathbb{R}^{N}\) for any \(N\geq3\).

The study of finite-time blow-up of solutions to parabolic problems under a homogeneous Dirichlet boundary condition and Neumann condition has earned great attention (see [4–10]). Recently, some papers began to consider the blow-up phenomena of these problems under the Robin boundary conditions (see [11–14]). Many methods have been used to study equations (1.1)-(1.3) (see [15–17]).

In this paper, Ω is a bounded star-shaped domain in \(\mathbb{R}^{N}\) (\(N\geq3\)) with smooth boundary ∂Ω. The operator ∇ is the gradient operator, and \(t^{*}\) is the possible blow-up time. Furthermore, i stands for the partial differentiation with respect to \(x_{i}\), \(i=1,2,3,\ldots,N\). The repeated index indicates Einstein’s summation convention over the indices. We assume that ρ is a positive \(C^{1}\) function that satisfies

$$ \rho(s)+2s\rho'(s)>0, \quad s>0, $$
(1.4)

so that \((\rho u_{,i})_{,i}\) is an elliptic operator. We also assume that ρ and f satisfy the conditions

$$ 0< f(s)\leq a_{1}+a_{2}s^{p}, \quad s>0, $$
(1.5)

and

$$ \rho(s)\geq b_{1}+b_{2}s^{q}, \quad s>0, $$
(1.6)

where \(p>1\) and \(0<2q<p-1\), and \(a_{1}\), \(a_{2}\), \(b_{1}\), \(b_{2}\) are positive constants. Using the maximum principle, we can get that u is nonnegative in x and \(t \in[0,t^{*})\).

In the further discussions, we will use the following Hölder inequality:

$$ \int_{\Omega}w^{x_{1}+x_{2}}\,dx\leq \biggl( \int_{\Omega}w^{\frac{x_{1}}{\alpha}}\,dx \biggr)^{\alpha} \biggl( \int_{\Omega}w^{\frac{x_{2}}{1-\alpha}}\,dx \biggr)^{1-\alpha}, $$
(1.7)

where \(0<\alpha<1\), and \(x_{1}\), \(x_{2}\) are positive constants.

2 Lower bound for the blow-up time

In this section, we define the auxiliary function \(\varphi=\varphi(t)\) as follows (see [1]):

$$ \varphi(t)= \int_{\Omega}u^{2(n-1)(q+1)+2}\,dx= \int_{\Omega}u^{\sigma }\,dx \quad\mbox{with } \sigma=2(n-1) (q+1)+2. $$
(2.1)

We establish the following theorem.

Theorem 1

Assume that \(u=u(x,t)\) is the classical nonnegative solution of the mixed problem (1.1)-(1.3) in a bounded domain \(\Omega\in\mathbb{R}^{N} \) (\(N\geq3\)). Then the quantity \(\varphi(t)\) defined in (2.1) satisfies the differential inequality

$$ \varphi'(t) \leq\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma }}+k_{1} \bigl[ \phi(t) \bigr]^{\frac{(N-2)\alpha}{N\alpha-2}}+ k_{2} \bigl[\phi(t) \bigr]^{\frac{(N-2)\alpha'}{N\alpha'-2}}, $$
(2.2)

which yields that the blow-up time \(t^{*}\) is bounded from below. We have

$$ t ^{*}\geq \int_{\phi(0)}^{+\infty}\frac{d\xi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\xi]^{\frac{\sigma-1}{\sigma}}+k_{1} [\xi ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\xi ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}}, $$
(2.3)

where \(|\Omega|\) is the volume of the domain Ω, and \(k_{1}\), \(k_{2}\) are positive constants that will be defined later.

Proof

First, we compute

$$\begin{aligned} \varphi'(t)={}&\sigma \int_{\Omega}u^{\sigma-1} \bigl[\bigl(\rho\bigl(|\nabla u|^{2}\bigr)u_{,i}\bigr)_{,i}+f(u) \bigr]\,dx \\ ={}&{-} \sigma(\sigma-1) \int_{\Omega}u^{\sigma-2}\rho\bigl(|\nabla u|^{2} \bigr)|\nabla u|^{2}\,dx+\sigma \int_{\Omega}u^{\sigma-1}f(u)\,dx \\ \leq{}& {-} \sigma(\sigma-1) \int_{\Omega}u^{2(n-1)(q+1)}|\nabla u|^{2} \bigl(b_{1}+b_{2}|\nabla u|^{2q}\bigr)\,dx \\ &{}+\sigma \int_{\Omega}u^{\sigma-1}\bigl(a_{1}+a_{2}u^{p} \bigr)\,dx. \end{aligned}$$
(2.4)

Using the equality

$$ \bigl|\nabla u^{n}\bigr|^{2(q+1)}=\bigl|nu^{n-1}\nabla u\bigr|^{2(q+1)}=n^{2(q+1)}u^{2(n-1)(q+1)}|\nabla u|^{2(q+1)} $$

and the Hölder inequality, we get

$$ \varphi'(t) \leq - \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}} \int_{\Omega}\bigl|\nabla u^{n}\bigr|^{2(q+1)}\,dx+\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma}}+\sigma a_{2} \int_{\Omega}u^{\sigma+p-1}\,dx. $$
(2.5)

If we set \(v=u^{n}\), then we obtain

$$ \varphi'(t) \leq - \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}} \int_{\Omega}|\nabla v|^{2(q+1)}\,dx+\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma}}+\sigma a_{2} \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx, $$
(2.6)

where \(\gamma=p-1-2q>0\). After application of the Hölder and Schwarz inequalities, it follows

$$\begin{aligned} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx& \leq(q+1)^{2} \biggl( \int_{\Omega}|\nabla v|^{2(q+1)}\,dx \biggr)^{\frac{1}{q+1}} \biggl( \int_{\Omega}| v|^{2(q+1)}\,dx \biggr)^{\frac{q}{q+1}} \\ &\leq(q+1) \int_{\Omega}|\nabla v|^{2(q+1)}\,dx+(q+1)q \int_{\Omega}| v|^{2(q+1)}\,dx. \end{aligned}$$
(2.7)

Combining (2.6) and (2.7), we easily obtain

$$\begin{aligned} \varphi'(t) \leq{}& {-} \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}(q+1)} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx+ \frac{q\sigma(\sigma-1)b_{2}}{n^{2(q+1)}} \int_{\Omega}v^{2(q+1)}\,dx+\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma}} \\ &{}+\sigma a_{2} \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx. \end{aligned}$$
(2.8)

We choose \(x_{1}\), \(x_{2}\), and α such that

$$ x_{1}+x_{2}=2(q+1), \qquad x_{1}\cdot \frac{1}{\alpha}=\frac{\sigma}{n},\qquad x_{2}\cdot \frac{1}{1-\alpha}=(q+1)\frac{2N}{N-2}, $$

so that

$$ \begin{aligned} &x_{1}=\frac{\sigma}{n}\frac{2(q+1)\frac{2}{N-2}}{2(q+1)\frac {N}{N-2}-\frac{\sigma}{n}}, \qquad x_{2}=2(q+1)- \frac{\sigma}{n}\frac{2(q+1)\frac{2}{N-2}}{2(q+1)\frac{N}{N-2}-\frac {\sigma}{n}}, \\ &\alpha=\frac{2(q+1)\frac{2}{N-2}}{2(q+1)\frac{N}{N-2}-\frac{\sigma}{n}}. \end{aligned} $$

Then the Hölder inequality (1.7) yields

$$ \int_{\Omega}v^{2(q+1)}\,dx\leq \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\alpha}\biggl( \int_{\Omega}v^{(q+1)\frac{2N}{N-2}}\,dx \biggr)^{1-\alpha}. $$
(2.9)

We follow the same procedure for \(x_{1}'\), \(x_{2}'\), and \(\alpha'\), that is, we choose them such that

$$ x_{1}'+x_{2}'=2(q+1)+ \frac{\gamma}{n}, \qquad x_{1}\cdot \frac{1}{\alpha'}= \frac{\sigma}{n}, \qquad x_{2}'\cdot\frac{1}{1-\alpha'}=(q+1)\frac{2N}{N-2}, $$

so that

$$ \begin{aligned} &x_{1}'=\frac{\sigma}{n} \frac{2(q+1)\frac{2}{N-2}-\frac{\gamma }{n}}{2(q+1)\frac{N}{N-2}-\frac{\sigma}{n}}, \\ &x_{2}'=2(q+1)+\frac{\gamma}{n}- \frac{\sigma}{n} \frac{2(q+1)\frac{2}{N-2}-\frac{\gamma}{n}}{2(q+1)\frac {N}{N-2}-\frac{\sigma}{n}}, \\ &\alpha'=\frac{2(q+1)\frac{2}{N-2}-\frac{\gamma}{n}}{2(q+1)\frac {N}{N-2}-\frac{\sigma}{n}}, \end{aligned} $$

and obtain

$$ \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx\leq \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\alpha'} \biggl( \int_{\Omega}v^{(q+1)\frac{2N}{N-2}}\,dx \biggr)^{1-\alpha'}. $$
(2.10)

Stressing the Sobolev inequality gives \(W_{0}^{1,2}\hookrightarrow L^{\frac{2N}{N-2}}\) for \(N\geq3\). Consequently, we get

$$ \bigl\| v^{q+1}\bigr\| _{L^{\frac{2N}{N-2}}}^{\frac{2N}{N-2}(1-\alpha)}\leq c_{1}^{\frac{2N}{N-2}(1-\alpha)}\bigl\| \nabla v^{q+1}\bigr\| _{L^{2}}^{\frac{2N}{N-2}(1-\alpha)} $$
(2.11)

and

$$ \bigl\| v^{q+1}\bigr\| _{L^{\frac{2N}{N-2}}}^{\frac{2N}{N-2}(1-\alpha')}\leq c_{1}^{\frac{2N}{N-2}(1-\alpha')}\bigl\| \nabla v^{q+1}\bigr\| _{L^{2}}^{\frac{2N}{N-2}(1-\alpha')}, $$
(2.12)

where \(c_{1}\) is the best embedding constant (see [18]).

A combination of (2.9) and (2.11) leads to

$$ \int_{\Omega}v^{2(q+1)}\,dx\leq c_{1}^{\frac{2N(1-\alpha)}{N-2}} \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\alpha}\biggl( \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx \biggr)^{\frac{N(1-\alpha)}{N-2}}. $$
(2.13)

An application of the Young inequality yields

$$\begin{aligned} \int_{\Omega}v^{2(q+1)}\,dx\leq{}& \frac{N\alpha-2}{N-2}c_{1}^{\frac{2N(1-\alpha)}{N\alpha-2}} \varepsilon _{1}^{-{\frac{N(1-\alpha)}{N\alpha-2}}} \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\frac{(N-2)\alpha}{N\alpha-2}} \\ &{}+ \frac {N(1-\alpha)}{N-2}\varepsilon_{1} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx, \end{aligned}$$
(2.14)

where \(\varepsilon_{1}\) is a positive constant to be determined later.

A combination of (2.9) and (2.11) also leads to

$$\begin{aligned} \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx\leq{}& \frac{N\alpha'-2}{N-2}c_{1}^{\frac{2N(1-\alpha')}{N\alpha'-2}} \varepsilon _{2}^{-{\frac{N(1-\alpha')}{N\alpha'-2}}} \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\frac{(N-2)\alpha'}{N\alpha'-2}} \\ &{}+\frac {N(1-\alpha')}{N-2}\varepsilon_{2} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx, \end{aligned}$$
(2.15)

where \(\varepsilon_{2}\) is a positive constant to be determined later.

Combining (2.8), (2.14), and (2.15), we obtain

$$\begin{aligned} \varphi'(t) \leq{}& {-} \biggl[\frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}(q+1)}- \frac{q\sigma(\sigma -1)b_{2}}{n^{2(q+1)}}\frac{N(1-\alpha)}{N-2}\varepsilon_{1}-\sigma a_{2}\frac{N(1-\alpha')}{N-2}\varepsilon_{2} \biggr] \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx \\ &{}+\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}\bigl[\phi(t) \bigr]^{\frac{\sigma-1}{\sigma}}+\frac {N\alpha-2}{N-2}c_{1}^{\frac{2N(1-\alpha)}{N\alpha-2}} \varepsilon_{1}^{-{\frac{N(1-\alpha)}{N\alpha-2}}}\frac{q\sigma (\sigma-1)b_{2}}{n^{2(q+1)}} \bigl[\phi(t) \bigr]^{\frac{(N-2)\alpha }{N\alpha-2}} \\ &{}+\frac{N\alpha'-2}{N-2}c_{1}^{\frac{2N(1-\alpha')}{N\alpha'-2}} \varepsilon_{2}^{-{\frac{N(1-\alpha')}{N\alpha'-2}}} \bigl[\phi(t) \bigr]^{\frac {(N-2)\alpha'}{N\alpha'-2}}. \end{aligned}$$
(2.16)

By choosing \(\varepsilon_{1}\) and \(\varepsilon_{2}\) small enough such that

$$ \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}(q+1)}-\frac{q\sigma(\sigma -1)b_{2}}{n^{2(q+1)}}\frac{N(1-\alpha)}{N-2} \varepsilon_{1}-\sigma a_{2}\frac{N(1-\alpha')}{N-2} \varepsilon_{2}\geq0 $$
(2.17)

we get the differential inequality

$$ \varphi'(t) \leq\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma }}+k_{1} \bigl[ \phi(t) \bigr]^{\frac{(N-2)\alpha}{N\alpha-2}}+ k_{2} \bigl[\phi(t) \bigr]^{\frac{(N-2)\alpha'}{N\alpha'-2}} $$
(2.18)

with \(k_{1}=\frac{N\alpha-2}{N-2}c_{1}^{\frac{2N(1-\alpha)}{N\alpha -2}}\varepsilon_{1}^{-{\frac{N(1-\alpha)}{N\alpha-2}}}\) and \(k_{2}=\frac{N\alpha'-2}{N-2}c_{1}^{\frac{2N(1-\alpha')}{N\alpha'-2}} \varepsilon_{2}^{-{\frac{N(1-\alpha')}{N\alpha'-2}}}\).

Inequality (2.18) can be rewritten as

$$ \frac{d\phi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\phi(t)]^{\frac{\sigma-1}{\sigma }}+k_{1} [\phi(t) ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\phi(t) ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}} \leq \,dt. $$
(2.19)

An integration of (2.19) from 0 to t leads to

$$ \int_{\phi(0)}^{\phi(t)}\frac{d\xi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\xi]^{\frac{\sigma-1}{\sigma}}+k_{1} [\xi ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\xi ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}} \leq t. $$
(2.20)

Taking the limit as \(t\longrightarrow t^{*}\), we obtain

$$ \int_{\phi(0)}^{+\infty}\frac{d\xi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\xi]^{\frac{\sigma-1}{\sigma}}+k_{1} [\xi ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\xi ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}} \leq t^{*}, $$
(2.21)

and the proof is complete. □