Well-Posedness for Fractional Cauchy Problems Involving Discrete Convolution Operators

Abstract

This work focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {D}}^\beta _t u(n,t)= B u(n,t) + f(n-ct,u(n,t)),\, &{}n\in {\mathbb {Z}}, \;t>0,\\ u(n,0)=\varphi (n),\; &{}n\in {\mathbb {Z}},\\ \end{array}\right. } \end{aligned}$$

under the assumptions that $\beta \in (0,1]$, $c>0$ some constant, and B is a discrete convolution operator with kernel $b\in \ell ^1(\mathbb {Z})$, which is the infinitesimal generator of the Markovian $C_0$-semigroup and suitable nonlinearity f. We present results concerning the existence and uniqueness of solutions, as well as establishing a comparison principle of solutions according to the respective initial values.

Appendix

In this appendix, we give the basic concepts of the special functions involved in the development of this study.

1.1 Wright Function $\Phi _{\gamma }$

The Wright type function with one parameter from [16, Formula (28)] (see also [30, 33, 45]) is given by

$$\begin{aligned} \Phi _{\alpha }(z):= \sum _{n=0}^{\infty } \frac{(-z)^n}{n! \Gamma (-\alpha n +1 -\alpha )} = \frac{1}{2\pi i} \int \limits _{\gamma } \mu ^{\alpha -1} e^{\mu - z \mu ^{\alpha }} \text {d}\mu , \quad 0< \alpha <1,\nonumber \\ \end{aligned}$$

(4.5)

where $\gamma$ is a contour which starts and ends at $-\infty$ and encircles the origin once counterclockwise. This special case has sometimes been called Mainardi function.

Remark 4.5

Let $z\in {\mathbb {C}}$, $t>0$ and $0<\alpha ,\gamma <1$. Then the following properties hold:

• (i) $\displaystyle E_{\gamma ,1}(z)=\int \limits _{0}^{\infty }\Phi _{\gamma }(t)e^{zt}\text {d}t$.

• (ii) $\displaystyle \Phi _{\alpha }(t) \ge 0$.

• (iii) $\displaystyle \int \limits _{0}^{\infty }\Phi _{\alpha }(t)\text {d}t =1$.

It follows from (ii) and (iii) that $\Phi _{\alpha }$ is a probability density function on ${\mathbb {R}}_0^+$. Actually, the Wright function has been used for models in stochastic processes [16, 17].

1.2 Lévy Function $f_{t,\alpha }$

We present the following function, called stable Lévy function, defined for $0<\alpha <1$ by

$$\begin{aligned} f_{t,\alpha }(\lambda ) = {\left\{ \begin{array}{ll} \displaystyle \frac{1}{2\pi i} \int _{\sigma - i \infty }^{ \sigma + i \infty } e^{ z \lambda - t z^{\alpha }} \text {d}z, \;\;\;\; \sigma>0, \quad t>0, &{}\lambda \ge 0, \\ 0 \quad &{}\lambda <0, \end{array}\right. } \end{aligned}$$

(4.6)

where the branch of $z^{\alpha }$ is taken so that $\text{ Re }(z^{\alpha })>0$ for $\text{ Re }(z)>0.$ This branch is single valued in the z-plane cut along the negative real axis. These functions were introduced by Bochner [8] in the study of certain stochastic processes. Yosida [47] used them systematically in the study of $C_0$-semigroups generated by fractional powers of uniformly bounded $C_0$-semigroups of linear operators. The Lévy functions are the density functions associated with the stable Lévy processes in the rotational invariant case and are related to the fractional Brownian motion.

Remark 4.6

The following properties hold:

• (i) $\displaystyle \int _0^{\infty } e^{-\lambda a} f_{t,\alpha }(\lambda ) \text {d}\lambda = e^{-ta^{\alpha }}, \quad t>0, \quad a>0,\quad 0<\alpha <1.$

• (ii) $\displaystyle f_{t,\alpha }(\lambda ) \ge 0, \quad \lambda>0, \,\, t>0, \quad 0<\alpha <1.$

• (iii) $\displaystyle \int _0^{\infty } f_{t,\alpha }(\lambda ) \text {d}\lambda = 1, \quad t>0, \quad 0<\alpha <1.$

For a proof of (i)–(iii), see [47, pp. 260–262].

1.3 Bessel Function $I_\nu$

For $\nu \in {\mathbb {R}}$, the modified Bessel functions of the first kind is defined by

$$\begin{aligned} I_{\nu }(x)=\sum \limits _{n=0}^{\infty }\frac{1}{\Gamma (n+\nu +1)n!}\left( \frac{x}{2}\right) ^{2n+\nu }. \end{aligned}$$

(4.7)

It is clear from the Definition 4.7 that $I_n(x) \ge 0, \quad n\in {\mathbb {Z}}, \,\, x\ge 0$. The following properties can be found in [18, Formula 8.511], [4].

• (1) $\sum _{n\in {\mathbb {Z}}} I_n(x)z^{n} = \displaystyle e^{\frac{x}{2}(z+\frac{1}{z})}, \quad z \in {\mathbb {C}}{\setminus }\{0\}$.

• (2) $I_{-n}(x)=I_{n}(x)=(-1)^nI_n(-x)$.

• (3) $I_n(x+y)=\sum _{k\in {\mathbb {Z}}}I_{n-k}(x)I_k(y)$.