# Some remarks on the notion of transitions

F. M. Ciaglia 1,3 , F. Di Cosmo1,2,4
1Depto. de Matemáticas, Univ. Carlos III de Madrid, Leganés, Madrid, Spain
2 ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)
3 e-mail: fciaglia[at]math.uc3m.es
4 e-mail: fcosmo[at]math.uc3m.es

###### Abstract

In this paper some reflections on the concept of transition are presented:
groupoids are introduced as models for the construction of a “generalized
logic” whose basic statements involve pairs of propositions which can be
conditioned. In this sense, we could distinguish between classical probability
theory where propositions can be conditioned if they have a non-zero
intersection, from cases where “non-local” conditioning are allowed. The
algebraic and geometrical properties of groupoids can be exploited to
construct models of such non-local description.

##  1 Introduction

The origin of information theory dates back to the work by Shannon on the
mathematical theory of communication [1]. His aim was to introduce a
mathematical description of the process of communication: a message elaborated
by an information source that reaches the destination via a channel.It is
clear that in this description both the concepts of states and transition
between states are primitive objects and it is necessary to ask oneself what
are their properties and what mathematical structure would properly encode
them. From a more philosophical point of view, we could summarize the previous
discussion by saying that a mathematical description of information sources,
and more generally of communication processes, would require a way to
implement both the concept of “being” and “becoming”. It is, then, evident
that these aspects will play a crucial role also when passing to the quantum
description but with some differences, since there are plenty of experiments
showing how distant quantum and classical mechanics could be.

In this paper we are going to present some reflections on the concept of
transition and how groupoids could actually encode their properties both from
a classical and a quantum point of view. Even if some of these ideas have
already been discussed in our previous works on the Schwinger’s picture of
Quantum Mechanics [7, 11], in this paper we are going to revisit those
considerations from the point of view of a “generalised logic”. Indeed, in a
previous work [8] the authors showed that the functions on a groupoid can be
endowed with a von Neumann algebra [2] structure and the associated lattice of
projections is an orthocomplemented lattice. This lattice, therefore,
represents an example of a propositional calculus satisfying the axioms
introduced by Birkhoff and von Neumann [4]. But what can one say about
transitions? The lattice approach, in this sense, seems to look at the logical
structure of the experimental propositions providing information about the
state of the system. How can it provide information about transitions?

As a first attempt towards a statistical definition of transition, one could
say that a transition between two events could happen if the occurrence of one
could be conditioned by the other. Therefore, one could introduce a
“generalized logic” whose basic propositions are pairs of “experimental
propositions” that can be conditioned. In classical probability theory one
implements the conditioning in a “local” way: the two events involved in the
transition must have a non-empty intersection. Therefore, if we assume that
the lattice of experimental proposition is atomic, the basic atoms cannot be
conditioned, since they have empty intersections. However, one could go beyond
this “local” approach introducing the concept of measurable groupoid.

From an algebraic point of view, a groupoid is a set $\mathcal{G}$ of
morphisms with a partial composition law, on which two maps are defined which
associate to each morphism two objects, its source and target. These objects
belong to another set, say $\Omega$. If we introduce a Boolean lattice of
measurable subsets of the groupoid, we have a measurable groupoid. Using the
composition law one can define a product among sets of the groupoid, whereas
the source and the target maps send every set of morphisms to some sets of
objects. Both these structures can be used to define a notion of “non-local”
conditioning among subsets of the set of objects. In this paper we are going
to show that using a measurable groupoid and introducing a generalized notion
of conditioning between sets of objects, a measure on the groupoid determines
a grade-2 measure on the set of objects [3]. Grade-2 measures where introduced
by Sorkin in his formulation of Quantum Mechanics as a generalization of
probability theory which could allow for the description of interference
phenomena. Interestingly, using a groupoid the above mentioned grade-2 measure
can be also associated with a state on the algebra of the functions on the
groupoid and from this perspective the sets of zero 2-measure can be seen as
elements in the Gelfand ideal of this state.

##  2 The lattice of propositions

In this section we are going to shortly recall the basic notions introduced by
Birkhoff and von Neumann to describe the logic of Quantum Mechanics, which is
the set $\mathcal{P}$ of all “experimental proposition” that can be inferred
about a system. These propositions are basic ingredients for the formulation
of every physical model, and consequently they will share some common
properties. In order to convey more clearly the idea we will restrict to a
situation where only a finite set of propositions are considered. The basic
relation existing on this set is an order relation, say $\subset$, which is
the mathematical implementation of the implication relation between
propositions. Once we have this order we assume that for every pair of
elements $a,b\in\mathcal{P}$ there is a greatest lower bound, $a\wedge b$, and
a least upper bound, $a\vee b$, which are, respectively, the greatest (with
respect to the previous order) of all the propositions which are implied by
$a$ and $b$, and the least of all the propositions which imply both $a$ and
$b$. Additionally, we can always find two special propositions: the one which
is always true, denoted by $\mathbb{I}$, and the one which is always false,
say $\emptyset$. The final property of the set $\mathcal{P}$ is that for every
proposition $a$ there is a complementary proposition $a^{\perp}$ satisfying
the conditions $a\vee a^{\perp}=\mathbb{I}$ and $a\wedge a^{\perp}=\emptyset$.
A set $\mathcal{P}$ endowed with the previous structures
$\left(\subset,\,\wedge,\,\vee,\,\perp\,\right)$ is called an
orthocomplemented lattice, and this is the basic structure implementing the
operations among “experimental propositions”. The property that distinguishes
the classical setting, however, is the distributive character of the two
operations, i.e., $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$. Then, when we
have a finite distributive lattice $\mathcal{P}_{c}$ we can identify
$\mathbb{I}$ with the set $\Gamma$ of all its atoms, and $\mathcal{P}_{c}$
will be isomorphic to the power set $\mathcal{P}(\Gamma)$. On the other hand,
when we relax the distributive requirement, we can describe a lattice whose
propositions are vector subspaces of a certain Hilbert space, which is the
departure point in the von Neumann description of Quantum Mechanics. In
particular, if one assumes a weaker form of the distributive identity, called
modular identity, which in formulas reads

$a\subset c\;\Rightarrow\;a\vee(b\wedge c)=(a\vee b)\wedge c\,$ (1)

it is possible to prove that any complemented modular lattice of finite
dimensions is isomorphic to the direct product of a finite Boolean algebra and
a finite number of projective geometries. Moreover, if one additionally
assumes irreducibility of the lattice, which means that the only elements
$x\in\mathcal{P}$ satisfying

$a=(a\wedge x)\vee(a\wedge x^{\perp})\;\;\forall a$ (2)

are $x=\emptyset,\,\mathbb{I}$, then the modular lattice is a finite
projective geometry. In general by a projective geometry is meant any system
satisfying the following axioms [5]:

  * •

Two distinct points are contained in one and only one line

  * •

If A, B, C are three points not on the same line and E$\neq$D are points such
that B, C, D are on a line and A, C, E are on a line,then there is a point F
such that both A, B, F are on a line and D, E, F are on a line

  * •

Every line contains at least three points

  * •

The points on lines through any k-dimensional element and a fixed point not in
the element are a (k+1)-element, and every (k+1)-element is defined in this
way.

The set $\mathcal{C}(\mathcal{H})$ of Hilbert subspaces of the Hilbert space
$\mathcal{H}$, is the prototype of a projective geometry. In this case
k-dimensional elements correspond to (k+1)-dimensional closed subspaces of
$\mathcal{H}$. On the other hand, one can replace any closed subspace with the
associated orthogonal projector, obtaining a new lattice where the operation
of meet and join, however, do not coincide with the sum and product of the
algebra, generically.

Once the structure of the experimental propositions is given, it is possible
to associate to every event its probability: for Boolean lattices, monotone
additive functions will play the role. For the modular case, in general, one
can always define a dimension function
$d\,\colon\,\mathcal{P}\,\rightarrow\,\left[0,1\right]$ obeying the following
properties:

  * •

$d(a)>d(b)\quad iffb\subset a$

  * •

$d(a)+d(b)=d(a\wedge b)+d(a\vee b)\,$.

This dimension function corresponds to the normalized trace of each projection
and represents the dimension of each closed subspace of the Hilbert space.
More generally, however, one can consider real valued functions $m$ on the
lattice which are additive only with respect to the meet of orthogonal
projections. In this case Gleason theorem [6] ensures the existence of a
density matrix $\rho$ such that $m(a)=\mathrm{Tr}(\rho P_{a})$ with $P_{a}$
the projection corresponding to the closed subspace $a$ in the Hilbert space
$\mathcal{H}$. This point of view, therefore, allows to give a statistical
interpretation to each event, but only to events: we are interpreting only the
“static” information. What about transitions? The standard approach introduced
by von Neumann, define the probability transitions between two states of the
system in terms of the square modulus of the scalar product of the vectors
corresponding to the two states. In the rest of this short paper we will try
to follow a different approach to the notion of transition.

##  3 Groupoid and grade 2-measures

In this section we are going to introduce a generalized model of conditioning
via the concept of groupoids. In order to avoid inessential technical
difficulties and for the sake of communicability we are going to limit the
discussion to finite groupoids and consider the extension to the non-finite
case in forthcoming works. The departure point, therefore, is the definition
of groupoids. From the algebraic point of view, a groupoid
$G\rightrightarrows\Omega$, is a set $G$ of morphisms, together with a pair of
maps $s,t\,\colon\,G\,\rightarrow\,\Omega$ from the set of morphisms to the
set of objects $\Omega$, these maps being called source and target,
respectively.

Given the groupoid $G$, we will denote $G^{j}$ the set of morphisms whose
target is $j\in\Omega$ (analogously we denote $G_{j}$ the set of morphisms
whose source is $j$). The set of morphisms having the same source and target
object $j\in\Omega$ is a group, the isotropy group at $j$, and will be denoted
$G^{j}_{j}$. Two morphisms $\alpha$ and $\beta$ will be said to be composable
if $s(\alpha)=t(\beta)$ and their composition will be denoted
$\alpha\circ\beta$, this operation being associative. Units in the groupoid
$G$ will be denoted as $1_{j}\colon j\to j$ and they satisfy the conditions

$\alpha\circ 1_{i}=\alpha\,,\quad 1_{j}\circ\alpha=\alpha$ (3)

provided that $\alpha\colon i\to j$. Finally there is an inverse operation
$\tau\colon\alpha\mapsto\alpha^{-1}$ such that

$\alpha^{-1}\circ\alpha=1_{i}\,,\quad\alpha\circ\alpha^{-1}=1_{j}\,.$ (4)

Some basic examples of groupoids, which will be used in the rest of the paper,
are provided by the following ones. Firstly, the groupoid of pairs
$G(\Omega)=\Omega\times\Omega\rightrightarrows\Omega$ of an arbitrary set
$\Omega$: it has source and target maps $s(j,i)=i$, $t(j,i)=j$, respectively,
composition law $(k,j)\circ(j,i)=(k,i)$, units $1_{i}=(i,i)$ and inverse
$(j,i)^{-1}=(i,j)$. Secondly, standard sets are groupoids, i.e., if $\Omega$
is a set we consider the groupoid $\Omega\,\rightrightarrows\Omega$ with only
units (corresponding to each point of the set), so that a morphism is
composable only with itself.

Since a groupoid $G$ is a set of morphisms, we can consider the lattice
$\mathcal{P}(G)$ of all its subsets and, as we have already discussed in the
previous section, $\mathcal{P}(G)$ is a Boolean algebra. However, apart from
this family of subsets, one can use the algebraic structure of the groupoid to
generalize the “local” concept of conditioning expressed in terms of non-
vanishing intersections. Indeed, if one considers two subsets $A,B\subset G$
it is possible to construct a new subset $C=B\circ A$ as follows:

$C=\left\\{G\ni\gamma=\beta\circ\alpha\;\mid\;\beta\in B\,,\;\alpha\in
A\right\\}\,.$ (5)

In other words, $c$ is the set of all morphisms obtained by composing one
morphism from the set $A$ and one from the set $B$. One can prove that this
product is distributive with respect to the union of disjoint sets, but it is
not commutative. In particular, one can take $A=s^{-1}(a)$ and
$B=\tau^{-1}(s^{-1}(b))=t^{-1}(b)$ with $a,b$ being two subsets of the set
$\Omega$ of all objects of the groupoid. Then, we can say that two events
$a,b\in\mathcal{P}(\Omega)$ are conditioned iff $\tau^{-1}(s^{-1}(b))\circ
s^{-1}(a)\neq\emptyset$. It can be straightforwardly proven that conditioned
pairs form the graphs of a reflexive and symmetric relation, which in general
fails to be transitive. In the two main examples of groupoids above
introduced, this definition can be specified as follows:

  * •

for the groupoid $G(\Omega)\rightrightarrows\Omega$ of pairs of points of the
set $\Omega$, one can consider only the products
$(j,i)=\tau^{-1}(s^{-1}(j))\circ s^{-1}(i)$ where $i,j\in\Omega$ are atoms in
$\mathcal{P}(\Omega)$, since the other sets are obtained via the distributive
property of the product with respect to the union of disjoint events. It is
then easily shown that $(j,i)=\left\\{(j,i)\right\\}$ which is the set
containing the morphisms with source at $i$ and target at $j$.

  * •

for the groupoid $\Omega\rightrightarrows\Omega$ made up of units only, the
product $c=\tau^{-1}(s^{-1}(b))\circ s^{-1}(a)=b\wedge a$.

Once we have the lattice $\mathcal{P}(G)$ we can introduce a measure on $G$,
say $\mu$. How can we use it to give a statistical interpretation to subsets
of the groupoid? First of all, due to the algebraic structure of the groupoid,
we will consider measures which are compatible with the composition and the
inversion, in a way that generalizes the usual Haar measure for a group.
Therefore, we will endow the groupoid $G$ with a measure $\mu$ which admits
the following disintegration

$\mu(A)=\sum_{j\in\Omega}\lambda(j)\nu^{j}(A)\,,$ (6)

with respect to a measure $\lambda$ on the space of objects $\Omega$. The
family $\left\\{\nu^{j}\right\\}$ is a family of measures each one
concentrated on the set $G^{j}$. It is called a left Haar system of measures
if they satisfy the following properties:

  * •

for $\alpha\colon i\rightarrow j$, $((L_{\alpha})_{*}\nu^{(i)})=\nu^{(j)}$
where $(L_{\alpha})_{*}$ is the pushforward under the map
$L_{\alpha}\,\colon\,G^{i}\,\rightarrow G^{j}$ representing the left
multiplication by the morphism $\alpha$, i.e.,
$L_{\alpha}(\beta)=\alpha\circ\beta$. This property generalizes the left-
invariance of the Haar measure to a framework where the composition among
morphisms is only partially defined.

  * •

$\tau_{*}\mu=\delta^{-1}\mu$, where $\delta^{-1}$ is called the modular
function and it is a homomorphism of the groupoid $G$ to the group of positive
real numbers $\mathbb{R}_{+}$. This property means that by inversion we cannot
transform a set of measure zero into a set of non-zero measure or viceversa.
Once more, the modular function for groupoids generalizes the modular function
for groups.

The triple $(G,\mathcal{P}(G),\left[\mu\right])$ with $\left[\mu\right]$
denoting the class of all measures equivalent to the left system of Haar
measure $\mu$, will be called a measure groupoid. Let us remark that in this
class, there is always one measure $\Lambda$ which is invariant with respect
to the inversion, i.e., $\tau_{*}(\Lambda)=\Lambda$. Instead of considering
the whole lattice $\mathcal{P}(G)$, let us focus, now, only on pairs of
conditioned subsets $(b,a)$ in $\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)$.
Then, we can define the following two-set function
$D\,\colon\,\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)\,\rightarrow\,\mathbb{R}$:

$D(b,a)=\Lambda(\tau^{-1}(s^{-1}(b))\circ s^{-1}(a))\,.$ (7)

It can be easily seen, that this function satisfies the following properties:

  * •

positivity $D(a,a)\geq 0$

  * •

bi-additivity $D(a,b\cup c)=D(a,b)+D(a,c)$ whenever $b\cap c=\emptyset$

  * •

symmetry $D(b,a)=D(a,b)$.

Let us remark that everything would work in the same manner if the measure
$\Lambda$ would be weighted with a phase factor $e^{iS}$ with the function
$S\,\colon\,G\,\rightarrow\mathbb{R}$ satisfying the “logarithmic” properties
$S(\beta\circ\alpha)=S(\beta)+S(\alpha)$ and $S(\alpha^{-1})=-S(\alpha)$. The
only difference would consists in the replacement of the symmetry condition
with the hermiticity, since the function will be complex valued.

The properties of the function $D$ characterize what is called a decoherence
functional and consequently the associated quadratic form $\mu_{2}(a)=D(a,a)$
determine a grade 2-measure. Sorkin[3] introduced the notion of grade
2-measure in his generalization of probability theory in order to account for
the description of quantum phenomena. Differently from a traditional measure,
a grade 2-measure $\mu_{2}$ on a Boolean lattice $\mathcal{P}(\Omega)$
satisfies $\mu_{2}(a\vee b)-\mu_{2}(a)-\mu_{2}(b)=I(a,b)$ for every pair
$(a,b)$ of disjoint subsets of $\Omega$. The two-set function $I(a,b)$ is the
interference functional, and in principle does not vanish. However, a grade
2-measure satisfies the condition $0=I^{(3)}(a,b,c)=\mu_{2}(a\vee b\vee
c)-\mu_{2}(a\vee b)-\mu_{2}(a\vee c)-\mu_{2}(b\vee
c)+\mu_{2}(a)+\mu_{2}(b)+\mu_{2}(c)$. Summarizing the above discussion, the
algebraic structure of groupoids permit to introduce a generalized notion of
conditioning between pairs of sets of a Boolean algebra. Once a suitable
measure is introduced on the groupoid, a grade 2-measure is determined on the
space of objects $\Omega$.

The above picture can be analyzed, also, from an algebraic point of view by
considering the space of functions on the groupoid. Let $C(G)$ be the space of
complex valued continuous functions on the finite groupoid $G$. On a finite
groupoid this set is isomorphic to some vector space, say $\mathbb{C}^{N}$
where $N$ is the number of morphisms in $G$. Generalizing the construction for
group-algebras, we can introduce a algebra structure on $C(G)$ as follows:

$\begin{split}f\star h=\left(\sum_{\alpha\in
G}f_{\alpha}\delta_{\alpha}\right)\star\left(\sum_{\beta\in
G}h_{\beta}\delta_{\beta}\right)=\\\ =\sum_{\gamma\in
G}\sum_{(\alpha,\beta)\in
G^{(2)}\mid\alpha\circ\beta=\gamma}f_{\alpha}h_{\beta}\mu(\gamma)\delta_{\gamma}\,,\end{split}$
(8)

where $\delta_{\alpha}$ is the function which takes value 1 at $\alpha$ and 0
everywhere else, $\mu$ denotes the left invariant Haar system in the class
$\left[\mu\right]$, and $G^{(2)}$ denotes the set of composable pairs. This
product, called convolution product, is associative. Moreover we can define
the following involutive operator on $C(G)$:

$f^{\dagger}=\sum_{\alpha\in
G}\delta^{-1}(\alpha)\overline{f}_{\alpha}\delta_{\alpha^{-1}}\,,$ (9)

so that $C(G)$ is endowed with a $\ast$-algebra structure. Here $\delta$ is
the modular function associated with the measure groupoid structure. Moreover,
it is possible to introduce a suitable topology[10] on this space so that we
eventually get a von Neumann algebra, which we will denote $\nu(G)$. Once we
have obtained this algebraic structure it is possible to express the previous
decoherence functional in an alternative way. Indeed, it is easy to see that
the product $A\circ B=C$ between subsets of $G$ can be expressed in terms of
characteristic functions as follows:

$\chi_{C}=\chi_{A}\star\chi_{B}\,,$ (10)

where $\chi_{C}$ is the characteristic function supported on the set $C=A\circ
B$, and analogously one defines $\chi_{A}$ and $\chi_{B}$. It can be proven
[9] that the integration of a function $f\in\nu(G)$ with respect to the
measure $\mu$ determines a positive linear functional on the von Neumann
algebra which is normalized whenever the measure $\lambda$ on the set of
objects $\Omega$ is a probability measure. Let us denote $\omega_{\mu}$ this
state. Then, by direct computation one can show that

$D(b,a)=\omega_{\mu}((\chi_{s^{-1}(b)})^{\dagger}\star\chi_{s^{-1}(a)})\,.$
(11)

Once this algebraic point of view is introduced, we can recognize that the
sets whose corresponding characteristic functions are in the Gelfand ideal [2]
of the state $\omega_{\mu}$ correspond to sets whose grade 2-measure $\mu_{2}$
is zero. Therefore, we can interpret the GNS construction as the construction
of a suitable Hilbert space associated with the set of events of the groupoid.

##  4 Conclusions

In this paper we have presented some reflections on the notion of transitions,
associated with the concept of pairs of conditioned events. After a brief
review of the approach to quantum logic introduced by Birkhoff and von
Neumann, we have discussed a different model where groupoids are the departure
point. Indeed, using the rich algebraic structure defining a groupoid, we can
introduce a way of conditioning sets which generalize the “local” approach of
classical probability in a straightforward way. Of course, groupoid are not
the only way to obtain this non-local conditioning. However, we have shown
that using a groupoid this notion can be related to the $C^{*}$-algebraic
approach to Quantum Mechanics and to the corresponding notion of state.

### Acknowledgements

FMC acknowledges that this work has been supported by the Madrid Government
(Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the
line of “Research Funds for Beatriz Galindo Fellowships” (C&QIG-BG-CM-UC3M),
and in the context of the V PRICIT (Regional Programme of Research and
Technological Innovation). FDC thanks the UC3M, the European Commission
through the Marie Sklodowska-Curie COFUND Action (H2020-MSCA-COFUND-2017- GA
801538) for their financial support through the CONEX-Plus Programme.

## References

  * [1] C. E. Shannon  A mathematical theory of communication  The Bell System Technical Journal 27(3), 379–423 (1948).
  * [2] Takesaki, M.  Theory of Operator Algebras I; Springer-Verlag, Berlin, 2002.
  * [3] Sorkin, R.D.  Quantum mechanics as quantum measure theory.  Modern Physics Letters A 9(33), 3119–3127 (1994).
  * [4] Birkhoff, G.; J., V.N.  The Logic of Quantum Mechanics.  Annals of Mathematics 37, 823–843 (1936).
  * [5] Birkhoff, G.  Combinatorial Relations in Projective Geometries.  Annals of Mathematics 36, 743–748 (1935).
  * [6] Gleason, A.M.  Measures on the closed subspaces of a Hilbert space.  Journal of Mathematics and Mechanics 6, 885–893 (1957).
  * [7] Ciaglia, F.M.; Di Cosmo, F.; Ibort, A.; Marmo, G.  Schwinger’s Picture of Quantum Mechanics.  International Journal of Geometric Methods in Modern Physics 17(14), 2050054 (2020).
  * [8] Ciaglia, F.M.; Di Cosmo, F.; Ibort, A.; Marmo, G.  Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics.  Entropy 22(11), 2050054 (2020).
  * [9] Ciaglia, F.M.; Ibort, A.; Marmo, G.  Schwinger’s Picture of Quantum Mechanics III: The Statistical Interpretation.  International Journal of Geometric Methods in Modern Physics 16(11), 1950165 (2019).
  * [10] Ibort, A.; Rodriguez, M.A.  An introduction to the theory of groups, groupoids and their representations; CRC, Boca Raton, 2019.
  * [11] Ciaglia, F.M.; Ibort, A.; Marmo, G.  A gentle introduction to Schwinger’s formulation of quantum mechanics: the groupoid picture.  Modern Physics Letters A, 33(8), 1850122 (2018).


∎

11institutetext: Dian Feng1,∗ 22institutetext: 1\. School of Mathematical
Sciences, Fudan University, Shanghai 200433, China
22email<EMAIL_ADDRESS>∗ corresponding author 33institutetext: Masahiro
Yamamoto2,3 44institutetext: 2\. Graduate School of Mathematical Sciences, The
University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
55institutetext: 3\. Department of Mathematics, Faculty of Science, Zonguldak
Bülent Ecevit University, Zonguldak 67100, Türkiye
55email<EMAIL_ADDRESS>
# Global and local existence of solutions for nonlinear systems of time-
fractional diffusion equations

Dian Feng1 Masahiro Yamamoto2,3

(Received: XX 2024 / Revised: … / Accepted: ……)

###### Abstract

In this paper, we consider initial-boundary value problems for two-component
nonlinear systems of time-fractional diffusion equations with the homogeneous
Neumann boundary condition and non-negative initial values. The main results
are the existence of solutions global in time and the blow-up. Our approach
involves the truncation of the nonlinear terms, which enables us to handle all
local Lipschitz continuous nonlinear terms, provided their sum is less than or
equal to zero. By employing a comparison principle for the corresponding
linear system, we establish also the non-negativity of the nonlinear system.

###### Keywords:

Nonlinear time-fractional system weak solutionglobal existenceblow-up

###### MSC:

26A33 (primary) 33E12 34A08 34K37 35R11 60G22 …

††journal: Fract. Calc. Appl. Anal.

##  1 Introduction

Let $\partial_{t}^{\alpha}$ be the fractional derivative of order
$\alpha\in(0,1)$ defined on the fractional Sobolev spaces, which is defined as
the closure of the classical Caputo derivative

$d_{t}^{\alpha}v(t):=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}\frac{dv}{ds}(s)ds,\quad\mbox{
for }v\in C^{1}[0,T]\mbox{ satisfying }v(0)=0$

(see Section 2 for the details). Let $\Omega\subset\mathbb{R}^{d},d=1,2,3$ be
a bounded domain with smooth boundary $\partial\Omega$. Moreover by $\nu$ we
denote the unit outward normal vector to $\partial\Omega$ at
$x\in\partial\Omega$.

We consider an initial-boundary value problem for the following nonlinear
system of time-fractional equations with the homogeneous Neumann boundary
condition

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u-a)=\Delta u+f(u,v),\quad
x\in\Omega,0<t<T,\\\ \partial_{t}^{\alpha}(v-b)=\Delta v+g(u,v),\quad
x\in\Omega,0<t<T,\\\ \partial_{\nu}u=\partial_{\nu}v=0,\quad\text{ on
}\partial\Omega\times(0,T),\end{array}\right.$ (1.1)

where we assume some conditions on the nonlinear terms $f$ and $g$, as
Assumption 1 in Section 2 describes.

When $\alpha=1$, the system is reduced to a classical reaction-diffusion
system, and can be regarded as a special case of the Klausmeier-Gray-Scott
model Kla ; WS . This system is considered to be a model equation for the
vegetation pattern formation, which describes the self-organization of
vegetation spatial patterns resulting from the interaction between water
source distribution and plant growth. In Pi  and Pi2 , it was proved that a
weak solution in $L^{1}$ exists in the case where the coefficients in front of
the spatial diffusion terms are not necessarily equal. However for the our
fractional derivative case $0<\alpha<1$, we can not apply the same technique
as for $\alpha=1$, so that we have to assume that the coefficients of spatial
diffusion terms are equal, that is, $1$. It should be a future work to discuss
more general elliptic operators in (1.1).

For $0<\alpha<1$, in contrast to the traditional reaction-diffusion process,
we introduce a fractional order in the time derivative, indicating that the
interaction between vegetation growth and water is influenced by soil medium
heterogeneity. The fractional derivatives enable the representation of time
memory effects, characterized by their nonlocal properties. Although there has
been some research on various nonlinear fractional-order equation or coupled
linear fractional-order equation systems, as seen in BP ; LHL ; FLY , to the
best of the authors’ knowledge, there has been no investigation concerning
nonlinear fractional-order systems.

Our approach involves the truncation of the nonlinear terms, which is inspired
by Pierre Pi ; Pi2 . Through a comparison principle and a compact mapping
property, we establish a convergent sequence of non-negative functions for the
linear case. Moreover, $u+v$ has the energy estimation in the sense of
$L^{1}$-norm, implying the convergence of the sequence in the $L^{1}$ sense.
Ultimately, we obtain weak solutions in the $L^{1}$ norm.

The rest of this paper is organized as follows. In Section 2, we present the
result regarding the global existence of weak solutions to the initial-
boundary value problems for the nonlinear system of time-fractional diffusion
equations. Section 3 is devoted to proving two key lemmas, which serve as the
basis for the proofs of the main results. These lemmas establish the non-
negativity of solution to the linear case, provided that the initial value and
source term are non-negative, and demonstrate that the mapping from initial
value and non-homogeneous term to solution of the corresponding linear system,
is compact with suitable norms. Section 4 completes the proof of the first
main result. Section 5 discusses the blow-up of solutions under other
conditions on the nonlinear terms, which rejects the global existence in time,
in general. Section 6 provides some conclusions and remarks.

##  2 Well-posedness results

In this section, we deal with the following initial-boundary value problem for
the nonlinear system of time-fractional equations (1.1) with the time-
fractional derivative of order $\alpha\in(0,1)$

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u-a)=\Delta u+f(u,v),\quad
x\in\Omega,\ 0<t<T,\\\ \partial_{t}^{\alpha}(v-b)=\Delta v+g(u,v),\quad
x\in\Omega,\ 0<t<T,\\\ \partial_{\nu}u=\partial_{\nu}v=0,\quad\text{ on
}\partial\Omega\times(0,T),\end{array}\right.$ (2.1)

along with the initial condition (2.2) formulated below.

For $\alpha\in(0,1)$, let $\mathrm{d}^{\alpha}_{t}$ denote the classical
Caputo derivative:

$\mathrm{d}^{\alpha}_{t}w(t):=\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}\frac{\mathrm{d}w}{\mathrm{d}s}(s)\mathrm{d}s,\
w\in W^{1,1}(0,T).$

Here $\Gamma(\cdot)$ denotes the gamma function. For a consistent treatment of
nonlinear time-fractional diffusion equations, we extend the classical Caputo
derivative $\mathrm{d}^{\alpha}_{t}$ as follows. First of all, we define the
Sobolev-Slobodeckij space $H^{\alpha}(0,T)$ with the norm
$\|\cdot\|_{H^{\alpha}(0,T)}$ for $0<\alpha<1$:

$\|w\|_{H^{\alpha}(0,T)}:=\left(\|w\|_{L^{2}(0,T)}+\int_{0}^{T}\int_{0}^{T}\frac{|w(t)-w(s)|^{2}}{|t-s|^{1+2\alpha}}\mathrm{d}t\mathrm{d}s\right)^{\frac{1}{2}}$

(e.g., AdamsAd ). Furthermore, we set $H^{0}(0,T):=L^{2}(0,T)$ and

$H_{\alpha}(0,T):=\left\\{\begin{array}[]{ll}H^{\alpha}(0,T),&0<\alpha<\frac{1}{2},\\\
\left\\{w\in
H^{\frac{1}{2}}(0,T);\int_{0}^{T}\frac{|w(t)|^{2}}{t}\mathrm{~{}d}t<\infty\right\\},&\alpha=\frac{1}{2},\\\
\left\\{w\in H^{\alpha}(0,T);w(0)=0\right\\},&\frac{1}{2}<\alpha\leq
1\end{array}\right.$

with the norms defined by

$\|w\|_{H_{\alpha}(0,T)}:=\left\\{\begin{array}[]{ll}\|w\|_{H^{\alpha}(0,T)},&\alpha\neq\frac{1}{2},\\\
\left(\|w\|_{H^{\frac{1}{2}(0,T)}}^{2}+\int_{0}^{T}\frac{|w(t)|^{2}}{t}\mathrm{~{}d}t\right)^{\frac{1}{2}},&\alpha=\frac{1}{2}.\end{array}\right.$

Moreover, for $\beta>0$, we set

$J^{\beta}w(t):=\int^{t}_{0}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}w(s)\mathrm{d}s,\
0<t<T,w\in L^{1}(0,T).$

It was proved, for instance, in Gorenflo, Luchko, and YamamotoGLY , that the
operator $J^{\alpha}:L^{2}(0,T)\to H_{\alpha}(0,T)$ is an isomorphism for
$\alpha\in(0,1)$.

Now we are ready to give the definition of the extended Caputo derivative

$\partial_{t}^{\alpha}:=\left(J^{\alpha}\right)^{-1},\
\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T).$

Henceforth $\mathcal{D}(\cdot)$ represents the domain of an operator under
consideration. It has been demonstrated that $\partial_{t}^{\alpha}$
represents the minimal closed extension of $\mathrm{d}^{\alpha}_{t}$, where
$\mathcal{D}(\mathrm{d}^{\alpha}_{t}):=\\{v\in C^{1}[0,T];v(0)=0\\}$. The
relation $\partial_{t}^{\alpha}w=\mathrm{d}^{\alpha}_{t}w$ is maintained for
$w\in C^{1}[0,T]$ with $w(0)=0$. For further details, we can refer to Gorenflo
et al. GLY  and YamamotoY4 .

Now we will define initial condition for problem (2.1) as follows:

$u(x,\cdot)-a(x)\in H_{\alpha}(0,T),v(x,\cdot)-b(x),\in H_{\alpha}(0,T)\
\text{ for almost all }x\in\Omega$ (2.2)

and write down a complete formulation of an initial-boundary value problem for
the nonlinear system of time-fractional equations (1.1):

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u-a)=\Delta u+f(u,v),\quad
x\in\Omega,0<t<T,\\\ \partial_{t}^{\alpha}(v-b)=\Delta v+g(u,v),\quad
x\in\Omega,0<t<T,\\\ \partial_{\nu}u=\partial_{\nu}v=0,\quad\text{ on
}\partial\Omega\times(0,T),\\\ u(x,\cdot)-a(x),v(x,\cdot)-b(x)\in
H_{\alpha}(0,T),\quad\text{ for almost all }x\in\Omega.\end{array}\right.$
(2.3)

It is worth mentioning that the terms $\partial_{t}^{\alpha}(u-a)$ and
$\partial_{t}^{\alpha}(v-b)$ in the first two lines of (2.3) are well-defined
for almost all $x\in\Omega$ and $0<t<T$, due to the inclusion formulated in
the last line of (2.3). Especially for $\frac{1}{2}<\alpha<1,$ noting that
$w\in H_{\alpha}(0,T)$ implies $w(0)=0$ by the trace theorem, we can
understand that the left-hand side means that $u(x,0)=a(x)$ and $v(x,0)=b(x)$
in the trace sense with respect to $t$. While for $\alpha<\frac{1}{2}$, we do
not need any initial conditions.

Now we are well prepared to investigate the initial-boundary value problem
(2.3). We first provide the definition of weak solutions for equations (2.3).
Henceforth we set

$Q:=\Omega\times(0,T)$

and

$(\partial_{t}^{\alpha})^{*}\psi(x,s):=\frac{-1}{\Gamma(1-\alpha)}\int^{T}_{s}(t-s)^{-\alpha}\frac{\partial\psi}{\partial
t}(x,t)\mathrm{d}t$

for $\psi\in C^{1}([0,T];L^{1}(\Omega))$. Then:

###### Definition 1

We call $(u,v)\in L^{1}(0,T;L^{1}(\Omega))^{2}$ a weak solution to (2.3) if

$\left\\{\begin{array}[]{l}\int_{Q}(u-a)(\partial_{t}^{\alpha})^{*}\psi\,\mathrm{d}x\mathrm{d}t=\int_{Q}u\Delta\psi\,\mathrm{d}x\mathrm{d}t+\int_{Q}f(u,v)\psi\,\mathrm{d}x\mathrm{d}t,\cr\\\
\int_{Q}(v-b)(\partial_{t}^{\alpha})^{*}\psi\,\mathrm{d}x\mathrm{d}t=\int_{Q}v\Delta\psi\,\mathrm{d}x\mathrm{d}t+\int_{Q}g(u,v)\psi\,\mathrm{d}x\mathrm{d}t,\end{array}\right.$
(2.4)

for all $\psi(x,t)\in C^{\infty}(\overline{Q})$ satisfying
$\partial_{\nu}\psi|_{\partial\Omega\times(0,T)}=0$ and $\psi(\cdot,T)=0$ in
$\Omega$.

If a weak solution $(u,v)\in L^{1}(0,T;L^{1}(\Omega))^{2}$ is sufficiently
smooth, then we can verify by the definition that $(u,v)$ satisfies (1.1)
pointwise in the classical sense.

Based on the definition, we can give a lemma immediately.

###### Lemma 1

If (2.3) has a solution $(u,v)\in L^{2}(0,T;H^{2}(\Omega))^{2}$ such that
$u-a,v-b\in H_{\alpha}(0,T;L^{2}(\Omega))$, then $(u,v)$ is a weak solution.

###### Proof

Since $(u,v)$ is a solution of (2.3), we have

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u-a)=\Delta u+f(u,v),\\\
\partial_{t}^{\alpha}(v-b)=\Delta v+g(u,v).\end{array}\right.$ (2.5)

Multiply both sides of (2.5) by $\psi(x,t)\in C^{\infty}(\overline{Q})$,
taking into account the boundary condition of $\psi(x,t)$ in Definition 1, and
integrate by parts. This leads us to the same formula as (2.4), thus
completing the proof.

###### Assumption 1

Let $f,g:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ be local Lipschitz
continuous. More precisely, for arbitrarily given $M>0$, there exists a
constant $C=C_{M}>0$ such that

$|f(\xi,\eta)-f(\widehat{\xi},\widehat{\eta})|+|g(\xi,\eta)-g(\widehat{\xi},\widehat{\eta})|\leq
C(|\xi-\widehat{\xi}|+|\eta-\widehat{\eta}|).$ (2.6)

for all $\xi,\eta,\widehat{\xi},\widehat{\eta}\in[-M,M]$. Moreover we assume

$f(0,\eta)=g(\xi,0)=0,\quad\mbox{ for all }\xi\geq 0\mbox{ and }\eta\geq 0.$

Finally we assume that we can find a constant $\lambda>0$ such that

$f(\xi,\eta)+\lambda g(\xi,\eta)\leq 0\quad\mbox{ for all }\xi,\eta\geq 0.$

Here, compared to the conditions required in Pi , we impose stronger
requirements on $f$ and $g$, which are crucial for the convergence of the
truncated non-linear terms in Sections 4. Unlike Pi , we no longer restrict
$\lambda$ to be less than or equal to 1.

Now we state the first main result in this article, which validates the well-
posedness for given $T>0$ of the initial-boundary value problem (2.3).

###### Theorem 2.1

In (2.3), we assume $a,b\geq 0$, $a,b\in H^{1}(\Omega)\cap
L^{\infty}(\Omega)$, with $f,g$ satisfy Assumption 1. Then, for arbitrarily
given $T>0$, there exists at least one weak solution $(u,v)\in
L^{1}(\Omega\times(0,T))^{2}$ such that $u,v\geq 0$ in $Q$.

We do not know the uniqueness of weak solution under our assumption, which is
the same as for the case $\alpha=1$.

Before starting with the proof of Theorem 2.1, we introduce some notations and
derive several results necessary for the proof.

For an arbitrary constant $p_{0}>0$, define an elliptic operator $A$ as
follows:

$\left\\{\begin{array}[]{l}\left(-Av\right)(x):=\Delta v(x)-p_{0}v(x),\quad
x\in\Omega,\\\ \mathcal{D}(A)=\left\\{v\in
H^{2}(\Omega);\partial_{\nu}v=0\text{ on
}\partial\Omega\right\\}.\end{array}\right.$ (2.7)

Henceforth, by $\|\cdot\|$ and $(\cdot,\cdot)$ we denote the standard norm and
the scalar product in $L^{2}(\Omega)$, respectively. It is well-konwn that the
operator $A$ is self-adjoint in $L^{2}(\Omega)$. Moreover, for a sufficiently
large constant $p_{0}>0$, we can verify that $A$ is positive definiteLY1 .
Therefore, by choosing a constant $p_{0}$ large enough, the spectrum of the
operator $A$ is comprised of discrete positive eigenvalues, herein represented
as $0<\lambda_{1}\leq\lambda_{2}\leq\cdots$, each uniquely designated by its
multiplicity. Additionally, $\lambda_{n}\to\infty$ as $n\to\infty$. Set
$\varphi_{n}$ be the eigenfunction corresponding to the eigenvalue
$\lambda_{n}$ such that $A\varphi_{n}=\lambda_{n}\varphi_{n}$, and
$(\varphi_{i},\varphi_{j})=\delta_{ij}$. Then the sequence
$\\{\varphi_{n}\\}_{n\in\mathbb{N}}$ is orthonormal basis in $L^{2}(\Omega)$.
For any $\gamma>0$, we can define the fractional power $A^{\gamma}$ of the
operator $A$ by the following relation (see, e.g.,Pa ):

$A^{\gamma}v=\sum_{n=1}^{\infty}\lambda^{\gamma}_{n}(v,\varphi_{n})\varphi_{n},$

where

$v\in\mathcal{D}(A^{\gamma}):=\left\\{v\in
L^{2}(\Omega):\sum_{n=1}^{\infty}\lambda_{n}^{2\gamma}(v,\varphi_{n})^{2}<\infty\right\\}$

and

$\|A^{\gamma}v\|=\left(\sum_{n=1}^{\infty}\lambda_{n}^{2\gamma}(v,\varphi_{n})^{2}\right)^{\frac{1}{2}}.$

We have $\mathcal{D}(A^{\gamma})\subset H^{2\gamma}(\Omega)$ for $\gamma>0$.
Since $\mathcal{D}(A^{\gamma})\subset L^{2}(\Omega)$, identifying the dual
$(L^{2}(\Omega))^{\prime}$ with itself, we have
$\mathcal{D}(A^{\gamma})\subset
L^{2}(\Omega)\subset\mathcal{D}(A^{\gamma})^{\prime}$. Henceforth we set
$\mathcal{D}(A^{-\gamma})=\mathcal{D}(A^{\gamma})^{\prime}$, which consists of
bounded linear functionals on $\mathcal{D}(A^{\gamma})$. For
$w\in\mathcal{D}(A^{-\gamma})$ and $v\in\mathcal{D}(A^{\gamma})$, by
${}_{-\gamma}\langle w,v\rangle_{\gamma}$, we denote the value which is
obtained by operating $w$ to $v$. We note that $\mathcal{D}(A^{-\gamma})$ is a
Hilbert space with norm:

$\|w\|_{\mathcal{D}(A^{-\gamma})}=\left\\{\sum_{n=1}^{\infty}\lambda^{-2\gamma}_{n}\left|{}_{-\gamma}\langle
w,v\rangle_{\gamma}\right|^{2}\right\\}^{\frac{1}{2}}.$

We further note that

${}_{-\gamma}\langle w,v\rangle_{\gamma}=(w,v)\text{ if }w\in
L^{2}(\Omega)\text{ and }v\in\mathcal{D}(A^{\gamma}).$

Moreover we define two other operators $S(t)$ and $K(t)$ as:

$S(t)a=\sum_{n=1}^{\infty}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n},\
a\in L^{2}(\Omega),t>0$

and

$K(t)a=\sum_{n=1}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n},\
a\in L^{2}(\Omega),t>0,$

where $E_{\alpha,\beta}(z)$ denotes the Mittag-Leffler function defined by a
convergent series as follows:

$E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)},\
\alpha>0,\beta\in\mathbb{C},z\in\mathbb{C}.$

It follows directly from the definitions given above that
$A^{\gamma}K(t)a=K(t)A^{\gamma}a$ and $A^{\gamma}S(t)a=S(t)A^{\gamma}a$ for
$a\in\mathcal{D}(A^{\gamma})$. Moreover, the following norm estimates were
proved inSY :

$\displaystyle\|A^{\gamma}S(t)a\|$ $\displaystyle\leq
Ct^{-\alpha\gamma}\|a\|,$ (2.8) $\displaystyle\|A^{\gamma}K(t)a\|$
$\displaystyle\leq Ct^{\alpha(1-\gamma)-1}\|a\|,\ a\in
L^{2}(\Omega),t>0,0\leq\gamma\leq 1.$

##  3 Key Lemma

For the proof of Theorem 2.1, we show several lemmas.

###### Lemma 2

(Non-negativity) We assume that $c\in L^{\infty}(Q)$, $a\in H^{1}(\Omega)$,
$a\geq 0$ in $\Omega$ and $F\in L^{2}(Q)$, $F\geq 0$ in $Q$. Let $u\in
L^{2}(0,T;H^{2}(\Omega))$ satisfy (3.1) and $u-a\in
H_{\alpha}(0,T;L^{2}(\Omega))$:

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u-a)=\Delta
u+c(x,t)u+F(x,t),\quad x\in\Omega,0<t<T,\\\ \partial_{\nu}u=0,\quad\text{ on
}\partial\Omega,0<t<T,\\\ u(x,\cdot)-a(x)\in H_{\alpha}(0,T),\quad\text{ for
almost all }x\in\Omega.\end{array}\right.$ (3.1)

Then $u\geq 0$ in $Q$.

The proof is simialr to Luchko and Yamamoto LY ; LY1  where the coefficient
$c$ is assumed to be more smooth than $L^{\infty}(Q)$. For the case $c\in
L^{\infty}(Q)$, approximating $c$ by functions in $C^{\infty}_{0}(Q)$ in the
space $L^{\kappa}(Q)$ with arbitrarily fixed $\kappa>1$. For completeness, we
provide the proof in Appendix.

Here and henceforth we set $A=-\Delta+p_{0}$. For $w_{0}\in H^{1}(\Omega)$ and
$F\in L^{2}(Q)$, we consider

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}\left(w-w_{0}\right)+Aw=F(x,t),\quad
x\in\Omega,0<t<T,\\\ \partial_{\nu}w=0,\quad x\in\partial\Omega,0<t<T,\\\
w(x,\cdot)-w_{0}(x)\in H_{\alpha}(0,T),\quad\text{ for almost all
}x\in\Omega.\end{array}\right.$ (3.2)

Then we know (e.g., KRY ) that there exists a unique solution $w\in
L^{2}(0,T;H^{2}(\Omega))$ to (3.2) satisfying $w-w_{0}\in
H_{\alpha}(0,T;L^{2}(\Omega))$. By $w(w_{0},F)$ we denote the solution.

We can prove a compactness property:

###### Lemma 3

Let $M_{1}>0$ be arbitrarily chosen constant. Then a set

$\left\\{w(w_{0}^{n},F_{n})|(w_{0}^{n},F_{n})\in H^{1}(\Omega)\times
L^{2}(Q),\ \|w_{0}^{n}\|_{L^{1}(\Omega)}+\|F_{n}\|_{L^{1}(Q)}\leq
M_{1}\right\\}$

contains a subsequence which converges in $L^{1}(Q)$.

###### Proof

We introduce an operator $\tau_{h}(f)$ for $h>0$ and $f\in L^{1}(0,T;X)$ with
Banach space $X$ by $\tau_{h}(f)(t)=f(t+h)$.

We show two lemmas for the proof of Lemma 3.

###### Lemma 4

(SimonSi ) Let $X\subset B\subset Y$ be Banach spaces and the embedding $X\to
Y$ be compact. We assume

$\|v\|_{B}\leq C\|v\|^{1-\theta}_{X}\|v\|^{\theta}_{Y},\quad\forall v\in X\cap
Y,\quad\text{ where }0\leq\theta\leq 1,$ $\mbox{ a set }U\mbox{ is bounded in
}L^{1}(0,T;X)$

and

$\|\tau_{h}u-u\|_{L^{1}(0,T-h;Y)}\to 0\mbox{ as }h\to 0\mbox{ uniformly for
}u\in U.$

Then $U$ is relatively compact in $L^{1}(0,T;B)$.

###### Lemma 5

Let $\Omega\in\mathbb{R}^{n}$, if $\gamma>\frac{d}{4}$, then we have

$\|A^{-\gamma}w\|_{L^{2}(\Omega)}\leq C\|w\|_{L^{1}(\Omega)}.$

Proof of Lemma 5

Since $\mathcal{D}(A^{\gamma})\subset H^{2\gamma}(\Omega)\subset
L^{\infty}(\Omega)$, we can get $\|v\|_{L^{\infty}(\Omega)}\leq
C\|A^{\gamma}v\|_{L^{2}(\Omega)}$. Moreover, we have

$\displaystyle\|A^{-\gamma}w\|_{L^{2}(\Omega)}$
$\displaystyle=\sup_{\|\varphi\|_{L^{2}(\Omega)}=1}|(A^{-\gamma}w,\varphi)_{L^{2}(\Omega)}|$
$\displaystyle=\sup_{\|\varphi\|_{L^{2}(\Omega)}=1}|(w,A^{-\gamma}\varphi)_{L^{2}(\Omega)}|$
$\displaystyle\leq
C\|w\|_{L^{1}(\Omega)}\|A^{-\gamma}\varphi\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq
C\|w\|_{L^{1}(\Omega)}\|A^{\gamma}A^{-\gamma}\varphi\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\|w\|_{L^{1}(\Omega)}.$

Then we finished the proof. $\blacksquare$

Now we proceed to the proof of Lemma 3. In Lemma 4, set
$X=\mathcal{D}(A^{1-\delta}),Y=\mathcal{D}(A^{-\delta})$ and
$B=L^{1}(\Omega)$. The intermediate spaces BL  between $X$ and $Y$ are given
by

$[X,Y]_{\theta}=\mathcal{D}(A^{1-\delta-\theta}),\quad\forall\theta\in[0,1].$

If we choose $\theta=1-\delta$, we obtain
$[X,Y]_{1-\delta}=\mathcal{D}(A^{0})=L^{2}(\Omega)$. Referring to the
interpolation result of Sobolev spaces, this implies

$\|w\|_{L^{2}(\Omega)}\leq
C\|w\|_{\mathcal{D}(A^{1-\delta})}^{\delta}\|w\|_{\mathcal{D}(A^{-\delta})}^{1-\delta}.$

Applying the embedding theory of $L^{p}(\Omega)$ for $p\in[1,\infty]$, we have
$\|w\|_{L^{1}(\Omega)}\leq C\|w\|_{L^{2}(\Omega)}$, which means

$\|w\|_{L^{1}(\Omega)}\leq
C\|w\|_{\mathcal{D}(A^{1-\delta})}^{\delta}\|w\|_{\mathcal{D}(A^{-\delta})}^{1-\delta}.$

In order to apply Lemma 4, we still need to prove the following inclusion
relation:

$\mathcal{D}(A^{1-\delta})\subset
L^{1}(\Omega)\subset\mathcal{D}(A^{-\delta}).$ (3.3)

To establish (3.3), we start by considering the following relationship:

$\mathcal{D}(A^{\delta})\subset H^{2\delta}(\Omega)\subset
H^{2\delta^{\prime}}(\Omega)\subset L^{\infty}(\Omega),$

where $\frac{d}{4}<\delta^{\prime}<\delta$. The Kondrachov embedding theorem
implies $H^{2\delta}(\Omega)\subset H^{2\delta^{\prime}}(\Omega)$ is compact.
Consequently, we obtain that $\mathcal{D}(A^{\delta})\subset
L^{\infty}(\Omega)$ is compact. By the definition,
$L^{1}(\Omega)\subset\left(L^{\infty}(\Omega)\right)^{\prime}\subset\left(\mathcal{D}(A^{\delta})\right)^{\prime}=\mathcal{D}(A^{-\delta})$.
Here $X^{\prime}$ denotes the dual space of a Banach space $X$. Hence, we have
proved (3.3), and the mapping
$\mathcal{D}(A^{1-\delta})\to\mathcal{D}(A^{-\delta})$ is compact.

We set $w_{n}:=w(w_{0}^{n},F_{n})$ for each $n\in\mathbb{N}$. In order to
prove Lemma 3, in terms of Lemma 4, it is sufficient to show that
$\|w_{n}\|_{L^{1}(0,T;\mathcal{D}(A^{1-\delta}))}$ is bounded and
$\|\tau_{h}w_{n}-w_{n}\|_{L^{1}(0,T;\mathcal{D}(A^{-\delta}))}\to 0$ as $h\to
0$ uniformly for each $n\in\mathbb{N}$.

Using the operator $S(t)$ and $K(t)$, the solution can be expressed by the
formulaLY1 ; LY2 :

$w_{n}(t)=S(t)w_{0}^{n}+\int_{0}^{t}K(t-s)F_{n}(s)\mathrm{d}s.$ (3.4)

Writing (3.4) as

$w_{n}(t)=S(t)w_{0}^{n}+\int_{0}^{t}A^{\gamma}K(t-s)A^{-\gamma}F_{n}(s)\mathrm{d}s.$

Then, multiplying both sides by $A^{1-\delta}$, we obtain

$A^{1-\delta}w_{n}(t)=AS(t)A^{-\delta}w_{0}^{n}+\int_{0}^{t}A^{1-\delta+\gamma}K(t-s)A^{-\gamma}F_{n}(s)\mathrm{d}s,$
(3.5)

where $\delta>\gamma>\frac{d}{4}$. Applying the norm estimates (2.8), we have
$\|A^{1-\delta+\gamma}K(t-s)\|\leq(t-s)^{\alpha(\delta-\gamma)-1}$ and
$\|AS(t)\|\leq t^{-\alpha}$. By combining this with Lemma 5, we can derive

$\left\|A^{1-\delta}w_{n}(t)\right\|_{L^{2}(\Omega)}\leq
C\int_{0}^{t}s^{-\alpha}\|w_{0}^{n}\|_{L^{1}(\Omega)}\mathrm{d}s+C\int_{0}^{t}(t-s)^{\alpha(\delta-\gamma)-1}\|F_{n}(s)\|_{L^{1}(\Omega)}\mathrm{d}s.$

Furthermore, applying Young’s inequality, we reach

$\left\|A^{1-\delta}w_{n}\right\|_{L^{1}(0,T;L^{2}(\Omega))}\leq
C\left(\|w_{0}^{n}\|_{L^{1}(\Omega)}+\|F_{n}\|_{L^{1}(0,T;L^{1}(\Omega))}\right).$

Multiply (3.2) by $A^{-\delta}$, and we have

$A^{-\delta}\partial_{t}^{\alpha}(w_{n}-w_{0}^{n})=-A^{1-\delta}w_{n}+A^{-\delta}F_{n}.$

Applying (3.5) and Lemma 5 again, we can obtain

$\displaystyle\|A^{-\delta}\partial_{t}^{\alpha}(w_{n}-w_{0}^{n})\|_{L^{1}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\left(\|w_{0}^{n}\|_{L^{1}(\Omega)}+\|F_{n}\|_{L^{1}(0,T;L^{1}(\Omega))}\right)+\|A^{-\delta}F_{n}\|_{L^{1}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\left(\|w_{0}^{n}\|_{L^{1}(\Omega)}+\|F_{n}\|_{L^{1}(0,T;L^{1}(\Omega))}\right).$

Here we have obtained

$\displaystyle\|A^{1-\delta}w_{n}\|_{L^{1}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq
C\left(\|w_{0}^{n}\|_{L^{1}(\Omega)}+\|F_{n}\|_{L^{1}(0,T;L^{1}(\Omega))}\right),$
(3.6)
$\displaystyle\|A^{-\delta}\partial_{t}^{\alpha}(w_{n}-w_{0}^{n})\|_{L^{1}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq
C\left(\|w_{0}^{n}\|_{L^{1}(\Omega)}+\|F_{n}\|_{L^{1}(0,T;L^{1}(\Omega))}\right).$

To utilize the Lemma 4, we now need to prove:

$\|w_{n}(t+h)-w_{n}(t)\|_{L^{1}(0,T-h;\mathcal{D}(A^{-\delta}))}\longrightarrow
0\quad\mbox{ as }h\to 0.$

Denote
$\widetilde{w}_{n}:=\partial_{t}^{\alpha}(A^{-\delta}(w_{n}-w_{0}^{n}))\in
L^{1}(0,T;L^{2}(\Omega))$. Applying the fractional integral $J^{\alpha}$ to
both sides, we have
$\widetilde{w}_{n}:=A^{-\delta}(w_{n}-w_{0}^{n})=J^{\alpha}\widetilde{w}_{n}$.
Noticing that
$w_{n}(t+h)-w_{n}(t)=(w_{n}(t+h)-w_{0}^{n})-(w_{n}(t)-w_{0}^{n})$, and

$\|w_{n}(t+h)-w_{n}(t)\|_{L^{1}(0,T-h;\mathcal{D}(A^{-\delta}))}=\|J^{\alpha}\widetilde{w}_{n}(t+h)-J^{\alpha}\widetilde{w}_{n}(t)\|_{L^{1}(0,T-h;L^{2}(\Omega))}.$

Now, we represent
$J^{\alpha}\widetilde{w}_{n}(t+h)-J^{\alpha}\widetilde{w}_{n}(t)$:

$\displaystyle
J^{\alpha}\widetilde{w}_{n}(t+h)-J^{\alpha}\widetilde{w}_{n}(t)$ (3.7)
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t+h-s)^{\alpha-1}\widetilde{w}_{n}(s)\mathrm{d}s-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\widetilde{w}_{n}(s)\mathrm{d}s$
$\displaystyle+$
$\displaystyle\int_{t}^{t+h}(t+h-s)^{\alpha-1}\widetilde{w}_{n}(s)\mathrm{d}s.$

Let

$I(t):=\int_{0}^{t}((t+h-s)^{\alpha-1}-(t-s)^{\alpha-1})\widetilde{w}_{n}(s)\mathrm{d}s,\quad
J(t):=\int_{t}^{t+h}(t+h-s)^{\alpha-1}\widetilde{w}_{n}(s)\mathrm{d}s.$

As for the first term concerning $I(t)$, we have

$\|I(t)\|_{\mathcal{D}(A^{-\delta})}\leq\int_{0}^{t}\left((t-s)^{\alpha-1}-(t+h-s)^{\alpha-1}\right)\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}s.$

Integrating both sides with respect to the time variable $t$, we obtain the
following estimate:

$\displaystyle\int_{0}^{T-h}\|I(t)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}t$
(3.8) $\displaystyle\leq$ $\displaystyle
C\int_{0}^{T-h}((T-h-s)^{\alpha}-(T-s)^{\alpha}+h^{\alpha})\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{T-h}h^{\alpha}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle
Ch^{\alpha}\|\widetilde{w}_{n}\|_{L^{1}(0,T;\mathcal{D}(A^{-\delta}))}.$

After that, we start the estimate of $J(t)$:

$\displaystyle\int^{T-h}_{0}\|J(t)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}t$
(3.9) $\displaystyle\leq$
$\displaystyle\int^{T-h}_{0}\int_{t}^{t+h}(t+h-s)^{\alpha-1}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}s\mathrm{d}t.$

Let the integrand
$(t+h-s)^{\alpha-1}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}$ be
denoted as $b(s,t)$. Then, by Fubini’s theorem, we can exchange the orders of
integration:

$\int^{T-h}_{0}\int_{t}^{t+h}b(s,t)\mathrm{d}s\mathrm{d}t=\int_{0}^{h}\int_{0}^{s}b(s,t)\mathrm{d}t\mathrm{d}s+\int_{h}^{T-h}\int_{s-h}^{s}b(s,t)\mathrm{d}t\mathrm{d}s+\int_{T-h}^{T}\int_{s-h}^{T-h}b(s,t)\mathrm{d}t\mathrm{d}s.$

Then the estimate for $J(t)$ can be divided into the following three parts:

$\displaystyle J_{1}$
$\displaystyle=\int_{0}^{h}\int_{0}^{s}(t+h-s)^{\alpha-1}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}t\mathrm{d}s$
(3.10)
$\displaystyle=\int_{0}^{h}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\left(\int_{0}^{s}(t+h-s)^{\alpha-1}\mathrm{d}t\right)\mathrm{d}s$
$\displaystyle\leq
Ch^{\alpha}\|\widetilde{w}_{n}\|_{L^{1}((0,T);\mathcal{D}(A^{-\delta}))}$

and

$\displaystyle J_{2}$
$\displaystyle=\int_{h}^{T-h}\int_{s-h}^{s}(t+h-s)^{\alpha-1}\|\widetilde{w}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}t\mathrm{d}s$
(3.11)
$\displaystyle=\int_{h}^{T-h}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\left(\int_{s-h}^{s}(t+h-s)^{\alpha-1}\mathrm{d}t\right)\mathrm{d}s$
$\displaystyle\leq
Ch^{\alpha}\|\widetilde{w}_{n}\|_{L^{1}((0,T);\mathcal{D}(A^{-\delta}))}$

and

$\displaystyle J_{3}$
$\displaystyle=\int^{T}_{T-h}\int_{s-h}^{T-h}(t+h-s)^{\alpha-1}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\mathrm{d}t\mathrm{d}s$
(3.12)
$\displaystyle=\int_{T-h}^{T}\|\widetilde{w}_{n}(s)\|_{\mathcal{D}(A^{-\delta})}\left(\int_{s-h}^{T-h}(t+h-s)^{\alpha-1}\mathrm{d}t\right)\mathrm{d}s$
$\displaystyle\leq
Ch^{\alpha}\|\widetilde{w}_{n}\|_{L^{1}((0,T);\mathcal{D}(A^{-\delta}))}.$

Consequently,

$\|\tau_{h}w_{n}-w_{n}\|_{L^{1}(0,T;\mathcal{D}(A^{-\delta}))}\leq
O(h^{\alpha}).$

Thus we have completed the proof of Lemma 3.

##  4 Completion of the proof of Theorem 2.1

###### Proof

Henceforth $C>0,C_{1}>0$, etc. denote generic constants which are independent
of $n\in\mathbb{N}$ and the choice of variables $\xi,\eta$. Henceforth when we
write $C=C_{0}(n)$, etc., we understand that the constant depends on $n$ such
as $C_{0}(n)$ above.

We begin by constructing approximating solutions $u_{n}$ and $v_{n}$. In order
to handle the nonlinear terms $f$ and $g$, we introduce the truncated
functions below. We choose $\psi_{1}\in C_{0}^{\infty}(\mathbb{R}^{2})$ such
that $0\leq\psi_{1}(\xi,\eta)\leq 1$ for all $\xi,\eta\in\mathbb{R}$ and

$\psi_{1}(\xi,\eta)=\left\\{\begin{array}[]{l}1,\quad\mbox{ if
}|\xi|,|\eta|\leq 1,\\\ 0,\quad\mbox{ if }|\xi|\geq 2\mbox{ or }|\eta|\geq
2.\end{array}\right.$

Next, we set
$\psi_{n}(\xi,\eta)=\psi_{1}\left(\frac{\xi}{n},\frac{\eta}{n}\right)$. With
this choice, $0\leq\psi_{n}\leq 1$ for all $n\in\mathbb{N}$, and $\psi_{n}$
tends pointwise to 1 as $n$ tends to $\infty$.

We define truncated nonlinear terms as follows

$\left\\{\begin{array}[]{l}f_{n}(\xi,\eta)=\psi_{n}(\xi,\eta)f(\xi,\eta),\quad\xi,\eta\in\mathbb{R},\\\
g_{n}(\xi,\eta)=\psi_{n}(\xi,\eta)g(\xi,\eta),\quad\xi,\eta\in\mathbb{R}.\end{array}\right.$

Then we can verify that there exists a constant $C_{0}(n)>0$ such that

$\left\\{\begin{array}[]{l}\|f_{n}\|_{L^{\infty}(\mathbb{R}^{2})},\,\|g_{n}\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C_{0}(n),\\\
\|\partial_{k}f_{n}\|_{L^{\infty}(\mathbb{R}^{2})},\,\|\partial_{k}g_{n}\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C_{0}(n),\quad k=1,2\end{array}\right.$ (4.1)

and

$f_{n}(0,\eta)=g_{n}(\xi,0)=0\quad\mbox{ for all }\xi,\eta\in\mathbb{R}.$
(4.2)

Indeed, (4.2) is verified directly. Moreover

$|f_{n}(r,s)|+|g_{n}(r,s)|\leq\sup_{|\xi|,|\eta|\leq
2n}|f(\xi,\eta)|+\sup_{|\xi|,|\eta|\leq 2n}|g(\xi,\eta)|,$

for all $r,s\in\mathbb{R}$, which means the first estimate in (4.1). The rest
estimates in (4.1) follow from the definition of $f_{n},g_{n}$. Furthermore we
note that $f_{n},g_{n}$ satisfy Assumption 1.

A truncated form of equation (2.3) is given by:

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u_{n}-a)=-Au_{n}+F_{n}(u_{n}(x,t),v_{n}(x,t)),\quad
x\in\Omega,0<t<T,\\\
\partial_{t}^{\alpha}(v_{n}-b)=-Av_{n}+G_{n}(u_{n}(x,t),v_{n}(x,t)),\quad
x\in\Omega,0<t<T,\\\ \partial_{\nu}u_{n}=\partial_{\nu}v_{n}=0,\quad\text{ on
}\partial\Omega\times(0,T),\\\ u_{n}(x,\cdot)-a(x)\in H_{\alpha}(0,T),\
v_{n}(x,\cdot)-b(x)\in H_{\alpha}(0,T),\quad\mbox{ for almost all
}x\in\Omega.\end{array}\right.$ (4.3)

Here we recall that $-Aw=\Delta w-p_{0}w$, $Q=\Omega\times(0,T)$, and for
$n\in\mathbb{N}$ we set

$F_{n}(u_{n},v_{n}):=f_{n}(u_{n},v_{n})+p_{0}u_{n},\
G_{n}(u_{n},v_{n}):=g_{n}(u_{n},v_{n})+p_{0}v_{n}.$

Now we can prove:

###### Lemma 6

Let $a,b\in H^{1}(\Omega)\cap L^{\infty}(\Omega)$ and $a,b\geq 0$ in $\Omega$.
(i) For $n\in\mathbb{N}$, there exists a unique solution $(u_{n},v_{n})\in
L^{2}(0,T;H^{2}(\Omega))^{2}$ such that $u_{n}-a,v_{n}-b\in
H_{\alpha}(0,T;L^{2}(\Omega))$.
(ii) $u_{n},v_{n}\geq 0$ in $Q$ for each $n\in\mathbb{N}$.

Proof of Lemma 6.

Since (4.1) implies the global Lipschitz contintuity of $f_{n},g_{n}$ over
$\mathbb{R}^{2}$ for each $n$, in terms of $a,b\in H^{1}(\Omega)\cap
L^{\infty}(\Omega)$, we can apply a usual iteration method to establish the
unique existence of $(u_{n},v_{n})\in L^{2}(0,T;H^{2}(\Omega))^{2}$ and
$u_{n}-a,v_{n}-b\in H_{\alpha}(0,T;L^{2}(\Omega))$, which complete the proof
of Lemma 6 (i). Further details can be found, for example, in FLY ; LY1 ; LY2
.

Next we will prove (ii). First we can find functions
$C^{1}_{n}(x,t),C^{2}_{n}(x,t)\in L^{\infty}(Q)$ such that

$\displaystyle|f_{n}(u_{n}(x,t),v_{n}(x,t))|\leq C^{1}_{n}(x,t)u_{n}(x,t),$
(4.4) $\displaystyle|g_{n}(u_{n}(x,t),v_{n}(x,t))|\leq
C^{1}_{n}(x,t)v_{n}(x,t),\quad\mbox{ for almost all }(x,t)\in Q.$

Actually, we can set
$C^{1}_{n}(x,t)=\operatorname{sgn}(u_{n}(x,t))\|\partial_{1}f_{n}\|_{L^{\infty}(Q)}$
and
$C^{2}_{n}(x,t)=\operatorname{sgn}(v_{n}(x,t))\|\partial_{2}f_{n}\|_{L^{\infty}(Q)}$,
where

$\operatorname{sgn}(\xi):=\left\\{\begin{array}[]{r}1,\quad\mbox{ if
}|\xi|\geq 0,\\\ -1,\quad\mbox{ if }|\xi|<0.\end{array}\right.$

Indeed, by (4.1) and (4.2), we apply the mean value theorem and obtain

$\displaystyle|f_{n}(u_{n}(x,t),v_{n}(x,t))|=|f_{n}(u_{n}(x,t),v_{n}(x,t))-f_{n}(0,v_{n}(x,t))|$
$\displaystyle\leq$
$\displaystyle\|\partial_{1}f_{n}\|_{L^{\infty}(\mathbb{R}^{2})}|u_{n}(x,t)|.$

The proof for $g_{n}$ is similar and so the verification of (4.4) is complete.

Then we can rewrite (4.3) as

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(u_{n}-a)=\Delta
u_{n}-C^{1}_{n}(x,t)u_{n}(x,t)+W_{n}(x,t),\quad(x,t)\in Q,\\\
\partial_{t}^{\alpha}(v_{n}-b)=\Delta
v_{n}-C_{n}^{2}(x,t)v_{n}(x,t)+V_{n}(x,t),\quad(x,t)\in Q,\\\
\partial_{\nu}u_{n}=\partial_{\nu}v_{n}=0,\quad\text{ on
}\partial\Omega\times(0,T),\\\ u_{n}(x,\cdot)-a(x)\in H_{\alpha}(0,T),\quad
v_{n}(x,\cdot)-b(x)\in H_{\alpha}(0,T),\end{array}\right.$ (4.5)

where we set

$\displaystyle
W_{n}(x,t):=C_{n}^{1}(x,t)u_{n}(x,t)+f_{n}(u_{n}(x,t),v_{n}(x,t)),$
$\displaystyle
V_{n}(x,t):=C_{n}^{2}(x,t)v_{n}(x,t)+g_{n}(u_{n}(x,t),v_{n}(x,t)).$

We note that $W_{n},V_{n}\geq 0$ in $Q$ by (4.4). Therefore we apply Lemma 2
to $u_{n}$ and $v_{n}$, so that $u_{n},v_{n}\geq 0$ in $Q$. Thus the proof of
Lemma 6 is complete. $\blacksquare$

Next, integrating the first two equations in (4.3) with respect to $x$, we
obtain

$\displaystyle\partial^{\alpha}_{t}\int_{\Omega}(u_{n}-a)(x,t)\mathrm{d}x$
$\displaystyle=\int_{\Omega}f_{n}(u_{n}(x,t),v_{n}(x,t))\mathrm{d}x,$ (4.6)
$\displaystyle\partial^{\alpha}_{t}\int_{\Omega}(v_{n}-b)(x,t)\mathrm{d}x$
$\displaystyle=\int_{\Omega}g_{n}(u_{n}(x,t),v_{n}(x,t))\mathrm{d}x.$

Here we use the fact that $\int_{\Omega}\Delta
u_{n}(x,t)\mathrm{d}x=\int_{\Omega}\Delta v_{n}(x,t)\mathrm{d}x=0$ because
$\partial_{\nu}u_{n}=\partial_{\nu}v_{n}=0$ on $\partial\Omega\times(0,T)$.
Summing up the above equations in $u_{n}$ and $\lambda v_{n}$, we can obtain

$\partial_{t}^{\alpha}\int_{\Omega}(u_{n}-a+\lambda v_{n}-\lambda
b)(x,t)\mathrm{d}x=\int_{\Omega}f_{n}(u_{n},v_{n})+\lambda
g_{n}(u_{n},v_{n})\mathrm{d}x\leq 0.$ (4.7)

By applying the fractional integral to both sides, by $u_{n},v_{n}\geq 0$ in
$Q$, we obtain

$\|u_{n}(\cdot,t)\|_{L^{1}(\Omega)}+\|v_{n}(\cdot,t)\|_{L^{1}(\Omega)}\leq
C(\|a\|_{L^{1}(\Omega)}+\|b\|_{L^{1}(\Omega)}).$ (4.8)

Integrating each equation of (4.3) with respect to $x$ and $t$ and using
$\partial_{\nu}u_{n}=\partial_{\nu}v_{n}=0$ on $\partial\Omega\times(0,T)$, we
derive

$\displaystyle\|f_{n}(u_{n},v_{n})\|_{L^{1}(0,T;L^{1}(\Omega))}=\|J^{1-\alpha}(u_{n}-a)\|_{L^{1}(0,T;L^{1}(\Omega))},$
(4.9)
$\displaystyle\|g_{n}(u_{n},v_{n})\|_{L^{1}(0,T;L^{1}(\Omega))}=\|J^{1-\alpha}(v_{n}-b)\|_{L^{1}(0,T;L^{1}(\Omega))}.$

Since $\|J^{1-\alpha}w(x,\cdot)\|_{L^{1}(0,T)}\leq
C\|w(x,\cdot)\|_{L^{1}(0,T)}$ for $w\in L^{1}(0,T;L^{1}(\Omega))$ by Young’s
inequality, we see

$\displaystyle\|J^{1-\alpha}(u_{n}-a)\|_{L^{1}(0,T;L^{1}(\Omega))}+\|J^{1-\alpha}(v_{n}-b)\|_{L^{1}(0,T;L^{1}(\Omega))}$
(4.10) $\displaystyle\leq$ $\displaystyle
C(\|u_{n}-a\|_{L^{1}(0,T;L^{1}(\Omega))}+\|v_{n}-b\|_{L^{1}(0,T;L^{1}(\Omega))}).$

Therefore, in terms of (4.8) - (4.10), we reach

$\displaystyle\|u_{n}(\cdot,t)\|_{L^{1}(\Omega)}+\|v_{n}(\cdot,t)\|_{L^{1}(\Omega)}+\|f_{n}(u_{n},v_{n})\|_{L^{1}(0,T;L^{1}(\Omega))}+\|g_{n}(u_{n},v_{n})\|_{L^{1}(0,T;L^{1}(\Omega))}$
(4.11) $\displaystyle\leq$ $\displaystyle
C(\|a\|_{L^{1}(\Omega)}+\|b\|_{L^{1}(\Omega)}),$

for all $n\in\mathbb{N}$. We emphasize that the constant $C>0$ is independent
of $n\in\mathbb{N}$. By (4.11), we can obtain

$\|f_{n}(u_{n},v_{n})\|_{L^{1}(Q)}+\|g_{n}(u_{n},v_{n})\|_{L^{1}(Q)}\leq C$
(4.12)

for all $n\in\mathbb{N}$.

Next we will establish the convergence of $f_{n}(u_{n},v_{n})$ and
$g_{n}(u_{n},v_{n})$. We set $a_{0}:=\|a+\lambda b\|_{L^{\infty}(\Omega)}$ and
$W:=u_{n}+\lambda v_{n}$ in $Q$. Hence (4.3) implies

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(W-a-\lambda b)=\Delta
W+(f_{n}(u_{n},v_{n})+\lambda g_{n}(u_{n},v_{n})),\quad\mbox{ in }Q,\\\
\partial_{\nu}W=0,\quad\mbox{ on
}\partial\Omega\times(0,T).\end{array}\right.$ (4.13)

By the definition of $f_{n},g_{n}$, we have

$f_{n}(\xi,\eta)+\lambda
g_{n}(\xi,\eta)=\psi_{n}(\xi,\eta)(f(\xi,\eta)+\lambda g(\xi,\eta))\leq
0\quad\mbox{ for all }\xi,\eta\in\mathbb{R}.$ (4.14)

On the other hand, $\widetilde{W}(x,t):=\|a+\lambda b\|_{L^{\infty}(\Omega)}$
satisfies

$\left\\{\begin{array}[]{l}\partial_{t}^{\alpha}(\widetilde{W}-\|a+\lambda
b\|_{L^{\infty}(\Omega)})=\Delta\widetilde{W},\quad\mbox{ in }Q,\\\
\partial_{\nu}\widetilde{W}=0,\quad\mbox{ on
}\partial\Omega\times(0,T).\end{array}\right.$

Applying the comparison principle(LY1, , (Corollary 1)) to $W$ and
$\widetilde{W}$, in terms of (4.14), we obtain

$W(x,t)\leq\widetilde{W}(x,t),\quad(x,t)\in Q.$

By Lemma 6 (ii), we have $W(x,t)=u_{n}(x,t)+\lambda v_{n}(x,t)\geq 0$ for
$(x,t)\in Q$. Therefore,

$0\leq u_{n}(x,t)+\lambda v_{n}(x,t)\leq\|a+\lambda
b\|_{L^{\infty}(\Omega)},\quad(x,t)\in Q.$

Since
$\xi+\eta\leq\max\left\\{1,\,\frac{1}{\lambda}\right\\}(\xi+\lambda\eta)$ for
$\xi,\eta\geq 0$, then we have

$\displaystyle 0$ $\displaystyle\leq
u_{n}+v_{n}\leq\max\left\\{1,\,\frac{1}{\lambda}\right\\}(u_{n}+\lambda
v_{n})$ (4.15) $\displaystyle\leq C\|a+\lambda
b\|_{L^{\infty}(\Omega)}=:C_{1},\quad\mbox{ almost everywhere in }Q.$

Because $u_{n},v_{n}$ are both non-negative, we deduce $u_{n},v_{n}\leq
C_{1}$. Therefore

$\displaystyle|f_{n}(u_{n}(x,t),v_{n}(x,t))|+|g_{n}(u_{n}(x,t),v_{n}(x,t))|$
(4.16) $\displaystyle\leq$ $\displaystyle\sup_{|\xi|,|\eta|\leq
C_{1}}|f_{n}(\xi,\eta)|+\sup_{|\xi|,|\eta|\leq C_{1}}|g_{n}(\xi,\eta)|$
$\displaystyle\leq$ $\displaystyle\sup_{|\xi|,|\eta|\leq
C_{1}}|f(\xi,\eta)|+\sup_{|\xi|,|\eta|\leq C_{1}}|g(\xi,\eta)|=:C_{2},$

for all $n\in\mathbb{N}$ and almost all $(x,t)\in Q$. We note that $C_{2}$
does not depend on $n$.

Then we can choose a subsequence $n^{\prime}\in\mathbb{N}$ such that

$\displaystyle\lim_{n^{\prime}\to\infty}f_{n^{\prime}}(u_{n^{\prime}}(x,t),\,v_{n^{\prime}}(x,t))=f(u(x,t),\,v(x,t)),\quad\mbox{
for almost all }(x,t)\in Q,$ (4.17)
$\displaystyle\lim_{n^{\prime}\to\infty}g_{n^{\prime}}(u_{n^{\prime}}(x,t),\,v_{n^{\prime}}(x,t))=g(u(x,t),\,v(x,t)),\quad\mbox{
for almost all }(x,t)\in Q.$

In order to prove this, we can assume that $n\geq\frac{C_{1}}{2}$. Then, by
the definition of $f_{n}$ and $g_{n}$, we deduce

$\displaystyle f_{n}(u_{n}(x,t),\,v_{n}(x,t))=f(u_{n}(x,t),\,v_{n}(x,t)),$
(4.18) $\displaystyle
g_{n}(u_{n}(x,t),\,v_{n}(x,t))=g(u_{n}(x,t),\,v_{n}(x,t)),$

for almost all $(x,t)\in Q:=\Omega\times(0,T)$ if $n\geq\frac{C_{1}}{2}$. On
the other hand, in terms of (4.11) we apply Lemma 3 to see that
$(u_{n},v_{n}),n\in\mathbb{N}$ contains a convergent subsequence in
$L^{1}(Q)$. Therefore, we can find a subsequence
$\\{n^{\prime}\\}\subset\mathbb{N}$ and some $(u,v)\in(L^{1}(Q))^{2}$ such
that $(u_{n^{\prime}},v_{n^{\prime}})\longrightarrow(u,v)$ in $L^{1}(Q)^{2}$
as $n^{\prime}\to\infty$. Hence, choosing further a subsequence of
$\\{n^{\prime}\\}$, denoted by the same notations, such that

$\displaystyle\lim_{n^{\prime}\to\infty}u_{n^{\prime}}(x,t)=u(x,t),$ (4.19)
$\displaystyle\lim_{n^{\prime}\to\infty}v_{n^{\prime}}(x,t)=v(x,t),\quad\mbox{
for almost all }(x,t)\in Q.$

Hence, by $f,g\in C(\mathbb{R}^{2})$ and (4.18)-(4.19), we verified (4.17).

Finally considering (4.16) and (4.17), we apply the Lebesgue convergence
theorem to verify

$\displaystyle\lim_{n\to\infty}\|f_{n}(u_{n},v_{n})-f(u,v)\|_{L^{1}(Q)}=0,$
$\displaystyle\lim_{n\to\infty}\|g_{n}(u_{n},v_{n})-g(u,v)\|_{L^{1}(Q)}=0.$

Multiplying both sides of (4.3) with $\psi$, after integrating, we obtain

$\displaystyle(u_{n}-a,(\partial_{t}^{\alpha})^{*}\psi)$
$\displaystyle=(u_{n},-A\psi)+(F_{n},\psi),$ (4.20)
$\displaystyle(v_{n}-b,(\partial_{t}^{\alpha})^{*}\psi)$
$\displaystyle=(v_{n},-A\psi)+(G_{n},\psi),$

for all $\psi\in C^{\infty}(\overline{Q})$ satisfying $\partial_{\nu}\psi=0$
in $\partial\Omega\times(0,T)$ and $\psi(\cdot,T)=0$ in $\Omega$. Passing $n$
in (4.20) to the limit, we can see

$\displaystyle(u-a,(\partial_{t}^{\alpha})^{*}\psi)$
$\displaystyle=(u,\Delta\psi)+(f(u,v),\psi),$ (4.21)
$\displaystyle(v-b,(\partial_{t}^{\alpha})^{*}\psi)$
$\displaystyle=(v,\Delta\psi)+(g(u,v),\psi),$

for all $\psi\in C^{\infty}(\overline{Q})$ satisfying $\partial_{\nu}\psi=0$
in $\partial\Omega\times(0,T)$ and $\psi(\cdot,T)=0$ in $\Omega$. This implies
that $(u,v)$ is a weak solution to equation (2.3), we have completed the proof
of Theorem 2.1.

##  5 Blow up for a system for diffusion equation with convex nonlinear terms

In this section, we show that the global existence does not hold true for some
nonlinear terms, which is our second main result. Here, we do not require the
coefficients in front of the spatial diffusion terms to be equal.
Specifically, for the second line in (1.1), we consider

$\partial_{t}^{\alpha}(v-b)=d\Delta v+g(u,v).$

For the statement, we introduce the following assumption for the nonlinear
terms $f$ and $g$.

###### Assumption 2

For $f,g\in C^{\infty}(\mathbb{R}^{2})$, there exist constants $\lambda>0,p>1$
and $C=C_{p,\lambda}>0$ such that

$f(\xi,\eta)+\lambda g(\xi,\eta)\geq C(\xi^{p}+\eta^{p}),\quad\text{ for all
}\xi,\eta\geq 0.$ (5.1)

Now we are ready to state our main result on the blow-up with upper bound of
the blow-up time for $p>1$.

###### Theorem 5.1

Let $f,g$ satisfy the Assumption 2 and $a,b\in H^{2\gamma}(\Omega)$,
$\frac{3}{4}<\gamma\leq 1$, satisfy $\partial_{\nu}a=\partial_{\nu}b=0$ on
$\partial\Omega$ and $a,b\geq 0,\not\equiv 0$ in $\Omega$. Then there exists
some $T=T_{\alpha,p,a,b}>0$ such that the solutions of (2.3) satisfying

$u,v\in C([0,T];H^{2}(\Omega)),\quad u-a,v-b\in
H_{\alpha}(0,T;L^{2}(\Omega)),$

exist for $0<t<T_{\alpha,p,a,b}$ and

$\lim_{t\uparrow
T_{\alpha,p,a,b}}\left(\|u(\cdot,t)\|_{L^{1}(\Omega)}+\|v(\cdot,t)\|_{L^{1}(\Omega)}\right)=\infty$

holds. Moreover, we can bound $T_{\alpha,p,a,b}$ from above as

$T_{\alpha,p,a,b}\leq\left(\frac{1}{(p-1)\Gamma(2-\alpha)C_{p,\lambda}^{-1}\left(\frac{1}{|\Omega|}\int_{\Omega}a+\lambda
b\mathrm{~{}d}x\right)^{p-1}}\right)^{\frac{1}{\alpha}}=:T^{*}(\alpha,p,a,b).$

###### Proof

Step 1. Our proof is similar to the one for a single equation (FLY ). We show
the following two lemmas from FLY .

###### Lemma 7

Let $h\in L^{2}(0,T)$ and $c\in C[0,T]$. Then there exists a unique solution
$y\in H_{\alpha}(0,T)$ to

$\partial_{t}^{\alpha}y-c(t)y=h,\quad 0<t<T.$

Moreover, if $h\geq 0$ in $(0,T)$, then $y\geq 0$ in $(0,T)$.

###### Lemma 8

Let $c_{0}>0,a_{0}\geq 0,p>1$ be constants and $y-a_{0},z-a_{0}\in
H_{\alpha}(0,T)\cap C[0,T]$ satisfy

$\partial_{t}^{\alpha}(y-a_{0})\geq
c_{0}y^{p},\quad\partial_{t}^{\alpha}(z-a_{0})\leq c_{0}z^{p},\quad\text{ in
}(0,T).$

Then $y\geq z$ in $(0,T)$.

Step 2. We set

$\theta(t):=\int_{\Omega}\left(u(x,t)+\lambda
v(x,t)\right)\mathrm{d}x=\int_{\Omega}\left((u-a)+\lambda(v-b)\right)(x,t)\mathrm{d}x+m_{0}$

where $m_{0}:=\int_{\Omega}(a+\lambda b)\mathrm{d}x$. Here we see that
$m_{0}>0$ because $a,b\geq 0,\not\equiv 0$ in $\Omega$ and $\lambda>0$ by the
assumption of Theorem 5.1. Then by (1.1) we have

$\displaystyle\partial_{t}^{\alpha}(\theta(t)-m_{0})$
$\displaystyle=\int_{\Omega}(\Delta u+\lambda d\Delta
v)(x,t)\mathrm{d}x+\int_{\Omega}(f(u,v)+\lambda g(u,v))(x,t)\mathrm{d}x$
$\displaystyle=\int_{\Omega}(f(u,v)+\lambda g(u,v))(x,t)\mathrm{d}x.$

On the other hand, we have

$\theta(t)=\int_{\Omega}(u+\lambda
v)\mathrm{d}x\leq\left(\int_{\Omega}\mathrm{d}x\right)^{\frac{1}{p^{\prime}}}\left(\int_{\Omega}(u+\lambda
v)^{p}\mathrm{d}x\right)^{\frac{1}{p}}=|\Omega|^{\frac{1}{p^{\prime}}}\left(\int_{\Omega}(u+\lambda
v)^{p}\mathrm{d}x\right)^{\frac{1}{p}}$

with $\frac{1}{p^{\prime}}+\frac{1}{p}=1$, and $(u+\lambda v)^{p}\leq
C_{p,\lambda}(u^{p}+v^{p})$ for $u,v\geq 0$. Hence, (5.1) yields

$\displaystyle\theta^{p}(t)$
$\displaystyle\leq|\Omega|^{\frac{p}{p^{\prime}}}\int_{\Omega}(u+\lambda
v)^{p}\mathrm{d}x\leq
C_{p,\lambda}|\Omega|^{\frac{p}{p^{\prime}}}\int_{\Omega}(u^{p}+v^{p})\mathrm{d}x$
$\displaystyle\leq
C_{p,\lambda}|\Omega|^{\frac{p}{p^{\prime}}}\int_{\Omega}(f(u,v)+\lambda
g(u,v))(x,t)\mathrm{d}x.$

So we obtain

$\partial_{t}^{\alpha}(\theta(t)-m_{0})\geq
C_{p,\lambda}^{-1}|\Omega|^{-\frac{p}{p^{\prime}}}\theta^{p}(t),\quad 0<t<T.$

Step 3. This step is devoted to the construction of a lower solution
$\underline{\theta}(t)$ satisfying

$\partial_{t}^{\alpha}(\underline{\theta}(t)-m_{0})\leq
C_{0}\underline{\theta}^{p}(t),\quad 0<t<T-\epsilon,\quad\lim_{t\uparrow
T}\underline{\theta}(t)=\infty.$ (5.2)

Here we set $C_{0}:=C_{p,\lambda}^{-1}|\Omega|^{-\frac{p}{p^{\prime}}}$. The
construction of $\underline{\theta}(t)$ is same as the proof in FLY . As a
possible lower solution, we consider

$\underline{\theta}(t):=m_{0}\left(\frac{T}{T-t}\right)^{m},\ m\in\mathbb{N}.$

By the definition
$\partial_{t}^{\alpha}(\underline{\theta}(t)-a_{0})(t)=\mathrm{d}^{\alpha}_{t}\underline{\theta}(t)$.
We can get

$\partial_{t}^{\alpha}(\underline{\theta}(t)-m_{0})(t)=\mathrm{d}^{\alpha}_{t}\underline{\theta}(t)=m_{0}T^{m}\mathrm{d}^{\alpha}_{t}\left(\frac{1}{(T-t)^{m}}\right)\leq\frac{m_{0}T^{m+1-\alpha}m}{\Gamma(2-\alpha)}\frac{1}{(T-t)^{m+1}}.$

This equation holds for arbitrary $m\in\mathbb{N},T>0$ and $0<t<T-\epsilon$.

Finally we claim that for any $p>1$ and $m_{0}>0$, there exist constants
$m\in\mathbb{N}$ and $T>0$ such that

$\frac{m_{0}T^{m+1-\alpha}m}{\Gamma(2-\alpha)}\frac{1}{(T-t)^{m+1}}\leq
C_{0}\underline{\theta}^{p}(t)=\frac{C_{0}m_{0}^{p}T^{mp}}{(T-t)^{mp}},\
0<t<T-\epsilon.$ (5.3)

Therefore, if

$\displaystyle T$
$\displaystyle\geq\left(\frac{m}{\Gamma(2-\alpha)C_{0}m_{0}^{p-1}}\right)^{\frac{1}{\alpha}}\geq\left(\frac{1}{(p-1)\Gamma(2-\alpha)C_{0}m_{0}^{p-1}}\right)^{\frac{1}{\alpha}}$
(5.4)
$\displaystyle=\left\\{(p-1)\Gamma(2-\alpha)C_{p,\lambda}^{-1}\left(\frac{1}{|\Omega|}\int_{\Omega}(a+\lambda
b)\mathrm{~{}d}x\right)^{p-1}\right\\}^{-\frac{1}{\alpha}}=:T^{*}(\alpha,p,a,b),$

then (5.3) is satisfied.

Consequently, we can obtain

$\underline{\theta}(t):=m_{0}\left(\frac{T^{*}(\alpha,p,a,b)}{T^{*}(\alpha,p,a,b)-t}\right)^{m}$

satisfies (5.2).

Now it suffices to apply Lemma 8 to get

$\theta(t)\geq\underline{\theta}(t),\quad 0\leq t\leq
T^{*}(\alpha,p,a,b)-\epsilon.$

Since $\epsilon>0$ was arbitrarily chosen, we obtain

$\int_{\Omega}\left(u(x,t)+\lambda
v(x,t)\right)\mathrm{d}x=\theta(t)\geq\underline{\theta}(t)=m_{0}\left(\frac{T^{*}(\alpha,p,a,b)}{T^{*}(\alpha,p,a,b)-t}\right)^{m}.$

This means that the solution $u+v$ cannot exist for $t>T^{*}(\alpha,p,a,b)$.
Hence, the blow-up time $T(\alpha,p,a,b)\leq T^{*}(\alpha,p,a,b)$. The proof
of Theorem 5.1 is complete.

##  6 Concluding remarks

It’s worth mentioning that the conditions on $f$ and $g$ in the Assumption 1
are $f(0,\eta)=g(\xi,0)=0$ for all $\xi\geq 0$ and $\eta\geq 0$, and $f,g$ are
local Lipschitz continuous. If we improve the regularity of $f$ and $g$ such
that $f,g\in C^{1}(\mathbb{R}^{2})$, we can obtain the non-negativity by
comparison principle. For simplicity, we choose an initial value $a=0$, and
denote $\widetilde{u}$ as the solution of

$\partial_{t}^{\alpha}\widetilde{u}=\Delta\widetilde{u}+f(\widetilde{u},\widetilde{v})-f(0,\widetilde{v})$

Noticing
$f(\widetilde{u},\widetilde{v})-f(0,\widetilde{v})=\partial_{1}f(\xi,\widetilde{v})\widetilde{u}$,
we can rewrite the above equation as

$\partial_{t}^{\alpha}\widetilde{u}=\Delta\widetilde{u}+\partial_{1}f(\xi,\widetilde{v})\widetilde{u}.$

Here $\xi$ is some number between $0$ and $\widetilde{u}(x,t)$. Similarly, we
obtain the equation for $\widetilde{v}$ as

$\partial_{t}^{\alpha}\widetilde{v}=\Delta\widetilde{v}+\partial_{2}g(\widetilde{u},\eta)\widetilde{v},$

where $\eta$ is some number between $0$ and $\widetilde{v}$. Therefore, by the
comparison principleLY2  and Lemma 2, we can derive the non-negativity of the
solutions $\widetilde{u}$ and $\widetilde{v}$.

Moreover, it can be observed that limited in $L^{1}(\Omega\times(0,T))$, we
can derive the global existence of solutions. For the uniqueness of solutions,
we need to impose stronger priori assumptions on the right-hand side term and
the initial values. Similarly to LY2 , we impose the following requirements on
the nonlinear term $g(u,v)$. For $g:\mathcal{D}(A_{0}^{\gamma}\times
A_{0}^{\gamma})\to L^{2}(\Omega)$, we assume that for some constant $m>0$,
there exists a constant $C_{g}=C_{g}(m)>0$ such that

$\left\\{\begin{array}[]{l}\text{ (i) }\|g(u,\cdot)\|+\|g(\cdot,v)\|\leq
C_{g},\quad\left\|g\left(u_{1},\cdot\right)-g\left(u_{2},\cdot\right)\right\|\leq
C_{g}\left\|u_{1}-u_{2}\right\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\\\
\left\|g\left(\cdot,v_{1}\right)-g\left(\cdot,v_{2}\right)\right\|\leq
C_{g}\left\|v_{1}-v_{2}\right\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)}\\\
\text{ if
}\|u\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\|v\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\left\|u_{1}\right\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\left\|u_{2}\right\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\left\|v_{1}\right\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\left\|v_{2}\right\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)}\leq
m\\\ \text{ and }\\\ \text{ (ii) there exists a constant
}\varepsilon\in\left(0,\frac{3}{4}\right)\text{ such that }\\\
\|g(u,\cdot)\|_{H^{2\varepsilon}(\Omega)}+\|g(\cdot,v)\|_{H^{2\varepsilon}(\Omega)}\leq
C_{g}(m)\quad\text{ if
}\|u\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\|v\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)}\leq
m,\end{array}\right.$

along with initial values $a,b$ satisfying
$\|a\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)},\|b\|_{\mathcal{D}\left(A_{0}^{\gamma}\right)}\leq
m$. After expressing solutions $u(t)$ and $v(t)$ as

$\displaystyle
u(t)=S(t)a+\int_{0}^{t}K(t-s)\left(p_{0}u-g(u(s),v(s))\right)\mathrm{d}s,$
$\displaystyle
v(t)=S(t)b+\int_{0}^{t}K(t-s)\left(p_{0}v+g(u(s),v(s))\right)\mathrm{d}s,$

we can apply the contraction theorem similarly to the approach demonstrated in
LY2 . In this situation, there exists a constant $T=T(m)>0$ such that the
solutions to the system (2.3) in $L^{2}(0,T;H^{2}(\Omega))\cap
C([0,T];\mathcal{D}(A^{\gamma}_{0}))$ are unique.

##  7 Appendix. Proof of Lemma 2

Since $1<\kappa<\infty$, the embedding $L^{\infty}(Q)\hookrightarrow
L^{\kappa}(Q)$ holds, thus we have $c\in L^{\kappa}(Q)$. Furthermore, given
that $C_{0}^{\infty}(Q)$ is dense in $L^{\kappa}(Q)$, there exist functions
$\\{c_{n}\\}\in C_{0}^{\infty}(Q)$ such that $\lim_{n\to\infty}c_{n}=c$ in
$L^{\kappa}(Q)$.

We define the operator $A_{n}$ regarding $c_{n}(x,t)$ in $L^{2}(\Omega)$
similar to LY1  as

$(A_{n}v)(x)=-\Delta v(x)-c_{n}(x,t)v(x),\quad x\in\Omega.$

Correspondingly, the operators $S(t)$ and $K(t)$ related to
$\left(-A_{0}v\right)(x):=\Delta v(x)-c_{0}v(x)$ with an arbitrary constant
$c_{0}>0$ remain the same

$S(t)a=\sum_{n=1}^{\infty}E_{\alpha,1}\left(-\mu_{n}t^{\alpha}\right)\left(a,\varphi_{n}\right)\varphi_{n},\quad
a\in L^{2}(\Omega),t>0,\\\ $

and

$K(t)a=\sum_{n=1}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}\left(-\mu_{n}t^{\alpha}\right)\left(a,\varphi_{n}\right)\varphi_{n},\quad
a\in L^{2}(\Omega),t>0.$

Additionally, the operators in (LY1, , (2.14)) are updated to

$\begin{array}[]{l}G(t):=\int_{0}^{t}K(t-s)F(s)ds+S(t)a,\\\
Q_{n}v(t)=Q_{n}(t)v(t):=\left(c_{0}+c_{n}(\cdot,t)\right)v(t).\end{array}$

In accordance on the previous modifications, the solution presented in (LY1, ,
(4.3)) has been updated to reflect the changes in the operators $A_{n}$ and
$Q_{n}$ as defined above. The updated solution is given by

$u_{n}(F,a)(t)=G(t)+\int_{0}^{t}K(t-s)Q_{n}u_{n}(s)\mathrm{d}s,\quad 0<t<T.$

Since $c_{n}\in C_{0}^{\infty}(Q)$, according to (LY1, , Theorem 2), we know
that $u_{n}(F,a)(x,t)\geq 0$ in $\Omega\times(0,T)$ and that $u_{n}(F,a)\in
L^{2}(0,T;H^{2}(\Omega))$ as well as $u_{n}(F,a)-a\in
H_{\alpha}(0,T;L^{2}(\Omega))$. Following a similar approach as in LY1 , we
can extract a subsequence ${u_{n^{\prime}}(F,a)}$ from ${u_{n}(F,a)}$ such
that ${u_{n^{\prime}}(F,a)}\to{u(F,a)}$ in $L^{2}(0,T;H^{2}(\Omega))$ and
${u_{n^{\prime}}(F,a)}-a\to{u(F,a)}-a$ in $H_{\alpha}(0,T;L^{2}(\Omega))$.
Finally, $u_{n}(F,a)(x,t)\geq 0$ implies that $u(F,a)\geq 0$ in $Q$.

###### Acknowledgements.

D. Feng is supported by Key-Area Research and Development Program of Guangdong
Province (No.2021B0101190003) and Science and Technology Commission of
Shanghai Municipality (23JC1400501). M. Yamamoto is supported partly by Grant-
in-Aid for Scientific Research (A) 20H00117 and Grant-in-Aid for Challenging
Research (Pioneering) 21K18142 of Japan Society for the Promotion of Science.

## Conflict of interest

The authors declare that they have no conflict of interest.

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