Coupled systems of nonlinear sequential proportional Hilfer-type fractional differential equations with multi-point boundary conditions

1.   Introduction and main results

Fractional derivatives are integro-differential operators which generalize integer-order differential and integral calculus. They can describe the property of memory and heredity of various materials and processes compared with integer-order derivatives. In recent years, many scholars are committed to the research of time-fractional or space-fractional partial differential equations, see ^{[1,2,3,4,5,6,7]}. On the other hand, fractional diffusion models are employed for some engineering problems ^[8,9] with power-law memory in time and physical models considering memory effects ^[10,11,12]. There are numerous works devoted to fractional diffusion equations. We only list several of the numerous papers on the analysis for fractional diffusion equations. In ^[13], the author discussed well-posedness of semilinear time-fractional diffusion equations using embedding relation among spaces. Eidelman and Kochubei ^[14] constructed fundamental solutions of time fractional evolution equations. In ^[15], the author established $L^r-L^q$ estimates and weighted estimates of fundamental solutions, and obtained existence and uniqueness of mild solutions of the Keller-Segel type time-space fractional diffusion equation. In ^[16], Wang and Zhou introduced and discussed four types special data dependences for a class of fractional evolution equations.

In this paper, we focus on the following nonlinear time-space fractional reaction-diffusion equations with fractional Laplacian

where $\Omega\subset\mathbb{R}^N(N\geqslant1)$ is a bounded open domain with smooth boundary $\partial\Omega$; $\alpha, \beta\in(0, 1)$ and $^c\mathbb{D}^{\alpha}_t\cdot$ is the Caputo time-fractional derivative of order $\alpha$ defined as

$\Gamma(\cdot)$ is the Gamma function; The spectral fractional Laplacian could be defined as

$f:\Omega\times[0, \infty)\times\mathbb{R}\rightarrow \mathbb{R}$ is the nonlinear function and the continuous initial data $u_0:\Omega\rightarrow \mathbb{R}$. We obtain the local uniqueness of mild solutions, the blowup alternative result for saturated mild solutions and Mittag-Leffler-Ulam-Hyers stability.

The main results of this paper are as following:

Theorem 1.1. Assume that nonlinear function $f:\Omega\times[0, \infty)\times\mathbb{R}\rightarrow \mathbb{R}$ is continuous and satisfies locally Lipschitz condition about the third variable, then there exists a constant $h > 0$ such that Eq (1.1) has a unique mild solution on $\Omega\times[0, h]$.

Theorem 1.2. Assume that all assumptions of Theorem 1.1 are satisfied, then the unique mild solution can be extended to a large time interval $[0, h^*]$ for some $h^* > h$ such that Eq (1.1) has a unique mild solution on $\Omega\times[0, h^*]$.

Theorem 1.3. Assume that all assumptions of Theorem 1.1 are satisfied, then there exists a maximal existence interval $[0, T_{\max})$ such that Eq (1.1) has a unique saturated mild solution $u\in C(\Omega\times[0, T_{\max}), \mathbb{R})$. Furthermore, if $T_{\max} < \infty$, then $\lim\sup_{t\rightarrow T^-_{\max}}\|u(t)\|_{\mathbb{H}^\beta(\Omega)} = \infty$, where $\mathbb{H}^\beta(\Omega)$ is Sobolev space introduced in the following section.

Theorem 1.4. Assume that all assumptions of Theorem 1.1 are satisfied, then there exists a constant $h > 0$ such that Eq (1.1) is Mittag-Leffler-Ulam-Hyers stable on $\Omega\times[0, h]$.

2.   Preliminaries

Throughout of this paper, we adopt spectral fractional Laplacian $(-\Delta)^\beta$ defined by (1.2). For each $\beta\in(0, 1)$, we define the fractional Sobolev space as

where $\lambda_j$ are the eigenvalues of $-\Delta$ with zero Dirichlet boundary conditions on $\Omega$, $\phi_j$ are eigenfunctions with respect to $\lambda_j$, $(\lambda_j, \phi_j)$ is the eigen pair of $-\Delta$, for the details one can see ^[17]. Denote $C([0, \infty), \mathbb{H}^\beta(\Omega))$ the Banach space of all continuous $\mathbb{H}^\beta(\Omega)$-value functions on $[0, \infty)$ with norm $\|u\|_C: = \sup_{t\in[0, \infty)}\|u(t)\|_{\mathbb{H}^\beta(\Omega)}$ and $A^\beta u = (-\Delta)^\beta u$. We know from ^[18] that $-A^\beta$ generates a Feller semigroup $T_\beta(t)(t\geqslant0)$.

We now define two operators $\mathcal{T}_{\alpha, \beta}(t)(t\geqslant0)$ and $\mathcal{S}_{\alpha, \beta}(t)(t\geqslant0)$ as follows

where $h_\alpha(s) = \frac{1}{\pi\alpha}\sum_{n = 1}^{\infty}(-s)^{n-1}\frac{\Gamma(n\alpha+1)}{n!}\sin(n\pi\alpha)$ is a function of Wright type ^[19] defined on $(0, \infty)$ which satisfies $h_\alpha(s)\geqslant0, \; s\in(0, \infty)$, $\int_0^{\infty}h_\alpha(s)ds = 1$.

Lemma 2.1. The operators $\mathcal{T}_{\alpha, \beta}(t)(t\geqslant0)$ and $\mathcal{S}_{\alpha, \beta}(t)(t\geqslant0)$ have the following properties ^[18]:

(i) The operators $\mathcal{T}_{\alpha, \beta}(t)(t\geqslant0)$ and $\mathcal{S}_{\alpha, \beta}(t)(t\geqslant0)$ are strongly continuous on $\mathbb{H}^\beta(\Omega)$;

(ii) $\|\mathcal{T}_{\alpha, \beta}(t)u\|_{\mathbb{H}^\beta(\Omega)}\leqslant\|u\|_{\mathbb{H}^\beta(\Omega)}$, $\|\mathcal{S}_{\alpha, \beta}(t)u\|_{\mathbb{H}^\beta(\Omega)}\leqslant\frac{1}{\Gamma(\alpha)}\|u\|_{\mathbb{H}^\beta(\Omega)}$;

(iii) $\mathcal{T}_{\alpha, \beta}(t)$ and $\mathcal{S}_{\alpha, \beta}(t)$ are compact operators for every $t > 0$.

Lemma 2.2. The Gamma function $\Gamma(z) = \int^\infty_0e^{-s}s^{z-1}\, ds$, $z > 0$ and Beta function $\mathrm{B}(p, q) = \int^1_0s^{p-1}(1-s)^{q-1}\; ds$, $p, q > 0$ have the following equality ^[20]:

Lemma 2.3. ${\bf{(Stirling's\; Formula)}}$ ^[21] For $x\rightarrow \infty$ we have

Lemma 2.4. Suppose that $a(t)$ is a nonnegative ^[16], nondecreasing function locally integrable on $[0, \infty)$ and $h(t)$ is a nonnegative, nondecreasing continuous function defined on $[0, \infty)$, $h(t)\leqslant\widetilde{M}$(constant), and suppose $u(t)$ is nonnegative and locally integrable on $[0, \infty)$ with

Then $u(t)\leqslant a(t)E_{\alpha}[h(t)\Gamma(\alpha)t^\alpha]$, where $E_{\alpha}$ is the Mittag-Leffer function defined by $E_{\alpha}[z] = \sum^\infty_{k = 0}\frac{z^k}{\Gamma(k\alpha+1)}$, $z\in\mathbb{C}$.

Let $u(t) = u(\cdot, t)$, $f(t, u(t)) = f(\cdot, t, u(\cdot, t))$, $u_0 = u_0(\cdot)$. Then the Eq (1.1) can be rewritten abstract form of fractional evolution equation in $C([0, \infty), \mathbb{H}^\beta(\Omega))$ as

If the nonlinear function $f:\Omega\times[0, \infty)\times\mathbb{R}\rightarrow \mathbb{R}$ satisfies locally Lipschitz condition about the third variable with Lipschitz constant $L$, one can derive

3.   Local existence and uniqueness, continuation and Blowup alternative results of solutions

Definition 3.1. A function $u\in C([0, \infty), {\mathbb{H}^\beta(\Omega)})$ is called a mild solution of (2.1) if it satisfies

Proof of Theorem 1.1. It follows discussions in Section 2 that Eq (1.1) can be transformed into the abstract evolution Eq (2.1) in $C([0, \infty), \mathbb{H}^\beta(\Omega))$. We now prove the local existence and uniqueness of the mild solution to the evolution Eq (2.1). Assume that nonlinear function $f$ is continuous in $\Theta = \{(t, u):0\leqslant t\leqslant a, \|u(t)-u_0\|_{\mathbb{H}^\beta(\Omega)}\leqslant b\}$ for $a > 0$ and $b > 0$, then there exists a unique mild solution to the evolution Eq (2.1) on $[0, h]$, where

Define ${\bf{P}}:C([0, h], \mathbb{H}^\beta(\Omega))\rightarrow C([0, h], \mathbb{H}^\beta(\Omega))$ as

From Definition 3.1, the mild solution to (2.1) on $[0, h]$ is equivalent to the fixed point of operator ${\bf{P}}$ defined by (3.1). Set $\Lambda = \{u\in C([0, h], \mathbb{H}^\beta(\Omega)):\|u(t)-u_0\|_{\mathbb{H}^\beta(\Omega)}\leqslant b, \; t\in[0, h]\}$ is a nonempty, convex and closed subset in $C([0, h], \mathbb{H}^\beta(\Omega))$. Now we show the operator ${\bf{P}}$ has a fixed point in $\Lambda$ by applying power compression mapping principle.

Step I. ${\bf{P}}:\Lambda\rightarrow \Lambda$. For any $u\in\Lambda$, $t\in[0, h]$, by (3.1) and Lemma 2.1 we have

Then, we get that ${\bf{P}}:\Lambda\rightarrow \Lambda$.

Step II. ${\bf{P}}:\Lambda\rightarrow \Lambda$ is a power compression mapping. For any $u, v\in\Lambda$, by (2.2), (3.1) and Lemma 2.1, we get

By (2.2), (3.1), (3.2), Lemma 2.1 and Lemma 2.2, we get

Suppose $n = k-1$ we have

Let $n = k$, by (2.2), (3.1), (3.3), Lemma 2.1 and Lemma 2.2, we get

Therefore, we have

for any $n\in\mathbb{N}^+$ and $t\in[0, h]$ by mathematical induction. By Lemma 2.3 we get

which implies

Hence, there exists $m\in\mathbb{N}$ such that

Combining (3.4) and (3.5) we have

which means that the operator ${\bf{P}}^m$ is compressive and ${\bf{P}}$ is a power compression operator. Therefore ${\bf{P}}$ has unique fixed point $u\in\Lambda$ by power compression mapping principle, the fixed point is the unique mild solution of (2.1) on $[0, h]$. Hence, Eq (1.1) has unique mild solution $u\in C(\Omega\times[0, h], \mathbb{R})$. This completes the proof of Theorem 1.1.

Definition 3.2. A function $u^*$ is a continuation mild solution of the unique mild solution $u\in C([0, h], \mathbb{H}^\beta(\Omega))$ to (2.1) on $(0, h^*]$ for some $h^* > h$ if it satisfies

Proof of Theorem 1.2. Let $u\in C([0, h], \mathbb{H}^\beta(\Omega))$ be the unique mild solution of (2.1), $h$ is the constant defined in Theorem 1.1. Fix $b^* = 2\|u_0\|_{\mathbb{H}^\beta(\Omega)}+2$, $M^* = \sup\{\|f(t, u^*(t))\|_{\mathbb{H}^\beta(\Omega)}:\|u(t)\|_{\mathbb{H}^\beta(\Omega)}\leqslant b^*, h\leqslant t\leqslant h+a^*\}$ for $a^* > 0$, we shall prove that $u^*:[0, h^*]\rightarrow \mathbb{H}^\beta(\Omega)$ is a mild solution of (2.1) for $h^* > h$. Set $\Lambda^* = \{u^*\in C([0, h^*], \mathbb{H}^\beta(\Omega)):\|u(t)-u(h)\|_{C([h, h^*], {\mathbb{H}^\beta(\Omega)})}\leqslant b^*, \; t\in[h, h^*]; u^*(t) = u(t), \; t\in[0, h]\}$, where

Define ${\bf{P}}:C([0, h^*], \mathbb{H}^\beta(\Omega))\rightarrow C([0, h^*], \mathbb{H}^\beta(\Omega))$ as (3.1). Now we show the operator ${\bf{P}}$ has a fixed point in $\Lambda^*$ via Banach fixed point theorem.

Step I. ${\bf{P}}:\Lambda^*\rightarrow \Lambda^*$. Let $u^*\in\Lambda^*$, if $t\in[0, h]$, from the proof of Theorem 1.1 we know equation (2.1) has unique mild solution and $u^*(t) = u(t)$. Thus ${\bf{P}}u^*(t) = {\bf{P}}u(t) = u(t)$ for all $t\in[0, h]$. Now we just consider $t\in[h, h^*]$, thus we have

Step II. ${\bf{P}}$ is a compression on $\Lambda^*$. Let $u^*, v^*\in\Lambda^*$, and we have that for $t\in[0, h^*]$,

Then,

This implies the operator ${\bf{P}}$ is compressive. By the Banach fixed point theorem it follows there exists a unique fixed point $u^*$ of ${\bf{P}}$ in $\Lambda^*$, which is a continuation of $u$. The fixed point is the unique mild solution of Eq (2.1) on $[0, h^*]$. Therefore, Eq (1.1) has unique mild solution $u$ on $\Omega\times[0, h^*]$. This completes the proof of Theorem 1.2.

Proof of Theorem 1.3. Repeating the methods and steps in the proof of Theorem 1.2, one can obtain that Eq (1.1) exists unique saturated mild solution on maximal interval $\Omega\times[0, T_{\max})$. Let $T_{\max}: = \sup\{h > 0:\text{the unique mild solution exits on}\; (0, h]\}$ and $u_0\in\mathbb{H}^\beta(\Omega)$. Assume that $T_{\max} < \infty$ and for some $b_0 > 0$, $M_0 = \sup\{\|f(t, u(t))\|_{\mathbb{H}^\beta(\Omega)}:\|u(t)\|_{\mathbb{H}^\beta(\Omega)}\leqslant b_0, \; 0\leqslant t\leqslant T_{\max}\}$. Suppose there exists a sequence $\{t_n\}_{n\in\mathbb{N}}\subset[0, T_{\max})$ such that $t_n\rightarrow T_{\max}$ and $\{u(t_n)\}_{n\in\mathbb{N}}\subset\mathbb{H}^\beta(\Omega)$. Let us demonstrate that $\{u(t_n)\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $\mathbb{H}^\beta(\Omega)$. Indeed, for any $\epsilon > 0$, fix $N\in\mathbb{N}$ such that for all $n, m > N$, $0 < t_n < t_m < T_{\max}$, we get

We choose $N: = N(\epsilon)\in\mathbb{N}^*$ with $m\geqslant n\geqslant N$ such that $t_m-t_n$ small enough following the sequence $\{t_n\}_{n\in\mathbb{N}^*}$ is convergent. By Lemma 2.1,

Clearly see $\|I_4\|_{\mathbb{H}^\beta(\Omega)} = 0$ for $t_n = 0$, $0 < t_m < T_{\max}$. For $t_n > 0$ and $0 < \epsilon < t_n$, by Lemma 2.1 we have

Therefore, for $\epsilon > 0$ there exists $N\in\mathbb{N}$ such that $\|u(t_m)-u(t_n)\|_{\mathbb{H}^\beta(\Omega)} < \epsilon$ when $m, n\geqslant N$. We arrive at that $\{u(t_n)\}_{t\in\mathbb{N}}\subset\mathbb{H}^\beta(\Omega)$ is a Cauchy sequences and for any $\{t_n\}_{n\in\mathbb{N}^*}$ the $\lim_{t\rightarrow T^-_{\max}}\|u(t)\|_{\mathbb{H}^\beta(\Omega)} < \infty$ exists. From result of Theorem 1.2 we know that the unique mild solution can be extended to larger interval. This means that $u$ can be continued beyond $T_{\max}$, and this contradict $u\in C([0, T_{\max}), \mathbb{H}^\beta(\Omega))$ is a saturated mild solution. Therefore, we arrive at if $T_{\max} < \infty$ then $\lim\sup_{t\rightarrow T^-_{\max}}\|u(t)\|_{\mathbb{H}^\beta(\Omega)} = \infty$. This complete the proof of Theorem 1.3.

4.   Mittag-Leffler-Ulam-Hyers stability

In this section, we consider the Mittag-Leffler-Ulam-Hyers stability of Eq (1.1). It follows discussions in Section 2 that Eq (1.1) can be transformed into the abstract evolution Eq (2.1) in $C([0, \infty), \mathbb{H}^\beta(\Omega))$, we now verify the stability of Eq (2.1) on $[0, h]$, $h$ is the constant defined in Theorem 1.1. Let $\varepsilon > 0$, we consider the following inequation

Definition 4.1. Eq (2.1) is Mittag-Leffler-Ulam-Hyers stable with respect to $E_\alpha$, if there exists a real number $\delta > 0$ such that for each $\varepsilon > 0$ and for each solution $v\in C^1([0, h], {\mathbb{H}^\beta(\Omega)})$ of inequation (4.1), there exists a mild solution $u\in C([0, h], {\mathbb{H}^\beta(\Omega)})$ of Eq (2.1) with $\|v(t)-u(t)\|_{\mathbb{H}^\beta(\Omega)}\leqslant\delta\varepsilon E_\alpha[t]$, $t\in[0, h]$.

Remark 4.1. A function $v\in C^1([0, h], {\mathbb{H}^\beta(\Omega)})$ is a solution of inequation (4.1) if and only if there exists a function $w\in C([0, h], {\mathbb{H}^\beta(\Omega)})$ (which depend on $v$) such that

(i) $\|w(t)\|_{\mathbb{H}^\beta(\Omega)}\leqslant\varepsilon$, for all $t\in[0, h]$;

(ii) $^c\mathbb{D}^{\alpha}_tu(t)+ A^{\beta}u(t) = f(t, u(t))+w(t)$, $t\in[0, h]$.

Remark 4.2. If $v\in C^1([0, h], {\mathbb{H}^\beta(\Omega)})$ is a solution of inequation (4.1), then $v$ is a solution of the following integral inequation

Proof of Theorem 1.4. Let $v\in C^1([0, h], \mathbb{H}^\beta(\Omega))$ be a solution of the inequation (4.1) and denote by $u\in C([0, h], {\mathbb{H}^\beta(\Omega)})$ the unique mild solution of the problem

We have

and by Remark 4.2 we get

It follows from (2.2) and (4.2) that

Applying Lemma 2.4 to inequality (4.3), we get

Hence, Eq (2.1) is Mittag-Leffler-Ulam-Hyers stable. This completes the proof of Theorem 1.4.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 12061063), the Outstanding Youth Science Fund of Gansu Province (No. 21JR7RA159) and Project of NWNU-LKQN2019-3. The authors would like to thank the referees for their valuable comments and suggestions which improve the quality of the manuscript.

Conflict of interest

The authors declare there is no conflicts of interest.