We consider the initial boundary value problem for a nonlocal quasilinear parabolic equation

${u}_{t}={\mathrm{\Delta}}_{p}u+{|u|}^{q-1}u-\frac{1}{m(\mathrm{\Omega})}{\int}_{\mathrm{\Omega}}{|u|}^{q-1}u\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0,$

(1.1)

with Neumann-Robin boundary and initial conditions

${|\mathrm{\nabla}u|}^{p-2}\frac{\partial u}{\partial n}=0,\phantom{\rule{1em}{0ex}}x\in \partial \mathrm{\Omega},t>0,$

(1.2)

$u(x,0)={u}_{0}(x),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},$

(1.3)

where $\mathrm{\Omega}\subset {\mathbf{R}}^{N}$ ($N\ge 1$) is a bounded domain with a smooth boundary, $m(\mathrm{\Omega})$ denotes the Lebesgue measure of the domain Ω, ${\mathrm{\Delta}}_{p}u=div({|\mathrm{\nabla}u|}^{p-2}\mathrm{\nabla}u)$ with $p\ge 2$, $q>p-1$, ${u}_{0}(x)\in {L}^{\mathrm{\infty}}(\mathrm{\Omega})\cap {W}^{1,p}(\mathrm{\Omega})$, ${u}_{0}(x)\not\equiv 0$, and ${\int}_{\mathrm{\Omega}}{u}_{0}\phantom{\rule{0.2em}{0ex}}dx=0$. It is easy to check that the integral of *u* over Ω is conserved. Meanwhile, since $u(x,t)$ is not required to be nonnegative, we use ${|u|}^{q-1}u$ instead of ${u}^{q}$ in equation (1.1).

Equation (1.1) arises naturally from the fluid mechanics, biology, and population dynamics. In particular, it is a possible model for the diffusion system of some biological species with a human-controlled distribution, in which

$u(x,t)$,

$div({|\mathrm{\nabla}u|}^{p-2}\mathrm{\nabla}u)$,

${|u|}^{q-1}u$, and

$-\frac{1}{m(\mathrm{\Omega})}{\int}_{\mathrm{\Omega}}{|u|}^{q-1}u\phantom{\rule{0.2em}{0ex}}dx$ represent the density of the species, the mutation, which we may view as the spread of the characteristic, the growth source of the species, and the human-controlled distribution at position

*x* and time

*t*, respectively. The arising of a nonlocal term denotes the evolution of the species at a point of space, which depends not only on nearby density, but also on the mean value of the total amount of species due to the effects of spatial inhomogeneity, see [

1–

3]. This equation can be also used to describe the slow diffusion of concentration of non-Newton flow in a porous medium or the temperature of some combustible substance (

*cf.* [

4–

6]). In addition, when

$p=q=2$ in (1.1), equation (

1.1) becomes

${u}_{t}=\mathrm{\Delta}u+{u}^{2}-\frac{1}{m(\mathrm{\Omega})}{\int}_{\mathrm{\Omega}}{u}^{2}\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0,$

which is one of the simplest equations with nonlocal terms and a homogeneous Neumann boundary condition, and the quantity ${\int}_{\mathrm{\Omega}}u(x,t)\phantom{\rule{0.2em}{0ex}}dx$ is conserved. This equation is also related to the Navier-Stokes equation on an infinite slab, which is explained in [7].

In recent years, blow-up theory for solutions of the initial boundary value problem of parabolic equations with local or nonlocal term has been rapidly developed, and there have been many delicate results. Especially, for the relations between initial energy and blow-up solution, see [

8–

14]. As for researches on the initial boundary value problem of semilinear parabolic equations, we refer the readers to [

8–

12]. For instances, Hu and Yin [

8] considered the nonlocal semilinear equation

${u}_{t}=\mathrm{\Delta}u+{|u|}^{q-1}u-\frac{1}{m(\mathrm{\Omega})}{\int}_{\mathrm{\Omega}}{|u|}^{q-1}u\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0$

(1.4)

with a homogeneous Neumann boundary condition

$\frac{\partial u}{\partial n}=0,\phantom{\rule{1em}{0ex}}x\in \partial \mathrm{\Omega},t>0$

(1.5)

and established a result of local existence for the negative initial energy by using a convexity argument. Soufi [

9] investigated a similar problem and established a relation between the finite time blow-up of solutions and the negativity of initial energy for

$1<q\le 2$ by using a gamma-convergence argument. They also conjectured that the relation might hold for all

$q>1$, and a positive answer to which was given by Jazar in [

10]. Lately, by using the energy method, Gao [

11] established a relation between the finite time blow-up of solutions and the positivity of initial energy of problem (1.4)-(1.5). In addition, Niculescu and Rovenţa [

12] considered a more general initial boundary value problem of nonlocal semilinear parabolic equation given by

${u}_{t}=\mathrm{\Delta}u+f(|u|)-\frac{1}{m(\mathrm{\Omega})}{\int}_{\mathrm{\Omega}}f(|u|)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0,$

with homogeneous Neumann boundary condition (1.5), and established a blow-up result, when

$f(|u|)$ belongs to a large class of nonlinearities and the initial energy was non-positive by using the convexity method. For the initial boundary value problem of quasilinear parabolic equations, Liu and Wang [

13] studied the local

*p*-Laplacian equation

${u}_{t}={\mathrm{\Delta}}_{p}u+f(u),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0,$

with homogeneous Dirichlet boundary condition, and built a relation between the finite time blow-up of solutions and the positivity of initial energy. Recently, Niculescu and Rovenţa [

14] considered the nonlocal quasilinear equation

${u}_{t}={\mathrm{\Delta}}_{p}u+f(|u|)-\frac{1}{m(\mathrm{\Omega})}{\int}_{\mathrm{\Omega}}f(|u|)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0,$

with the Neumann-Robin boundary condition (1.2), and established a relation between the finite time blow-up solutions and the negative initial energy, when $p\ge 2$ and *f* belongs to a large class of nonlinearities by virtue of a convexity argument.

In those works mentioned above, most problems assumed that the initial energy was negative or non-positive to ensure the occurrence of blow-up. But, to the best of our knowledge, the positive initial energy can also ensure the occurrence of blow-up in local or nonlocal problems. It is difficult to determine whether the solutions of the initial boundary value problem of nonlocal equation (1.1) will blow up in finite time, since the comparison principle, which is the most effective tool to show blow-up of solutions, is invalid. The aim of our work is to find a relation between the finite time blow-up of solutions and the positive initial energy of problem (1.1)-(1.3) by the improved convexity method.