The global solution of a diffusion equation with nonlinear gradient term
© Zhan; licensee Springer 2013
Received: 4 September 2012
Accepted: 7 March 2013
Published: 26 March 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.
where is a bounded open domain, , , , , , , , and ∇ is the spatial gradient operator.
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.
- (i)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 thatin 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.and(1.19)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
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.
We get the desired result. □
where c is a constant independent of l.
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.
By Lemma 3.1 in , we can get (2.10); we omit details here. □
3 The estimation of the gradient
which means (3.2) is true. □
4 The proof of Theorem 1.5
then (4.5),(4.6) and (1.8) are true.
which means (4.9) is true, and so (1.8) is true.
Secondly, we are to prove (1.9).
which means (4.17) is true.
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.
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.
Hence, we have proved Theorem 1.6.
6 The proof of Theorem 1.7
which gives the information of provided that .
where , and .
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.
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. □
for all .
on account of the non-positivity of the damping term .
The proof of the proposition is complete.
Theorem 1.7 is a direct corollary of the proposition.
The paper is supported by NSF (no. 2012J01011) of Fujian Province, supported by SF of Xiamen University of Technology, China.
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