Existence of Positive Solution to the Cauchy Problem for a Fractional Diffusion Equation with a Singular Nonlinearity

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.


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 D α t is a regularized fractional derivative the Caputo derivative of order 0 < α ≤ 1 with respect to t, defined by x is the Riemann-Liouville fractional integral of order 1 − α with respect to t; D is a positive constant, n ≥ 1, φ min min R n φ > 0, there exist k ≥ 0 and M > 0 only such that |x| −k φ x ≤ Λ Λ > 0 for |x| ≥ M .

1.3
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 .

Main Result
In this section, we will establish the local existence of positive solutions to the Cauchy problem 1.1 .In what follows, C C • • • 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 φ (locally Hölder continuous if n > 1) and a bounded jointly continuous and locally Hölder continuous, in x, function q, has the following integral representation of solution:

2.3
H 20 12 is an H-function defined by where C is an infinite contour [12].Moreover, the functions Z 0 and Y 0 are nonnegative.Z 0 satisfies the relation Remark 2.2.From 4 , Y 0 t, x is the Riemann-Liouville derivative of Z 0 t, x in t of the order 1 − α for x / 0, Z 0 t, x → 0 as t → 0, so that the Riemann-Liouville derivative D 1−α t Z 0 t, x coincides in this case with the regularized fractional derivative I α t ∂Z 0 t, x /∂t and that

2.6
Therefore, together with the formulas is summable with respect to t, we obtain that the solution 2.2 has the following form: where φ y dy.

2.8
Consequently, from the Lemma 2.1 and Remark 2.2, we see that the solution of 1.1 has the form

2.9
Remark 2.3.In 2.9 , it follows from the definition of E tΔ , t ≥ 0 and 0 ≤ s ≤ t, that E t−s Δ is well defined.Definition 2.4.We call a function u a local positive solution of 1.
The next theorem is the main result of this paper.Proof.Defining ρ φ min > 0, we will find 0 < t 0 ≤ T and ρ > 0 depending on ρ, ν, and n such that 1.1 has a positive solution v t, x satisfying

2.11
For 0 < ρ < ρ, which will be determined below, we construct a sequence {u k } as follows

2.13
Also, we can obtain that , by 2.5 , we have

2.15
Moreover, it is obvious that

6 ISRN Mathematical Analysis
Hence, it holds that where t 0 min{T, Γ 1 α ρ 1 ν 1/α }.From the previous arguments, it is very easy to show that Now, we show that

2.19
In fact, we have that

2.20
If we assume that 2.19 holds for k m − 1 m > 2 , next, we will show that 2.19 is valid for k m.Indeed, since where ξ su m−1 1 − s u m−2 with s ∈ 0, 1 , it follows from 2.18 that C t, x ≤ ν ρ − ν 1 .Hence, we can obtain that

2.28
This mean that v t, x is a local positive to the Cauchy problem 1.1 .Thus, the proof is complete.
Remark 2.6.Under the assumption that 29 by a similar proof to that of Theorem 2.5, we can claim that problem 1.1 has positive solution on the interval 0, T × R n .In fact, we may let 2.30 then we can obtain that

2.31
The remaining proofs may be finished by the same way as the Theorem 2.5.

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 u t t, x should be replace by D α t u t, x , 0 < α ≤ 1.

Theorem 2 . 5 .
Let φ ∈ C LB R n .Then, 1.1 has a local positive solution v.
1 j t jα 0 /Γ 1 jα , we can claim that ζ k → ζ uniformly in 0, t 0 ×R n as k → ∞ and ζ ∈ C 0, t 0 ×R n .In fact, from the Stirling formula, we have that ISRN Mathematical Analysisu 0 ζ : v t, x uniformly in 0, t 0 × R n , as k → ∞, and v t, x ∈ C 0, t 0 × R n .Combining with 2.18 , we have then, by the convergence principle of D'Alembert, we can claim thatζ k → ζ uniformly in 0, t 0 × R n as k → ∞ and ζ ∈ C 0, t 0 × R n .Moreover, we can obtain that u k u 0 ζ k →