Existence of Positive Solution to the Cauchy Problem for a Fractional Diffusion Equation with a Singular Nonlinearity
Ailing Shi1and Shuqin Zhang2
Academic Editor: D. Kouznetsov
Received16 Nov 2011
Accepted17 Jan 2012
Published07 Mar 2012
Abstract
Fractional diffusion equations describe an anomalous diffusion on fractals. In this paper,
by means of the successive approximation method and other analysis technique, we present a local positive solution to Cauchy problem for a fractional diffusion equation with singular nonlinearity. The fractional derivative is described in the Caputo sense.
1. Introduction
In this paper, we consider the existence of a local positive solution to the Cauchy problem for fractional diffusion equation with singular nonlinearity
where is a regularized fractional derivative (the Caputo derivative) of order with respect to , defined by is the Riemann-Liouville fractional integral of order with respect to ; is a positive constant, ,
Fractional diffusion equations describe an anomalous diffusion on fractals (physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials); see [1, 2]. We know that there is a few papers dealing with solutions to some initial value problems for linear fractional diffusion equations, by means of the Laplace transform, Fourier transform, and so forth; see [3β6]. There are also some papers concerned with initial value problems for nonlinear fractional diffusion equations, in which the existence of solutions is considered, using some estimates and the monotone iterative method; see [7β9]. For the details for fractional diffusion equations, please see articles [10, 11] written by Kilbas and Trujillo. However, as far as we know there are few papers which consider the existence of local positive solution to the Cauchy problem (1.1). In this paper, we consider the local existence of positive solutions to singular problem (1.1).
2. Main Result
In this section, we will establish the local existence of positive solutions to the Cauchy problem (1.1). In what follows, denotes positive constants, besides the arguments inside the parenthesis, which vary from line to line.
Lemma 2.1 (see [4]). The linear initial value problem
with a bounded continuous function (locally HΓΆlder continuous if ) and a bounded jointly continuous and locally HΓΆlder continuous, in , function , has the following integral representation of solution:
where
is an -function defined by
where is an infinite contour [12]. Moreover, the functions and are nonnegative. satisfies the relation
Remark 2.2. From [4], is the Riemann-Liouville derivative of in of the order (for , as , so that the Riemann-Liouville derivative coincides in this case with the regularized fractional derivative and that
Therefore, together with the formulas and , is summable with respect to , we obtain that the solution (2.2) has the following form:
where
Consequently, from the Lemma 2.1 and Remark 2.2, we see that the solution of (1.1) has the form
Remark 2.3. In (2.9), it follows from the definition of and , that is well defined.
Definition 2.4. We call a function a local positive solution of (1.1) on if(i) with for any ,(ii) for all .
The next theorem is the main result of this paper.
Theorem 2.5. Let . Then, (1.1) has a local positive solution .
Proof. Defining , we will find and depending on , , and such that (1.1) has a positive solution satisfying
Let
For , which will be determined below, we construct a sequence as follows
with . For , by (2.5), we have
Also, we can obtain that
where On the other hand, if , by (2.5), we have
Moreover, it is obvious that
Hence, it holds that
where . From the previous arguments, it is very easy to show that
Now, we show that
In fact, we have that
If we assume that (2.19) holds for (), next, we will show that (2.19) is valid for . Indeed, since
where with , it follows from (2.18) that . Hence, we can obtain that
Therefore, (2.19) is valid for all . Thus, we see that, for , it holds that
Define . Since , we can claim that uniformly in as and . In fact, from the Stirling formula, we have that
where . We let ; thus, we have that
that is,
then, by the convergence principle of DβAlembert, we can claim that uniformly in as and . Moreover, we can obtain that uniformly in , as , and . Combining with (2.18), we have
and, letting in (2.12), we have
This mean that is a local positive to the Cauchy problem (1.1). Thus, the proof is complete.
Remark 2.6. Under the assumption that
by a similar proof to that of Theorem 2.5, we can claim that problem (1.1) has positive solution on the interval . In fact, we may let
then we can obtain that
The remaining proofs may be finished by the same way as the Theorem 2.5.
3. Conclusion
In this paper, we obtain the existence of a local (or nonlocal) positive of problem (1.1), that is, our main results: Theorem 2.5 and Remark 2.2. It is well known that fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Problem (1.1) may appear in several applications in mechanics and physics and in particular can be used to model the electrostatic micro-electromechanic System devices, in which the derivatived should be replace by .
Acknowlegment
This paper is supported by the NNSF of china (10971221), the Ministry of Education for New Century Excellent Talent Support Program (NCET-10-0725), and the Fundamental Research Funds for the Central Universities (2009QS06).
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