A note on blow-up of solutions for the nonlocal quasilinear parabolic equation with positive initial energy
© Fang et al.; licensee Springer 2013
Received: 2 March 2013
Accepted: 23 July 2013
Published: 8 August 2013
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.
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).
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 .
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).
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.
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 in t.
and hence, is non-increasing in t. □
The following second lemma gives a lower bound estimate for the solution in the -norm:
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 .
which is impossible by Lemma 1, and hence, inequality (3.1) is established.
from which inequality (3.2) follows. □
we have the following lemma.
which guarantees (3.5). □
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 .
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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