Chapter 5 Two-sided generalized equations of RL and Caputo type
6.1 Introduction
The generalized fractional evolution equations of Caputo type studied in this chap- ter can be thought of as classical evolution equations wherein the first-order time derivative has been replaced by the non-local operator of Caputo type−Da(ν+∗).
Based on the notion ofGreen’s functions for differential operators, we shall study: i) the nonhomogeneous generalized fractional evolution equation
−tD(aν+∗)u(t, x) =Axu(t, x) −g(t, x), t∈ (a, b], x∈Rd,
for given functionsg andφa defined on[a, b] ×Rdand Rd, respectively;
ii) the generalized fractional nonlinear equation
−tD(aν+∗)u(t, x) =Axu(t, x) +f(t, x, u(t, x)), t∈ (a, b], x∈Rd,
u(a, x) =φa(x), x∈Rd (6.1.2)
wheref is a given function on [a, b] ×Rd×R.
Notation−tDa(ν+∗) means the Caputo type operator−D(aν+∗) acting on the (time) vari-
ablet, whereas−Axstands for the generator of a Feller process acting on the (space)
variablex.
Since Caputo derivatives are special cases of the operators −D(aν+∗), the generalized
equations in (6.1.1)-(6.1.2) include, as particular cases, a variety of equations stud- ied in the theory of fractional partial differential equations (FPDE’s). The latter equations have been successfully used for describing diffusions in disordered media, also called anomalous diffusions, which include subdiffusions as well as enhanced diffusions (orsuperdiffusions). Subdiffusion phenomena are usually related to time- FPDE’s, whereas superdiffusions are related to space- FPDE’s.
In the classical fractional setting, the fractional Cauchy problems are special cases of equation (6.1.1). Fractional Cauchy problems are initial value problems involving the Caputo derivative of orderβ ∈ (0,1):
tDβ0+∗u(t, x) =Axu(t, x), (t, x) ∈ [0, b] ×Rd,
u(0, x) =φ0(x), x∈Rd (6.1.3)
Equations of the type in (6.1.3) have been actively studied in the literature. Amongst the standard analytical approaches to solve FPDE’s,the Laplace-Fourier transform
technique plays an important role (see, e.g., [15], [18], [45], [73], [76], and references therein). From a probabilistic point of view, interesting connections have been found
between the solution of (time-) FPDE’s and the transition densities of time-changed Markov processes (see, e.g., [26], [52], [53], [67], [71], [78]).
A very standard example of the equation (6.1.3) is given by the (time-) fractional diffusion equation (or fractional-kinetic equation) [10], [62], [67] corresponding to the case Ax = −12∆x, where ∆x denotes the Laplace operator. Its fundamental
solution was first studied by Schneider and Wyss [79] and Kochubei [51]. In this case the fundamental solution corresponds to the time-changed transition probability function of the Brownian motion by the hitting time of aβ-stable subordinator. Another example of equation (6.1.3) was studied in [19], wherein the authors con- sider the second-order differential operator given by
Ax= d ∑ i,j aij(x) ∂2 ∂xi∂xj + d ∑ j=1 bj(x) ∂ ∂xj +c(x).
As for nonhomogeneous equations, the multi-time fractional differential equation: n
∑
k=1
λk tDβ0+∗k u(t, x) −∆xu(t, x) =g(t, x), λk, t∈Rn, x∈Rd.
was investigated in [74].
More recently, the regularity of the nonhomogenous time-space fractional linear equation for the fractional Laplacian operatorAx= −(−∆)α/2:
tDβ0+∗u(t, x) = −c(−∆) α/2u
(t, x) +g(t, x), x∈Rd, t≥0,
u(0, x) =φ0(x), x∈Rd
as well as the well-posedness for the fractional HJB type equation
tDβ0+∗u(t, x) = −c(−∆)α/2u(t, x) +H(t, x,∇u(t, x)), x∈Rd, t≥0,
were addressed in [54], forβ∈ (0,1),α∈ (1,2], and a positive constantc>0.
Evolution equations of the type (6.1.3) arise, for example, as the limiting evolution of an uncoupled and properly scaled continuous time random walk (CTRW) with the waiting times in the domain of attraction of β−stable laws. This probabilistic
model and some of its extension have been widely studied (see, e.g., [67], [78], [53], and references therein). Yet another extension of equation (6.1.3) can be obtained, for instance, by considering the limiting evolution of properly scaledcoupled CTRW (or semi-Markov processes) with power law waiting times depending on the state of the system. This procedure yields the generalized evolution equation with the Caputo type operator of variable ordertDβ0+∗(t,x), i.e.,
tD0β+∗(t,x)u(t, x) =A(xt)u(t, x), (t, x) ∈ [0, b] ×Rd, β(⋅) ∈ (0,1). (6.1.4)
Using the results presented here, we are able to deduce some of the results known for the previous cases, as well as to extend the analysis to more general situations. Some specific equations of the type in (6.1.4) will be discussed in a forthcoming paper in preparation.
We highlight the fact that the (analytical) approach used in this chapter is different from the one used in chapters 3-5. Namely, in analogy with the standard analytical methods to solve classical evolution equations, we obtain well-posedness results for the nonhomogeneous equation (6.1.1) by transforming (6.1.1) into anabstract gen- eralized fractional linear equation on a suitable Banach space. Then, we construct the solutions via the concept of Green’s function. For the stochastic representation for the solutions, we use Dynkin’s martingale and Doob’s stopping theorem as usual.
As for the nonlinear case, we study the well-posedness for the ’ordinary’ equation (6.1.2) following a similar strategy to the one used for the nonlinear equation in
(4.1.1). Namely, by means of the the integral representation (mild form) of the solution to the linear problem (6.1.1), we reduce the analysis of (6.1.2) to a fixed point problem for a suitable operator. Let us mention that, even though in this work we do not include the HJB type case, our results for the generalized nonlinear equation (6.1.2) can be used to extend the well-posedness for the corresponding equations of HJB type.