Abstract
In this short note, we consider a nonlocal quasilinear parabolic equation in a bounded domain with the NeumannRobin boundary condition. We establish a blowup result for a certain solution with positive initial energy.
1 Introduction
We consider the initial boundary value problem for a nonlocal quasilinear parabolic equation
with NeumannRobin boundary and initial conditions
where () is a bounded domain with a smooth boundary, denotes the Lebesgue measure of the domain Ω, with , , , , and . It is easy to check that the integral of u over Ω is conserved. Meanwhile, since is not required to be nonnegative, we use instead of 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 humancontrolled distribution, in which , , , and 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 humancontrolled 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 [13]. This equation can be also used to describe the slow diffusion of concentration of nonNewton flow in a porous medium or the temperature of some combustible substance (cf.[46]). In addition, when in (1.1), equation (1.1) becomes
which is one of the simplest equations with nonlocal terms and a homogeneous Neumann boundary condition, and the quantity is conserved. This equation is also related to the NavierStokes equation on an infinite slab, which is explained in [7].
In recent years, blowup 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 blowup solution, see [814]. As for researches on the initial boundary value problem of semilinear parabolic equations, we refer the readers to [812]. For instances, Hu and Yin [8] considered the nonlocal semilinear equation
with a homogeneous Neumann boundary condition
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 blowup of solutions and the negativity of initial energy for by using a gammaconvergence argument. They also conjectured that the relation might hold for all , 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 blowup 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
with homogeneous Neumann boundary condition (1.5), and established a blowup result, when belongs to a large class of nonlinearities and the initial energy was nonpositive by using the convexity method. For the initial boundary value problem of quasilinear parabolic equations, Liu and Wang [13] studied the local pLaplacian equation
with homogeneous Dirichlet boundary condition, and built a relation between the finite time blowup of solutions and the positivity of initial energy. Recently, Niculescu and Rovenţa [14] considered the nonlocal quasilinear equation
with the NeumannRobin boundary condition (1.2), and established a relation between the finite time blowup solutions and the negative initial energy, when 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 nonpositive to ensure the occurrence of blowup. But, to the best of our knowledge, the positive initial energy can also ensure the occurrence of blowup 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 blowup of solutions, is invalid. The aim of our work is to find a relation between the finite time blowup of solutions and the positive initial energy of problem (1.1)(1.3) by the improved convexity method.
2 Preliminaries and the main result
Since , equation (1.1) is degenerate on , there is no classical solution in general. Hence, it is reasonable to find a weak solution of problem (1.1)(1.3). To this end, we first give the following definition of the weak solution of problem (1.1)(1.3).
Definition 1 If a function satisfies the following conditions:
where and , then is called a weak solution of problem (1.1)(1.3).
Remark 1 The existence of local nonnegative solutions in time to problem (1.1)(1.3) can be obtained by using a fixed point theorem or a parabolic regular theory to get a suitable estimate in a standard limiting process, see [6,15,16]. The proof is standard, and so it is omitted here. Moreover, for convenience, we may assume that the appropriate weak solution is smooth, and no longer consider approximation problem.
Let denote a subspace of , and we assume that the functions u in satisfy . We also define a norm on by
It is easy to see that this norm is equivalent to the classical norm on by using the Poincaré inequality. Set B be the optimal constant of the embedding inequality
which is equivalent to
where
and
We now introduce our main result on the blowup solutions with the positive initial energy below.
Theorem 1 (Sufficient condition for blowup)
Set, , whenand, when. Suppose thatis a solution of (1.1)(1.3), and the initial datumis chosen to ensure thatand. Then the solutionblows up in a finite time.
Remark 2 Choose , and ; one can easily verify that satisfies , and , therefore, conditions in Theorem 1 are valid.
Remark 3 Our result improves the results of Gao [11] and Niculescu and Rovenţa [14].
3 The proof of Theorem 1
To prove our main result, we first establish the following three lemmas obtained by applying the idea of Liu and Wang in [13], where a different type of problem was discussed.
Lemma 1defined in (2.3) is nonincreasing int.
Proof A direct computation with the integration by parts yields
and hence, is nonincreasing in t. □
The following second lemma gives a lower bound estimate for the solution in the norm:
Lemma 2Letbe a solution of (1.1)(1.3) with initial data satisfying
Then there exists a positive constantsuch that
and
Proof By (2.1) and (2.3), we notice that
where . It can be easily seen that g is increasing for , and decreasing for , as , and , where and are constants defined in (2.2). Therefore, there exists a constant such that , since .
Setting , we have by (3.3), which implies that , since and .
To establish (3.1), we assume that there exists a constant such that . Because of the continuity of , we can choose such that . From (3.3), we deduce that
which is impossible by Lemma 1, and hence, inequality (3.1) is established.
It also follows from (2.3) that
We then obtain that
from which inequality (3.2) follows. □
Setting
we have the following lemma.
Lemma 3For all, we have the inequalities
Proof By Lemma 1, we have
and so
From (2.3) and (3.4), we get
It then follows from (3.1) and (3.3) that
which guarantees (3.5). □
Proof of Theorem 1 Setting and differentiating it, we obtain that
From (2.2) and (3.2), we deduce that
Substituting (3.7) into (3.6), we obtain
By Hölder’s inequality, we get
where . Combining (3.8) and (3.9) with Lemma 3, we have
Integrating (3.10) over , we obtain
which implies that blows up at a finite time , and so does . The proof is completed. □
Remark 4 Due to the restriction of our method, we cannot get the blowup result for , when . We conjecture that Theorem 1 will hold for all .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
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