In this short note, we consider a nonlocal quasilinear parabolic equation in a bounded domain with the Neumann-Robin boundary condition. We establish a blow-up result for a certain solution with positive initial energy.
We consider the initial boundary value problem for a nonlocal quasilinear parabolic equation
with Neumann-Robin 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 human-controlled 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 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 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 Navier-Stokes equation on an infinite slab, which is explained in .
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  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  investigated a similar problem and established a relation between the finite time blow-up of solutions and the negativity of initial energy for by using a gamma-convergence argument. They also conjectured that the relation might hold for all , and a positive answer to which was given by Jazar in . Lately, by using the energy method, Gao  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  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 blow-up result, when 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  studied the local p-Laplacian equation
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  considered the nonlocal quasilinear equation
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 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.
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
We also define , , and as
We now introduce our main result on the blow-up solutions with the positive initial energy below.
Theorem 1 (Sufficient condition for blow-up)
Set , , when and , when . Suppose that is a solution of (1.1)-(1.3), and the initial datum is chosen to ensure that and . Then the solution blows up in a finite time.
Remark 2 Choose , and ; one can easily verify that satisfies , and , therefore, conditions in Theorem 1 are valid.
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 , where a different type of problem was discussed.
Lemma 1 defined in (2.3) is non-increasing int.
Proof A direct computation with the integration by parts yields
and hence, is non-increasing in t. □
The following second lemma gives a lower bound estimate for the solution in the -norm:
Lemma 2Let be a solution of (1.1)-(1.3) with initial data satisfying
Then there exists a positive constant such that
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. □
we have the following lemma.
Lemma 3For all , we have the inequalities
Proof By Lemma 1, we have
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 blow-up result for , when . We conjecture that Theorem 1 will hold for all .
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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.
Furter, J, Grinfield, M: Local vs. non-local interactions in populations dynamics. J. Math. Biol.. 27, 65–80 (1989). Publisher Full Text
Allegretto, W, Fragnelli, G, Nistri, P: Coexistence and optimal control problems for a degenerate predator-prey model. J. Math. Anal. Appl.. 378, 528–540 (2011). Publisher Full Text
Budd, CJ, Dold, JW, Stuart, AM: Blow-up in a system of partial differential equations with conserved first integral. Part II: problems with convection. SIAM J. Appl. Math.. 54(3), 610–640 (1994). Publisher Full Text
Hu, B, Yin, HM: Semi-linear parabolic equations with prescribed energy. Rend. Circ. Mat. Palermo. 44, 479–505 (1995). Publisher Full Text
El Soufi, A, Jazar, M, Monneau, R: A gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 24(1), 17–39 (1995)
Jazar, M, Kiwan, R: Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 25, 215–218 (2008). Publisher Full Text
Gao, WJ, Han, YZ: Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl. Math. Lett.. 24(5), 784–788 (2011). Publisher Full Text
Liu, WJ, Wang, MX: Blow-up of the solution for a p-Laplacian equation with positive initial energy. Acta Appl. Math.. 103, 141–146 (2008). Publisher Full Text
Niculescu, CP, Rovenţa, J: Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem. Nonlinear Anal. TMA. 75, 270–277 (2012). Publisher Full Text
Zhao, JN: Existence and nonexistence of solutions for . J. Math. Anal. Appl.. 172, 130–146 (1993). Publisher Full Text
Li, FC, Xie, CH: Global and blow-up solutions to a p-Laplace equation with nonlocal source. Comput. Math. Appl.. 46, 1525–1533 (2003). Publisher Full Text