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 [3–6]. 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 [7–9]. 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 Hölder continuous if ) and a bounded jointly continuous and locally Hölder 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 D’Alembert, 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).