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The global solution of a diffusion equation with nonlinear gradient term
Journal of Inequalities and Applications volume 2013, Article number: 125 (2013)
Consider the viscosity solution to the initial boundary value problem of the diffusion equation
with , , , , its initial value , and its boundary value , . If , by considering the regularized problem and using Moser’s iteration technique, we get the locally uniformly bounded property of the solution and the locally bounded property of the -norm of the gradient. By the compactness theorem, the existence of the viscosity solution of the equation is obtained provided that
If , the existence of solution is obtained in a similar way, and the extinction of the solution is proved in this case.
MSC:35K55, 35K65, 35B40.
The objective of the paper is to study the nonnegative weak solution of the following nonlinear parabolic equation:
where is a bounded open domain, , , , , , , , and ∇ is the spatial gradient operator.
The equation of the form (1.1) was suggested as a mathematical model for a variety of problems in mechanics, physics and biology, which can be seen in [1–4]etc. It has been widely researched, whether it is linear (i.e., , , , ) or nonlinear, fast diffusion () or slow diffusion (). For example, the existence of a nonnegative solution of (1.1)-(1.3) without the damping term , defined in some weak sense, is well established (see [5, 6]). For other examples, Bertsh et al.  and Zhou et al.  discussed the existence and properties of viscosity solution for the equation
where γ is a positive constant. Zhang et al.  discussed the existence and properties of the viscosity solution for the equation
where is a known function.
The most important characteristic of equation (1.4) or (1.5) is in that, generally, the uniqueness of the solutions is not true; one can refer to [8–12]. Thus, for the equation of the type (1.1), we should mainly consider the existence of the viscosity solution (see Definition 1.2 below) and the related properties such as large time behaviors; one can refer to [13–16]etc. for some progress on this problem.
Now, we quote the following definition.
Definition 1.1 A nonnegative function is called a weak solution of (1.1)-(1.3) if u satisfies
We will get the solution of (1.1)-(1.3) by considering the regularized problem(1.10)
with the initial value (1.2) and the homogeneous boundary value (1.3). The solutions of the regularized equation (1.10) are denoted by .
Definition 1.2 If is a solution of the initial boundary value problem of (1.10)-(1.2)-(1.3), , a.e. in S, such that u is a weak solution of (1.1)-(1.3), then u is said to be a viscosity solution.
The main aim of the paper is to show how the damping term affects the equation, including how the damping term affects the existence of the solution and how the damping term affects the properties such as the extinction of the solution. By considering the solution of the regularized problem (1.10) and using Moser’s iteration technique, we get ’s local bounded properties and the local bounded properties of the -norm of the gradient . By the compactness theorem, we get the existence of the viscosity solution of the diffusion equation itself. Apart from the general process of the proof such as in [3–5, 7, 9]etc., in which the main difficulty is how to prove that
in our paper, in addition to overcoming the above difficulty, we have to solve another difficulty lying in how to prove that
Also, we need to overcome the difficulty which comes from the damping term when we prove the uniqueness of the viscosity solutions of (1.1)-(1.3).
In order to get the desired results, some important relationships among the exponents , , q, p, m, N are imposed. We also need the following lemmas.
Lemma 1.3  (Gagliardo-Nirenberg)
If , , , suppose that , then(1.11)
Lemma 1.4 
Let be a nonnegative function on . If it satisfies(1.12)
where , , , , then(1.13)
We will prove the following theorems. As usual, the constants c in what follows may be different from one to another.
Theorem 1.5 If and(1.14)(1.15)(1.16)(1.17)
then (1.1)-(1.3) has a weak viscosity solution which satisfies(1.18)
where . Moreover, if , then(1.20)
where , .
The condition (1.17) is only used to prove (1.9); if , this is a natural condition. We conjecture that this condition can be weakened.
Theorem 1.6 Let u be a weak solution of (1.1)-(1.3). If , , then(1.21)
for all s, t with .
Theorem 1.7 If ,(1.22)
then (1.1)-(1.3) has a weak solution which satisfies (1.18), and there exists a positive such that(1.23)
If the damping term disappears in (1.1), say, if (1.1) without by , then we know that the extinction of the solution as Theorem 1.7 is true. For other related works on equation (1.1), one can refer to the references [20–31]etc. We use some ideas in  and .
2 The estimate of the solution
Consider the regularized problem
where is a suitably smooth function such that
Then, if , since ,
by Chapter 8 of , viewing (2.1) as a divergent form of a quasilinear parabolic equation, we know that (2.1)-(2.3) has a unique nonnegative classical solution . In what follows, in the proof of the related lemmas, we only denote as u for simplicity.
Lemma 2.1 If , is the solution of (2.1)-(2.3), then and
Proof Let , and
The condition assures that defined above is nonnegative. If we multiply (2.1) by and integral on Ω, then we have
From the above calculation, we have
By the Poincare inequality, we have
Let in (2.7). We can deduce that
By the Jessen inequality, from (2.8) we get
We get the desired result. □
Lemma 2.2 If , is the solution of (2.1)-(2.3), then
Proof Multiply (2.1) by and integral on Ω, then
which deduces that
Set . Then
where c is a constant independent of l.
Now, if we choose , , , , , , by Lemma 1.3, we have
If we choose in (2.11), by (2.12) we have
We will prove that there exist two bounded sequences , such that
If , by Lemma 2.1, , makes (2.14) sure. If (2.14) is true for , then from (2.13),
We can choose
by Lemma 1.4 and (2.15), (2.14) is true.
Moreover, by Lemma 1.4, as , . It is easy to see that is bounded. Thus (2.9) is true.
To prove (2.10), we set , , . By (2.11), we have
By Lemma 3.1 in , we can get (2.10); we omit details here. □
3 The estimation of the gradient
Lemma 3.1 If , is the solution of (2.1)-(2.3), then
Proof If we multiply (3.1) by and integral on Ω, then
By (3.3)-(3.5), we have
If we multiply (3.1) by and integral on Ω, then
Setting , for , if we notice that , we have
If , let . By Lemma 1.3,
where , and when , , when . By Lemma 2.1 and Lemma 2.2, from (3.8), we have
At the same time, if we choose in Lemma 2.1, we have
By (3.7), we have
If and , then
If and , then
The inequalities (3.13) and (3.14) mean that the inequality (3.12) is still true when . Using Lemma 1.4, we get
which means (3.1) is true. Now, we will prove (3.2). For , by (2.10) we obtain
By (3.7), using (3.15)-(3.17) yields
and using the Young inequality gives
which means (3.2) is true. □
Lemma 3.2 If , is the solution of (2.1)-(2.3), then
Proof From (2.9), (3.1) and (3.7), (3.10), we have
4 The proof of Theorem 1.5
The proof of Theorem 1.5 From Lemma 2.1, Lemma 2.2, Lemma 3.1 and Lemma 3.2, using the compactness theory (cf. ), there is a sequence (still denoted as) of such that when , we have
where and every is a function in , . (4.1) and (4.2) are clearly true. In what follows, we only need to prove that
It is easy to know that
So, if we can prove that
then (4.5),(4.6) and (1.8) are true.
First, for any , , , we have
If we multiply by the two sides of (2.1), then we have
Noticing that when , we have
and when , we get
By (4.10), (4.11), we have
if we let in (4.12), we have
Now, we choose in (4.7),
From this formula and (4.13), we have
Let , , . Then
Let . We obtain
Moreover, if we choose , we are able to get
Now, if we choose ψ such that , and on suppφ, , then from (4.15)-(4.16), we can get (4.8). By (4.7) and (4.8), we have
which means (4.9) is true, and so (1.8) is true.
Secondly, we are to prove (1.9).
For small , denote . For any , let
For any given small and large enough k, l, we declare that
where is independent of , and . By (2.1) we have
Suppose that such that
and choose in (4.18), then
If we notice that the third term on the left-hand side of (4.19) tends to zero when , then we have
At the same time,
By (4.20) and (4.21), we have
By Lemma 2.2 and Lemma 3.1, if ,
which means (4.17) is true.
Now, for any given small r, if k, l are large enough, by (4.17), we have
Letting , we get (1.9). □
5 The uniqueness of the viscosity solution
As we have said in the introduction, the uniqueness of the solutions of (1.1)-(1.3) is not true generally. But we are able to prove the uniqueness of the viscosity solution.
Theorem 5.1 If , in addition,, , then the viscosity solution of (1.1)-(1.3) is unique.
Proof Let u, v be two viscosity solutions of (1.1)-(1.3). Then there are two sequences and , which are the solutions of (1.10)-(1.2)-(1.3), such that
Clearly, since ,
and since , using the convexity of the function , by (5.2), we have
By Chapter 8 of , we know that
Let , we know that the uniqueness of the viscosity solution (1.1)-(1.3) is true. □
Suppose that the viscosity solution of (1.1)-(1.3) is unique in what follows. Then, by considering the regularized problem (1.10) with (1.2)-(1.3), we easily get the following lemma.
Lemma 5.2 Let u be a weak solution of (1.1)-(1.3). If v satisfies
Now, we will prove Theorem 1.6. Let
Noticing that we supposed
which implies that
and using the argument similar to that in the proof Lemma 3.5 of , we can prove
It follows that
Letting , we get
Hence, we have proved Theorem 1.6.
6 The proof of Theorem 1.7
If , from the process of the proof of Lemma 2.1, we also have (2.8), i.e.,
But, since , the Jessen inequality is invalid now, and (2.4) may not be true. However, in this case, (6.1) implies that
which gives the information of provided that .
Lemma 6.1 Suppose that and
If is the solution of (2.1)-(2.3), then
where , and .
Proof Similarly as in the proof of Lemma 2.2, we multiply (2.1) by and integral on Ω, and then we get the following inequality (6.6), which is just the same as (2.11).
Let , be two sequences just the same as those in the proof of Lemma 2.2. Since (6.3) implies that and , we can deduce the conclusions (6.4) similarly as in Lemma 2.2.
To prove (6.5), we also set , , . By (6.6), we have
By Gagliardo-Nirenberg Lemma 1.3, let . Then
If we choose , then from the above inequality, we have
By (6.7), (6.8), we have
Now, we choose the constant , i.e.,
By (6.9), we have
Since , , by Lemma 1.4, we have
which implies that
If , which implies that , then we can get the conclusions of Lemma 3.1 in a similar way. As in the proof of Theorem 1.5, we get the existence of the solution for the system (1.1)-(1.3) in this case. □
Proposition 6.2 Let u be a weak solution of (1.1)-(1.3). If , then there exists a finite time T such that
for all .
To prove this proposition, we use the idea of the proof of Theorem 1.1 in , in which the extinction of the solution for the equation
was studied. In detail, we define an auxiliary function
Then we have
on account of the non-positivity of the damping term .
If we notice that
applying Lemma 6.1, by (6.12)-(6.13), we have
for all . By the definition of , we have
The proof of the proposition is complete.
Theorem 1.7 is a direct corollary of the proposition.
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The paper is supported by NSF (no. 2012J01011) of Fujian Province, supported by SF of Xiamen University of Technology, China.
The author declares that they have no competing interests.
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Zhan, H. The global solution of a diffusion equation with nonlinear gradient term. J Inequal Appl 2013, 125 (2013). https://doi.org/10.1186/1029-242X-2013-125
- diffusion equation
- Moser iteration
- viscosity solution