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

Authors: Ailing Shi; Shuqin Zhang
Journal: ISRN Mathematical Analysis
 (2012)

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## 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.


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## Body

## 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(1.1)Dtαu(t,x)=D2Δu-u-ν,x∈Rn,t∈(0,T],ν>0,u(0,x)=ϕ(x)∈CLB(Rn),x∈Rn,
where Dtα is a regularized fractional derivative (the Caputo derivative) of order 0<α≤1 with respect to t, defined by(1.2)Dtαu(t,x)=1Γ(1-α)[∂∂t∫0t(t-τ)-αu(τ,x)dτ-t-αu(0,x)∫0t(t-τ)-α],0<α≤1,1/Γ(1-α)∫0t(t-τ)-αu(τ,x)dτ=It1-αu(t,x) is the Riemann-Liouville fractional integral of order 1-α with respect to t; D is a positive constant, n≥1,(1.3)CLB(Rn)={M|x|-kϕ∈C(Rn):ϕ>0inRnonlywithMϕmin⁡=min⁡Rnϕ>0,thereexistk≥0andM>0onlysuchthat|x|-kϕ(x)≤Λ(Λ>0)for|x|≥M}.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, 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(2.1)Dtαu-D2Δu=q(t,x),t∈(0,T],x∈Rn,D>0,u(0,x)=ϕ(x),x∈Rn,
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:
(2.2)u(t,x)=∫RnZ0(t,x-ξ;ξ)ϕ(ξ)dξ+∫0tdτ∫RnY0(t-τ,x-ξ;ξ)q(τ,ξ)dξ,
where
(2.3)Z0(t,x-y)=π-n/2D|x-y|-nH1220×[14t-α|x-y|2|(1,α)(n/2,1),(1,1)],Y0(t,x-y)=π-n/2D|x-y|-ntα-1H1220×[14t-α|x-y|2|(α,α)(n/2,1),(1,1)].H1220 is an H-function defined by
(2.4)H1220(z)=H1220[z|(1,α)(n/2,1),(1,1)]=12πi∫CΓ(n/2+s)Γ(1+s)Γ(1+αs)z-sds,
where 𝒞 is an infinite contour [12]. Moreover, the functions Z0 and Y0 are nonnegative. Z0 satisfies the relation
(2.5)∫RnZ0(t,x-y)dy=1.Remark 2.2.
From [4], Y0(t,x) is the Riemann-Liouville derivative of Z0(t,x) in t of the order 1-α (for x≠0, Z0(t,x)→0 as t→0, so that the Riemann-Liouville derivative Dt1-αZ0(t,x) coincides in this case with the regularized fractional derivative Itα∂Z0(t,x)/∂t and that
(2.6)Dt1-αZ0(t,x)=π-n/2D|x-y|-ntα-1H1220×[14t-α|x-y|2|(α,α)(n/2,1),(1,1)].
Therefore, together with the formulas ItαItβf(t,x)=ItβItαf(t,x) and It1Dt1f(t,x)=f(t,x)-f(0,x), f is summable with respect to t, we obtain that the solution (2.2) has the following form:
(2.7)u(t,x)=∫RnZ0(t,x-ξ;ξ)ϕ(ξ)dξ+1Γ(α)∫0t(t-τ)α-1dτ×∫RnZ0(t-τ,x-ξ;ξ)q(τ,ξ)dξ=EtΔϕ+1Γ(α)∫0t(t-τ)α-1E(t-τ)Δq(τ,x)dτ,
where
(2.8)EtΔϕ=π-n/2D∫Rn|x-y|-nH1220×[14t-α|x-y|2|(1,α)(n/2,1),(1,1)]ϕ(y)dy.
Consequently, from the Lemma2.1 and Remark 2.2, we see that the solution of (1.1) has the form
(2.9)u(t,x)=EtΔϕ-1Γ(α)∫0t(t-s)α-1E(t-s)Δu-ν(s,x)ds,forall(t,x)∈[0,T]×Rn.Remark 2.3.
In (2.9), it follows from the definition of EtΔ,t≥0 and 0≤s≤t, that E(t-s)Δ is well defined.Definition 2.4.
We call a functionu a local positive solution of (1.1) on Rn×[0,T] if(i)
u∈C([0,T′]×Rn) with umin:=min[0,T′]×Rnu>0 for any 0<T′≤T,(ii)
u(t,x)=EtΔϕ-(1/Γ(α))∫0t(t-s)α-1E(t-s)Δu-ν(s,x)ds for all (t,x)∈[0,T′]×Rn.The next theorem is the main result of this paper.Theorem 2.5.
Letϕ∈CLB(Rn). Then, (1.1) has a local positive solution v.Proof.
Definingρ=ϕmin>0, we will find 0<t0≤T and ρ̃>0 depending on ρ, ν, and n such that (1.1) has a positive solution v(t,x) satisfying
(2.10)v(t,x)≥ρ̃for(t,x)∈[0,t0]×Rn.
Let
(2.11)F(u)=EtΔϕ-1Γ(α)∫0t(t-s)α-1E(t-s)Δu-ν(s,x)ds.
For 0<ρ̃<ρ, which will be determined below, we construct a sequence {uk} as follows
(2.12)uk(t,x)=EtΔϕ-1Γ(α)∫0t(t-s)α-1E(t-s)Δuk-1-ν(s,x)ds,k=0,1,2,…,
with u-1(t,x)=ρ̃. For 0<ρ̃<min{1,ρ/2}, by (2.5), we have
(2.13)EtΔϕ=π-n/2D∫Rn|x-y|-nH1220×[14t-α|x-y|2|(1,α)(n/2,1),(1,1)]ϕ(y)dy≥π-n/2ρD∫Rn|x-y|-nH1220×[14t-α|x-y|2|(1,α)(n/2,1),(1,1)]dy=ρ≥2ρ̃.
Also, we can obtain that
(2.14)EtΔϕ=π-n/2D∫Rn|x-y|-nH1220×[14t-α|x-y|2|(1,α)(n/2,1),(1,1)]ϕ(y)dy≤π-n/2D∫Rn|x-y|-nH1220[14t-α|x-y|2×|(1,α)(n/2,1),(1,1)]C(1+|y|)kdy=π-n/2D∫Rnη-nH1220[14|η|2|(1,α)(n/2,1),(1,1)]×C(1+|x|+tα/2|η|)kdη≤2kC(1+|x|)k+2kCBtkα/2≤C(1+|x|)k,
where B=(π-n/2/D)∫Rnηk-nH1220[(1/4)|η|2|(1,α)(n/2,1),(1,1)]dη.
On the other hand, if0<t≤(Γ(1+α)ρ̃1+ν)1/α, by (2.5), we have
(2.15)u0(t,x)=EtΔϕ-1Γ(α)∫0t(t-s)α-1E(t-s)Δρ̃-νds≥2ρ̃-ρ̃-νΓ(1+α)tα≥ρ̃.
Moreover, it is obvious that
(2.16)u0(t,x)≤C(1+|x|)k,for0<t≤T,x∈Rn.
Hence, it holds that
(2.17)ρ̃≤u0(t,x)≤C(1+|x|)kfor(t,x)∈[0,t0]×Rn,
where t0=min{T,(Γ(1+α)ρ̃1+ν)1/α}. From the previous arguments, it is very easy to show that
(2.18)ρ̃≤uk(t,x)≤C(1+|x|)kfor(t,x)∈[0,t0]×Rnandallk≥0.
Now, we show that
(2.19)|uk-uk-1|≤νk-1(2ρ̃ν+1)k1Γ(1+kα)tkα,∀k≥1.
In fact, we have that
(2.20)|u1-u0|≤1Γ(α)∫0t(t-s)α-1E(t-s)Δ|u0-ν-ρ̃-ν|ds≤2ρ̃ν1Γ(1+α)tα≤2ν0ρ̃ν+11Γ(1+α)tα.
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
(2.21)um-1-ν-um-2-ν=-νξ-(ν+1)(um-1-um-2)=-C(t,x)(um-1-um-2),
where ξ=sum-1+(1-s)um-2 with s∈(0,1), it follows from (2.18) that C(t,x)≤νρ̃-(ν+1).
Hence, we can obtain that(2.22)|um-um-1|≤1Γ(α)∫0t(t-s)α-1E(t-s)Δ|um-1-ν-um-2-ν|ds≤1Γ(α)∫0t(t-s)α-1E(t-s)ΔC(t,x)|um-1-um-2|ds≤νρ̃ν+11Γ(α)∫0t(t-s)α-1E(t-s)Δ|um-1-um-2|ds≤νρ̃ν+1νm-2(2ρ̃ν+1)m-11Γ(1+(m-1)α)1Γ(α)×∫0t(t-s)α-1E(t-s)Δs(m-1)αds=νm-12m-1(ρ̃ν+1)mΓ(1+(m-1)α)Γ(1+(m-1)α)Γ(1+mα)tmα≤νm-1(2ρ̃ν+1)m1Γ(1+mα)tmα.
Therefore, (2.19) is valid for all k=1,2,…. Thus, we see that, for 0<t≤t0,x∈Rn, it holds that
(2.23)|uk-uk-1|≤νk-1(2ρ̃ν+1)k1Γ(1+kα)t0kα.
Defineζk=∑j=1k(uj-uj-1). Since |ζk|≤∑j=1kνj-1(2/ρ̃ν+1)j(t0jα/Γ(1+jα)), we can claim that ζk→ζ uniformly in [0,t0]×Rn as k→∞ and ζ∈C([0,t0]×Rn). In fact, from the Stirling formula, we have that
(2.24)Γ(1+nα)=2πnα(nαe)nαeθ/12nα,
where 0<θ<1. We let ∑j=1∞νj-1(2/ρ̃ν+1)j(t0jα/Γ(1+jα))=∑j=1∞xj; thus, we have that
(2.25)xj+1xj=2νt0αΓ(1+jα)ρ̃ν+1Γ(1+(j+1)α)=2νt0αρ̃ν+12πjα(jα/e)jαeθ/12jα2π(j+1)α((j+1)α/e)(j+1)αeθ/12(j+1)α≤2νt0αρ̃ν+12π(j+1)α((j+1)α/e)jαeθ/12jα2π(j+1)α((j+1)α/e)(j+1)αeθ/12(j+1)α=2νt0αρ̃ν+1eθ/12j(j+1)α((j+1)αe)-α,0≤lim⁡j→∞xj+1xj≤lim⁡j→∞2νt0αρ̃ν+1eθ/12j(j+1)α((j+1)αe)-α=0,
that is,
(2.26)lim⁡j→∞xj+1xj=0;
then, by the convergence principle of D’Alembert, we can claim that ζk→ζ uniformly in [0,t0]×Rn as k→∞ and ζ∈C([0,t0]×Rn). Moreover, we can obtain that uk=u0+ζk→u0+ζ:=v(t,x) uniformly in [0,t0]×Rn, as k→∞, and v(t,x)∈C([0,t0]×Rn). Combining with (2.18), we have
(2.27)v(t,x)≥ρ̃in[0,t0]×Rn,
and, letting k→∞ in (2.12), we have
(2.28)v(t,x)=EtΔϕ-1Γ(α)∫0t(t-s)α-1E(t-s)Δv-ν(s,x)ds.
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(2.29)ϕmin⁡=ρ>2(TαΓ(1+α))1/(1+v),
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]×Rn.  In fact, we may let
(2.30)(TαΓ(1+α))1/(1+v)<ρ̃<{1,ρ2};
then we can obtain that
(2.31)u0(t,x)=EtΔϕ-1Γ(α)∫0t(t-s)α-1E(t-s)Δρ̃-νds≥2ρ̃-ρ̃ρ̃-1-νΓ(1+α)tα≥2ρ̃-ρ̃ρ̃-1-νΓ(1+α)Tα≥2ρ̃-ρ̃=ρ̃.
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 ut(t,x) should be replace by Dtαu(t,x),0<α≤1.



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*Source: 868295-2012-03-07.xml*