Blow-Up Theorems for p-Sub-Laplacian Heat Operators on Stratified Groups

Abstract

In this paper, we present blow-up results for solutions to the Dirichlet initial value problem for the p-sup-Laplacian heat operators on the stratified (Lie) groups. A stratified group version of the Poincaré inequality from Ruzhansky and Suragan (J Differ Equ 262:1799–1821, 2017) plays a key role in our proofs.

1 Introduction

A Lie group ${\mathbb {G}}=({\mathbb {R}}^{n},\circ )$ is called a stratified (Lie) group if it satisfies the following conditions:

(a) For some integer numbers $N_1+N_{2}+\cdots +N_{r}=n$, the decomposition ${\mathbb {R}}^{n}={\mathbb {R}}^{N_1}\times \cdots \times {\mathbb {R}}^{N_{r}}$ is valid, and for any $\lambda >0$ the dilation

$$\begin{aligned} \delta _{\lambda }(x):=(\lambda x',\lambda ^{2}x^{(2)},\ldots ,\lambda ^{r}x^{(r)}) \end{aligned}$$

is an automorphism of ${\mathbb {G}}.$ Here $x'\equiv x^{(1)}\in {\mathbb {R}}^{N_1}$ and $x^{(k)}\in {\mathbb {R}}^{N_{k}}$ for $k=2,\ldots ,r.$

(b) Let $N_1$ be as in (a) and let $X_{1},\ldots ,X_{N_1}$ be the left-invariant vector fields on ${\mathbb {G}}$ such that $X_{k}(0)=\frac{\partial }{\partial x_{k}}|_{0}$ for $k=1,\ldots ,N_1.$ Then the Hömander rank condition must be satisfied, that is,

$$\begin{aligned} \mathrm{rank}(\mathrm{Lie}\{X_{1},\ldots ,X_{N_1}\})=n, \end{aligned}$$

for every $x\in {\mathbb {R}}^{n}.$

Then, we say that the triple ${\mathbb {G}}=({\mathbb {R}}^{n},\circ , \delta _{\lambda })$ is a stratified (Lie) group.

Recall that the standard Lebesgue measure dx on ${\mathbb {R}}^{n}$ is the Haar measure for ${\mathbb {G}}$ (see, e.g. [8]). The left-invariant vector field $X_{j}$ has an explicit form:

$$\begin{aligned} X_{k}=\frac{\partial }{\partial x'_{k}}+ \sum _{l=2}^{r}\sum _{m=1}^{N_{l}}a_{k,m}^{(l)}(x',\ldots ,x^{(l-1)}) \frac{\partial }{\partial x_{m}^{(l)}}, \end{aligned}$$

(1.1)

see e.g. [8]. The following notations are used throughout this paper:

$$\begin{aligned} \nabla _{{\mathbb {G}}}:=(X_{1},\ldots , X_{N_1}) \end{aligned}$$

for the horizontal gradient,

$$\begin{aligned} {\mathcal {L}}_{p}f:=\nabla _{{\mathbb {G}}}\cdot (|\nabla _{{\mathbb {G}}} f|^{p-2}\nabla _{{\mathbb {G}}}f),\quad 1<p<\infty , \end{aligned}$$

(1.2)

for the p-sub-Laplacian. When $p=2$, that is, the second order differential operator

$$\begin{aligned} {\mathcal {L}}=\sum _{k=1}^{N_1}X_{k}^{2}, \end{aligned}$$

(1.3)

is called the sub-Laplacian on ${\mathbb {G}}$. The sub-Laplacian ${\mathcal {L}}$ is a left-invariant homogeneous hypoelliptic differential operator and it is known that ${\mathcal {L}}$ is elliptic if and only if the step r of ${\mathbb {G}}$ is equal to 1.

Let $\Omega \subset {\mathbb {G}}$ be an open set, then we define the functional spaces

$$\begin{aligned} S^{1,p}(\Omega ) =\{ u: \Omega \rightarrow {\mathbb {R}}; u, |\nabla _{{\mathbb {G}}} u| \in L^p(\Omega ) \}. \end{aligned}$$

(1.4)

We consider the following functional

$$\begin{aligned} J_p(u):= \left( \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x)|^p dx\right) ^{\frac{1}{p}}. \end{aligned}$$

Thus, the functional class $\mathring{S}^{1,p}(\Omega )$ can be defined as the completion of $C_0^1(\Omega )$ in the norm generated by $J_p$, see e.g. [3].

Let $\Omega$ be a bounded domain on the stratified Lie groups with the smooth boundary $\partial \Omega$. We consider the Dirichlet initial value problem for the p-sub-Laplacian heat operator

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t(x,t) - {\mathcal {L}}_p u(x,t) = f(u(x,t)), \,\,\, &{} (x,t) \in \Omega \times (0,+\infty ), \\ u(x,t) =0, \,\,\,&{} (x,t) \in \partial \Omega \times [0,+\infty ), \\ u(x,0) = u_0(x)\ge 0,\,\,\, &{} x \in {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(1.5)

Here f is a locally Lipschitz continuous function on ${\mathbb {R}}$, $f(0)=0$, and such that $f(u)>0$ for $u>0$. Furthermore, we suppose that $u_0$ is a non-negative and non-trivial function in $L^{\infty }(\Omega ) \cap \mathring{S}^{1,p}(\Omega )$ and that $u_0(x)=0$ on the boundary $\partial \Omega$ of $\Omega$.

In the Euclidean setting, it is well-known that there often exists a solution of the p-Laplacian parabolic equation as the one in (1.5) for all times. There is a large literature on the sufficient conditions for the local existence of solutions to the p-Laplacian parabolic equation. For example, the sufficient conditions for the local existence of solutions to the p-Laplacian parabolic equations are derived by Ball [1] and Zhao [10] for $p=2$ and $p>2$, respectively. Then, the blow-up solutions have been investigated by many authors such as Levine [6], Philippin and Proytcheva [7], Ding and Hu [5], Bandle and Brunner [2], with a more detailed review of their works presented in [4].

In the present paper, our aim is to extend Abelian (Euclidean) to general stratified (Lie) groups. Thus, in this paper, the blow-up theorems for the p-sub-Laplacian heat operators are proved on the stratified Lie groups. We use the concavity method and the Poincaré inequality from [9] as the main tool. However, these results cannot be directly extended to the general stratified groups since there are no those spectral inequalities on the general stratified groups. On the other hand, in general, our stratified group theorems cannot completely cover the Heisenberg group results due to the restriction $N_{1}\ne p$ (see Theorem 2.5).

2 Main Results

2.1 Blow-Up Solutions for the Sub-Laplacian Heat Operators

We consider the blow-up solutions to the sub-Laplacian heat equation on the stratified groups ${\mathbb {G}}$, that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t(x,t) - {\mathcal {L}} u(x,t) = f(u(x,t)), \,\,\, &{} (x,t) \in \Omega \times (0,+\infty ), \\ u(x,t) =0, \,\,\,&{} (x,t) \in \partial \Omega \times [0,+\infty ), \\ u(x,0) = u_0(x)\ge 0,\,\,\, &{} x \in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(2.1)

where f is locally Lipschitz continuous on ${\mathbb {R}}$, $f(0)=0$, and such that $f(u)>0$ for $u>0$. Furthermore, we suppose that $u_0$ is a non-negative and non-trivial function in $C^1({\overline{\Omega }})$ and that $u_0(x)=0$ on the boundary $\partial \Omega$.

Theorem 2.1

Let $\Omega$ be a bounded domain of the stratified group ${\mathbb {G}}$ with $N_1\ge 3$ being the dimension of the first stratum and smooth boundary $\partial \Omega$. Let a function f satisfy the condition that there exist constants $\alpha >2$ and $\gamma$ such that for all $u>0$ we have

$$\begin{aligned} \alpha F(u) \le uf(u) + \beta u^2 +\alpha \gamma , \end{aligned}$$

(2.2)

where $F(u)=\int _{0}^{u}f(s)ds$ and $0<\beta \le \frac{(\alpha -2)(N_1-2)^2}{8R^2}$. If $u_0 \in C^1({\overline{\Omega }})$ with $u_0=0$ on $\partial \Omega$ satisfies the inequality

$$\begin{aligned} -\frac{1}{2}\int _{\Omega } |\nabla _{{\mathbb {G}}} u_0(x)|^2 dx + \int _{\Omega } \left( \int _{0}^{u_0(x)} f(s)ds -\gamma \right) dx >0, \end{aligned}$$

(2.3)

then the nonnegative solution to the Eq. (2.1) blows up at a finite time $T^*$ for

$$\begin{aligned} M:=\frac{\left( 1+\sqrt{\frac{\alpha }{2}}\right) \left( \int _{\Omega } u_0^2(x)dx \right) ^2}{2(\alpha -2)[-\frac{1}{2} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^2 dx + \int _{\Omega } (\int _{0}^{u_0(x)} f(s)ds-\gamma )dx] }, \end{aligned}$$

(2.4)

such that

$$\begin{aligned} 0<T^*\le \frac{M}{\sigma \int _{\Omega }u_0^2(x)dx}, \end{aligned}$$

(2.5)

that is,

$$\begin{aligned} \lim _{t\rightarrow T^*} \int _{0}^t \int _{\Omega } u^2 (x,\tau ) dx d\tau = +\infty . \end{aligned}$$

(2.6)

Lemma 2.2

[9] Let $\Omega$ be a bounded domain on the stratified group ${\mathbb {G}}$ with $N_1\ge 3$ being the dimension of the first stratum. For every function $u \in C_0^{\infty }(\Omega \backslash \{x'=0\})$ we have

$$\begin{aligned} \int _{\Omega } |\nabla _{{\mathbb {G}}} u|^2 dx \ge \frac{(N_1-2)^2}{4R^2} \int _{\Omega } |u|^2 dx, \end{aligned}$$

(2.7)

where $R = \sup _{x \in \Omega } |x'|$.

Remark 2.3

Note that Lemma 2.2 will be useful in the proof of Theorem 2.1.

Proof of Theorem 2.1

Let $\Gamma$ be denoted by

$$\begin{aligned} \Gamma (t) := \int _{0}^t \int _{\Omega } u^2(x, \tau ) dx d\tau + M, \,\, t\ge 0, \end{aligned}$$

(2.8)

where $M>0$ is a constant to be determined later. By Leibniz’s integral rule we get

$$\begin{aligned} \Gamma '(t) =\frac{d}{dt} \Gamma (t) = \frac{d}{dt} \left( \int _{0}^t \int _{\Omega } u^2(x,\tau ) dx d\tau \right) = \int _{\Omega } u^2(x,t)dx, \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega } \int _{0}^t 2u(x,\tau ) u_{\tau }(x,\tau ) d\tau dx&= \int _{\Omega } \int _{0}^t\frac{d}{d\tau } u^2(x,\tau ) d\tau dx \\&= \int _{\Omega } u^2(x,t) dx - \int _{\Omega } u^2_0(x) dx. \end{aligned}$$

This gives the relation

$$\begin{aligned} \Gamma '(t)=\int _{\Omega } u^2(x,t) dx = \int _{\Omega } \int _{0}^t 2u(x,\tau ) u_{\tau }(x,\tau ) d\tau dx + \int _{\Omega } u^2_0(x) dx. \end{aligned}$$

(2.9)

Using the above computations, the condition (2.2) and Lemma 2.2, we compute the second derivative of $\Gamma (t)$ with respect to time

$$\begin{aligned} \Gamma ''(t)&=2\int _{\Omega } u(x,t) u_t(x,t) dx\\&=2 \int _{\Omega } u(x,t) {\mathcal {L}} u(x,t) + 2\int _{\Omega } u(x,t) f(u(x, t))dx \\&\ge -2\int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,t)|^2 dx + 2\int _{\Omega } \left[ \alpha F(u(x,t)) -\beta u^2 (x,t) -\alpha \gamma \right] dx\\&\ge 2\alpha \left[ -\frac{1}{2} \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,t)|^2 dx + \int _{\Omega } (F(u(x,t)) -\gamma ) dx \right] \\&\quad +\left( \frac{(\alpha -2)(N_1-2)^2}{4R^2}-2\beta \right) \int _{\Omega } u^2 (x, t) dx\\&\ge 2\alpha \left[ -\frac{1}{2} \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,t)|^2 dx + \int _{\Omega } (F(u(x,t))-\gamma ) dx \right] \\&= 2\alpha \left[ -\frac{1}{2} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^2 dx + \int _{\Omega } (F(u_0)-\gamma )dx \right] + 2\alpha \int _{0}^t \int _{\Omega } u_{\tau }^2(x,\tau ) dx d\tau . \end{aligned}$$

Then we have

$$\begin{aligned} \Gamma ''(t) \Gamma (t)&\ge \! 2\alpha \left( \left[ \!-\!\frac{1}{2} \!\int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^2 dx \!+\! \int _{\Omega } (F(u_0) \!-\!\gamma )dx\right] \!+\! \int _{0}^t \int _{\Omega } u_{\tau }^2(x,\tau ) dx d\tau \right) \\&\quad \times \left( \int _{0}^t \int _{\Omega } u^2(x,\tau ) dx d\tau +M \right) \\&\ge 2\alpha M \left[ -\frac{1}{2} \int _{\Omega } | \nabla _{{\mathbb {G}}} u_0|^2 dx + \int _{\Omega } (F(u_0)-\gamma )dx\right] \\&\quad + 2\alpha \left( \int _{\Omega } \int _{0}^t u^2(x, \tau ) d \tau dx \right) \left( \int _{\Omega } \int _{0}^t u^2_{\tau }(x, \tau ) d\tau dx \right) . \end{aligned}$$

Also, we compute by making use of Hölder and Schwartz’s inequalities,

$$\begin{aligned} ( \Gamma '(t) )^2&\le 4(1+\delta )\left( \int _{\Omega } \int _{0}^t u(x,\tau ) u_{\tau }(x, \tau ) d\tau dx \right) ^2 + \left( 1+ \frac{1}{\delta }\right) \left( \int _{\Omega } u_0^2(x)dx \right) ^2 \\&\le 4(1+\delta )\left( \int _{\Omega } \left( \int _{0}^t u^2(x,\tau ) d \tau \right) ^{\frac{1}{2}} \left( \int _{0}^t u^2_{\tau }(x, \tau ) d\tau \right) ^{\frac{1}{2}} dx \right) ^2 \\&\quad + \left( 1+ \frac{1}{\delta }\right) \left( \int _{\Omega } u_0^2(x)dx \right) ^2\\&\le 4(1+\delta )\left( \int _{\Omega } \int _{0}^t u^2(x, \tau ) d\tau dx \right) \left( \int _{\Omega } \int _{0}^t u^2_{\tau }(x, \tau ) d\tau dx \right) \\&\quad + \left( 1+ \frac{1}{\delta }\right) \left( \int _{\Omega } u_0^2(x)dx \right) ^2, \end{aligned}$$

where $\delta >0$. Then by combining the above expressions and taking $\sigma =\delta =\sqrt{\alpha /2}-1>0$, we establish the following estimate

$$\begin{aligned}&\Gamma ''(t)\Gamma (t) - (1+\sigma ) (\Gamma '(t))^2 \\&\quad \ge 2\alpha M \left[ -\frac{1}{2} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^2 dx + \int _{\Omega } (F(u_0)-\gamma )dx\right] - \frac{\alpha }{\sqrt{2\alpha }-2}\left( \int _{\Omega } u_0^2(x)dx \right) ^2. \end{aligned}$$

Then we choose $M>0$ as large enough to satisfy

$$\begin{aligned} \Gamma ''(t)\Gamma (t) - (1+\sigma ) (\Gamma '(t))^2 >0. \end{aligned}$$

(2.10)

We can see that the above expression for $t\ge 0$ implies

$$\begin{aligned} \frac{d}{dt} \left[ \frac{\Gamma '(t)}{\Gamma ^{\sigma +1}(t)} \right] >0 \Rightarrow {\left\{ \begin{array}{ll} \Gamma '(t) \ge \left( \frac{\int _{\Omega } u_0^2(x) dx}{M^{\sigma +1}} \right) \Gamma ^{1+\sigma }(t),\\ \Gamma (0)=M. \end{array}\right. } \end{aligned}$$

Then we arrive at

$$\begin{aligned} \Gamma (t) \ge \left( \frac{1}{M^{\sigma }}-\frac{\int _{\Omega } u_0^2(x) dx}{M^{\sigma +1}} t\right) ^{-\frac{1}{\sigma }}. \end{aligned}$$

From here we see that the solutions blow up in the finite time $T^*$ which is

$$\begin{aligned} 0< T^* \le \frac{M}{\sigma \int _{\Omega } u^2_0(x)dx }, \end{aligned}$$

where M can be estimated from (2.10), that is,

$$\begin{aligned} M:= \frac{\left( 1+\sqrt{\frac{\alpha }{2}}\right) \left( \int _{\Omega } u_0^2(x)dx \right) ^2}{2(\alpha -2)[-\frac{1}{2} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^2 dx + \int _{\Omega } (F(u_0)-\gamma )dx] }. \end{aligned}$$

Therefore, it follows that $\Gamma (t)$ cannot remain finite for all $t>0$. In other words, the solutions u blows up in finite time $T^*$. $\square$

2.2 Blow-Up Solutions for p-Sub-Laplacian Heat Operators

We consider now the blow-up solutions to the p-sub-Laplacian heat equation on the stratified group ${\mathbb {G}}$, that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t(x,t) - {\mathcal {L}}_p u(x,t) = f(u(x,t)), \,\,\, &{} (x,t) \in \Omega \times (0,+\infty ), \\ u(x,t) =0, \,\,\,&{} (x,t) \in \partial \Omega \times [0,+\infty ), \\ u(x,0) = u_0(x)\ge 0,\,\,\, &{} x \in {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$

(2.11)

where f is locally Lipschitz continuous on ${\mathbb {R}}$, $f(0)=0$, and such that $f(u)>0$ for $u>0$. Furthermore, we suppose that $u_0$ is a non-negative and non-trivial function in $L^{\infty }(\Omega ) \cap \mathring{S}^{1,p}(\Omega )$ and that $u_0(x)=0$ on the boundary $\partial \Omega$.

Lemma 2.4

[9] Let $\Omega$ be a bounded domain on the stratified group ${\mathbb {G}}$ with $N_1$ being the dimension of the first stratum. Let $1<p<\infty$ with $p\ne N_1$. For every function $u \in C_0^{\infty }(\Omega \backslash \{x'=0\})$ we have

$$\begin{aligned} \int _{\Omega } |\nabla _{{\mathbb {G}}} u|^p dx \ge \frac{|N_1-p|^p}{(pR)^p} \int _{\Omega } |u|^p dx, \end{aligned}$$

(2.12)

where $R = \sup _{x \in \Omega } |x'|$.

Theorem 2.5

Let $\Omega$ be a bounded domain of the stratified group ${\mathbb {G}}$ with $N_1$ being the dimension of the first stratum and smooth boundary $\partial \Omega$. Let a function f satisfy the condition that there exist constants $\alpha > p$ and $\gamma$ such that for all $u>0$ we have

$$\begin{aligned} \alpha F(u) \le uf(u) + \beta u^p +\alpha \gamma , \end{aligned}$$

(2.13)

where $F(u)=\int _{0}^{u}f(s)ds$ and $0<\beta \le \frac{(\alpha -p)|N_1-p|^p}{p(pR)^p}$ with $N_1\ne p$. If $u_0 \in L^{\infty }(\Omega )\cap \mathring{S}^{1,p}(\Omega )$ satisfies

$$\begin{aligned} -\frac{1}{p}\int _{\Omega } |\nabla _{{\mathbb {G}}} u_0(x)|^p dx + \int _{\Omega } \left( \int _{0}^{u_0(x)} f(s)ds -\gamma \right) dx >0, \end{aligned}$$

(2.14)

then the nonnegative solution to the Eq. (2.11) blows up at a finite time $T^*$ for

$$\begin{aligned} M:=\frac{\left( 1+\sqrt{\frac{\alpha }{2}}\right) \left( \int _{\Omega } u_0^2(x)dx \right) ^2}{2(\alpha -2)\left[ -\frac{1}{p} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^p dx + \int _{\Omega } (\int _{0}^{u_0(x)} f(s)ds-\gamma )dx\right] }, \end{aligned}$$

(2.15)

such that

$$\begin{aligned} 0<T^*\le \frac{M}{\sigma \int _{\Omega }u_0^2(x)dx}, \end{aligned}$$

(2.16)

that is,

$$\begin{aligned} \lim _{t\rightarrow T^*} \int _{0}^t \int _{\Omega } u^2 (x,\tau ) dx d\tau = +\infty . \end{aligned}$$

(2.17)

Lemma 2.6

Let u be the weak solution to the Eq. (2.11) with $|\nabla _{{\mathbb {G}}} u_0| \in L^p(\Omega )$. Then

$$\begin{aligned} \frac{1}{2} \int _{0}^t \int _{\Omega } (u^2(x, \tau ))_{\tau } dx d\tau&= \frac{1}{2} \int _{\Omega } [u^2(x,t)-u_0^2(x)] dx \nonumber \\&= \int _{0}^t \int _{\Omega } [-|\nabla _{{\mathbb {G}}} u(x, t)|^p+ u(x,t) f(u(x,t)) ] dx d\tau , \end{aligned}$$

(2.18)

and

$$\begin{aligned} \int _{0}^t \int _{\Omega } u_{\tau }^2(x, \tau ) dx d\tau&= -\frac{1}{p} \int _{\Omega } [|\nabla _{{\mathbb {G}}} u(x,t)|^p - |\nabla _{{\mathbb {G}}} u_0(x)|^p]dx \nonumber \\&\quad + \int _{\Omega } [F(u(x,t))-F(u_0(x))]dx, \end{aligned}$$

(2.19)

where $F(u):= \int _{0}^{u}f(s)ds$.

Proof of Lemma 2.6

We first prove the equality (2.18) by using the equation (2.11), that is,

$$\begin{aligned} \frac{1}{2} \int _{0}^t \int _{\Omega } (u^2(x,\tau ))_{\tau } dx d\tau = \frac{1}{2} \int _{\Omega } u^2(x,\tau )|_0^t dx = \frac{1}{2} \int _{\Omega }[u^2(x,t)-u^2_0(x)] dx, \end{aligned}$$

and

$$\begin{aligned}&\frac{1}{2}\int _{0}^t \int _{\Omega } (u^2(x,\tau ))_{\tau } dx d\tau = \int _{\Omega } \int _{0}^t u(x,\tau )u_{\tau }(x,\tau ) dx d\tau \\&\quad =\int _{0}^t \int _{\Omega } {\mathcal {L}}_p u(x,\tau ) u(x,\tau ) dx d\tau + \int _{0}^t \int _{\Omega } f(u(x,\tau )) u(x,\tau ) dx d\tau \\&\quad = - \int _{0}^t \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,\tau )|^p dx d\tau + \int _{0}^t \int _{\Omega } f(u(x,\tau )) u(x,\tau ) dx d\tau , \end{aligned}$$

which proves the expression (2.18). Now we prove inequality (2.19) by using the Leibniz integral rule, as follows

$$\begin{aligned} \int _{0}^t \int _{\Omega } u^2_{\tau }(x,\tau ) dx d\tau&= \int _{0}^t \int _{\Omega } {\mathcal {L}}_p u(x,\tau ) u_{\tau }(x,\tau ) dx d\tau \\&\quad + \int _{0}^t \int _{\Omega } f(u(x,\tau )) u_{\tau }(x,\tau ) dx d\tau \\&= -\int _{0}^t \int _{\Omega } \langle |\nabla _{{\mathbb {G}}} u(x,\tau )|^{p-2}\nabla _{{\mathbb {G}}} u(x,\tau ), \nabla _{{\mathbb {G}}} u_{\tau }(x,\tau ) \rangle dx d\tau \\&\quad + \int _{0}^t \int _{\Omega } f(u(x,\tau )) u_{\tau }(x,\tau ) dx d\tau \\&= - \frac{1}{2}\int _{\Omega } \int _{0}^t \frac{d}{d\tau } \left( \int _0^{|\nabla _{{\mathbb {G}}} u(x,\tau )|^2} s^{\frac{p-2}{2}} ds \right) d\tau dx \\&\quad + \int _{\Omega } \int _{0}^t\frac{d}{d\tau } \left( \int _{0}^{u(x,\tau )}f(s)ds\right) d\tau dx \\&= -\frac{1}{2} \int _{\Omega } \int _{0}^{|\nabla _{{\mathbb {G}}} u(x,\tau )|^2} s^{\frac{p-2}{2}} ds dx|_0^t + \int _{\Omega }F(u(x, \tau )) dx|_0^t \\&= -\frac{1}{2} \int _{\Omega } \left( \frac{2}{p}s^{\frac{p}{2}}|_0^{|\nabla _{{\mathbb {G}}} u(x,\tau )|^2} \right) dx|_0^t + \int _{\Omega }F(u(x,\tau )) dx|_0^t\\&= -\frac{1}{p} \int _{\Omega } \left[ |\nabla _{{\mathbb {G}}} u(x,t)|^p - |\nabla _{{\mathbb {G}}} u_0(x)|^p \right] dx \\&\quad + \int _{\Omega }[F(u(x,t)) -F(u_0(x))] dx, \end{aligned}$$

which proves the expression (2.19). $\square$

Proof of Theorem 2.5

Let us define the function $\Gamma _p$ by

$$\begin{aligned} \Gamma _p(t) := \int _{0}^t \int _{\Omega } u^2(x,\tau ) dx d\tau + M, \,\, t\ge 0, \end{aligned}$$

(2.20)

where M is a positive constant. Then we compute the derivative of $\Gamma (t)$ with respect to time, which gives that

$$\begin{aligned} \Gamma '_p(t)&= \frac{d}{dt} \int _{0}^t \int _{\Omega } u^2(x,\tau ) dx d \tau = \int _{\Omega } \frac{d}{dt} \left( \int _{0}^{t} u^2(x,\tau ) d \tau \right) dx \\&= \int _{\Omega } u^2(x,t) dx\\&= \int _{0}^t \int _{\Omega } 2u(x,\tau ) u_{\tau }(x,\tau ) d\tau dx + \int _{\Omega } u_0^2(x)dx. \end{aligned}$$

The second derivative of $\Gamma _p(t)$ with respect to time t can be calculated by Lemma 2.6, using the condition (2.13), and Lemma 2.4, so that we get the estimate for $\Gamma _p''(t)$ as follows:

$$\begin{aligned} \Gamma _p''(t)&= 2\int _{\Omega } u(x,t)u_t(x,t) dx \\&= - 2 \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,t)|^p dx + 2\int _{\Omega } u(x,t)f(u(x,t)) dx\\&\ge - 2 \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,t)|^p dx + 2\int _{\Omega } \left( \alpha F(u(x,t))-\beta u^p(x,t) -\alpha \gamma \right) dx \\&\ge 2\alpha \left[ -\frac{1}{p} \int _{\Omega } |\nabla _{{\mathbb {G}}} u(x,t)|^p dx + \int _{\Omega } [F(u(x,t))-\gamma ] dx \right] \\&\quad +2\left( \frac{(\alpha -p)|N_1-p|^p}{p^{p+1}R^p} - \beta \right) \int _{\Omega } u^p(x,t)dx \\&\ge 2\alpha \left[ -\frac{1}{p}\int _{\Omega } | \nabla _{{\mathbb {G}}} u(x,t)|^p dx + \int _{\Omega } [F(u(x,t)) -\gamma ] dx \right] \\&= 2\alpha \left[ -\frac{1}{p}\int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^p dx + \int _{\Omega } [F(u_0)-\gamma ] dx \right] + 2\alpha \int _{0}^t \int _{\Omega } u_{\tau }^2(x,\tau )dx d\tau . \end{aligned}$$

Then we have

$$\begin{aligned} \Gamma _p''(t)\Gamma _p(t)&\ge \! 2\alpha \left( \left[ \!-\!\frac{1}{p} \!\int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^p dx \!+\! \int _{\Omega } [F(u_0) \!-\!\gamma ] dx \right] \!+\! \int _{0}^t \int _{\Omega } u_{\tau }^2(x,\tau )dx d \tau \right) \\&\quad \times \left( \int _{0}^t \int _{\Omega } u^2(x,\tau ) dx d\tau + M\right) \\&\ge 2\alpha M \left[ -\frac{1}{p}\int _{\Omega } | \nabla _{{\mathbb {G}}} u_0|^p dx + \int _{\Omega } [F(u_0)-\gamma ] dx \right] \\&\quad + 2\alpha \left( \int _{0}^t \int _{\Omega } u_{\tau }^2(x,\tau )dx d\tau \right) \left( \int _{0}^t \int _{\Omega } u^2(x,\tau ) dx d\tau \right) . \end{aligned}$$

By making use of Schwartz’s inequality and for arbitrary $\delta >0$ as in the case $p=2$ we arrive at

$$\begin{aligned} \left( \Gamma _p'(t)\right) ^2&\le 4(1+\delta )\left( \int _{\Omega } \int _{0}^t u^2(x,\tau ) d\tau dx\right) \left( \int _{\Omega } \int _{0}^t u_{\tau }^2(x,\tau ) d\tau dx \right) \nonumber \\&\quad + \frac{1+\delta }{\delta } \left( \int _{\Omega } u^2_0(x) dx\right) ^2. \end{aligned}$$

(2.21)

Now we use estimates of $\Gamma _p'(t)$ and $\Gamma ''_p(t)$ to obtain

$$\begin{aligned}&\Gamma _p''(t)\Gamma _p(t) -(1+\sigma )(\Gamma '_p(t))^2 \nonumber \\&\quad \ge 2\alpha M \left[ -\frac{1}{p} \int _{\Omega } | \nabla _{{\mathbb {G}}} u_0|^p dx + \int _{\Omega } (F(u_0)-\gamma )dx \right] - \frac{\alpha }{\sqrt{2\alpha }-2}\left( \int _{\Omega } u_0^2(x)dx \right) ^2. \end{aligned}$$

(2.22)

Note that from (2.14) we have

$$\begin{aligned} -\frac{1}{p} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^p dx + \int _{\Omega } (F(u_0)-\gamma )dx >0, \end{aligned}$$

and taking $M>0$ as large as necessary, we obtain the following estimate

$$\begin{aligned} \Gamma _p''(t)\Gamma _p(t) -(1+\sigma )(\Gamma '_p(t))^2>0. \end{aligned}$$

(2.23)

For $t\ge 0$ the above expression can be written as

$$\begin{aligned} \frac{d}{dt} \left( \frac{\Gamma '_p(t)}{\Gamma _p^{1+\sigma }(t)} \right) >0, \end{aligned}$$

which implies

$$\begin{aligned} {\left\{ \begin{array}{ll} \Gamma '_p(t) \ge \left( \frac{\int _{\Omega } u_0^2(x) dx}{M^{\sigma +1}} \right) \Gamma _p^{1+\sigma }(t), \,\, t>0,\\ \Gamma _p(0)=M. \end{array}\right. } \end{aligned}$$

(2.24)

Then we arrive at

$$\begin{aligned} \Gamma _p(t) \ge \left( \frac{1}{M^{\sigma }}-\frac{\int _{\Omega } u_0^2(x) dx}{M^{\sigma +1}} t\right) ^{-\frac{1}{\sigma }}, \end{aligned}$$

From here we see that the solutions blow up in the finite time $T^*$ which is

$$\begin{aligned} 0< T^* \le \frac{M}{\sigma \int _{\Omega } u^2_0(x)dx }, \end{aligned}$$

where M can be estimated from (2.22) as follows

$$\begin{aligned} M = \frac{\left( 1+\sqrt{\frac{\alpha }{2}}\right) \left( \int _{\Omega } u^2_0(x) dx\right) ^2 }{2(\alpha -2) \left[ -\frac{1}{p} \int _{\Omega } |\nabla _{{\mathbb {G}}} u_0|^p dx + \int _{\Omega } (F(u_0)-\gamma )dx\right] }. \end{aligned}$$

(2.25)

$\square$