Local existence and nonexistence for fractional in time weakly coupled reaction-diffusion systems

Abstract

We study a fractional in time weakly coupled reaction-diffusion system in a bounded domain with the Dirichlet boundary condition. The domain is imbedded in an N-dimensional space and it has $C^2$ boundary, and fractional derivatives are meant in a generalized Caputo sense. The system can be referred to as a standard reaction-diffusion system in two components with polynomial growth. We obtain integrability conditions on the initial state functions which determine the existence/nonexistence of a local in time mild solution.

1 Introduction

Let $\Omega$ be a bounded domain in ${\mathbb {R}^N}$, $N\ge 1$, with $C^2$ boundary. We study existence and nonexistence of a local in time solution of the fractional in time weakly coupled reaction-diffusion system

$${\left\{ \begin{array}{ll} \partial ^{\alpha _1}_t u=\Delta u+f_1(x,t,v) &{} \text {in} \,\,\Omega \times (0,T), \\ \partial ^{\alpha _2}_t v=\Delta v+f_2(x,t,u) &{} \text {in} \,\,\Omega \times (0,T), \\ u(x,t)=v(x,t)=0 &{} \text {on} \,\,\partial \Omega \times (0,T), \\ u(x,0)=u_0 (x), v(x,0)=v_0 (x) & {} \text {in} \,\,\Omega , \end{array}\right. }$$

(1.1)

where $0<\alpha _1\le \alpha _2<1$ and $T>0$. The fractional derivatives $\partial ^{\alpha _1}_t$ and $\partial ^{\alpha _2}_t$ are meant in a generalized Caputo sense, i.e.,

$$\begin{aligned} \partial ^{\alpha }_t u(t)=\frac{1}{\Gamma (1-\alpha )}\displaystyle \int _0^t (t-s)^{-\alpha }\partial _s u(s)ds \ \ \text {for} \,\, 0<\alpha <1, \end{aligned}$$

where $\Gamma$ denotes the usual Gamma function. In the present paper we suppose the following:

Assumption A

Let $1<p_1, p_2<\infty$, $q_1, q_2\in (1,\infty ]\cap \left( \dfrac{N}{2},\infty \right]$, $m_1\in [0,\alpha_1)\, \textrm{ and }\, m_2\in [0,\alpha_2)$ be given constants. There exist nonnegative functions $c_1\in L^{q_1}(\Omega )$ and $c_2\in L^{q_2}(\Omega )$ such that the following hold:

(F1):

for $i=1,2$, $f_i(x,t,\cdot ):\mathbb {R}\rightarrow \mathbb {R}$ is a measurable function such that

$$\begin{aligned} |f_i(x,t,\xi )|\le c_i(x)\cdot t^{-m_i}(1+|\xi |)^{p_i} \, \, {\text {for}}\, \, \xi \in \mathbb {R}, \, \,{\text {a.e.}}\,\, (x,t)\in \Omega \times (0,\infty ), \end{aligned}$$

(F2):

for $i=1,2$, $f_i$ satisfies the local Lipschitz condition

$$\begin{aligned} |f_i(x,t,\xi )-f_i(x,t,\eta )|\le c_i(x)\cdot t^{-m_i}(1+|\xi |+|\eta |)^{p_i-1}|\xi -\eta | \,\, {\text {for}}\,\, \xi ,\eta \in \mathbb {R}, \, \, {\text { a.e.}}\, \, (x,t)\in \Omega \times (0,\infty ). \end{aligned}$$

Let us start with classical equations, $\alpha =1$. We consider the scalar problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u=\Delta u+f(u) &{} {\text {in}} \,\, \Omega \times (0,T), \\ u(x,t)=0 &{} {\text {on}} \,\,\partial \Omega \times (0,T), \\ u(x,0)=u_0 (x) &{} {\text {in}} \,\,\Omega , \end{array}\right. } \end{aligned}$$

(1.2)

where $f\in C^1$ and $\Omega$ is a (possibly unbounded) smooth domain. It is well known that the problem (1.2) possesses a local in time classical solution for a general nonlinear term f if $u_0 \in L^{\infty }(\Omega )$ (cf. [7, 16]). On the other hand, in the case where $u_0 \not \in L^{\infty }(\Omega )$, the existence of solutions heavily depends on the balance between the growth rate of f and the singularity of $u_0$ (cf. [7]). In Weissler [20], (1.2) was studied when $f(u)=|u|^{p-1}u$, $p>1$, and $\Omega$ is bounded. A local in time solution was constructed when $u_0 \in L^r (\Omega )$ for $r>\dfrac{N}{2}(p-1)$ and $r\ge 1$, or $r=\dfrac{N}{2}(p-1)$ and $r>1$. It was also shown that if $1\le r<\dfrac{N}{2}(p-1)$, then there exists a nonnegative initial function $u_0 \in L^r (\Omega )$ such that, for every $T>0$, (1.2) admits no nonnegative solution.

Next, we consider the reaction-diffusion system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u=\Delta u+f_1(u,v) &{} {\text {in}} \,\,\Omega \times (0,T), \\ \partial _t v=\Delta v+f_2(u,v) &{} {\text {in}} \,\,\Omega \times (0,T), \\ u(x,t)=v(x,t)=0 &{} {\text {on}}\,\, \partial \Omega \times (0,T), \\ u(x,0)=u_0 (x), v(x,0)=v_0 (x) &{} {\text {in}} \,\,\Omega . \end{array}\right. } \end{aligned}$$

(1.3)

Quittner–Souplet [15] studied (1.3), where $f_1(u,v)=|v|^{p_1-1}v$, $f_2(u,v)=|u|^{p_2-1}u$ $(p_1,p_2>0)$ and $\Omega$ is bounded. This is called a weakly coupled system. The existence (resp. nonexistence) of a local in time solution was proved when $(u_0,v_0)\in L^{r_1}(\Omega )\times L^{r_2}(\Omega )$ and

$$\begin{aligned} \frac{N}{2}\max \left\{ \frac{p_1}{r_2}-\frac{1}{r_1},\frac{p_2}{r_1}-\frac{1}{r_2}\right\} \le 1,\ p_1,p_2>1\ \text {and}\ r_1,r_2>1 \end{aligned}$$

(resp. $\dfrac{N}{2}\max \left\{ \dfrac{p_1}{r_2}-\dfrac{1}{r_1},\dfrac{p_2}{r_1}-\dfrac{1}{r_2}\right\} > 1$, $p_1,p_2>0$ and $r_1,r_2\ge 1$). Moreover, weakly and strongly coupled variants of system (1.3) were studied in [4,5,6, 11, 13, 19]. They assumed that initial functions may have singularities.

We obtain integrability conditions of $(u_0,v_0)$ which determine the existence/nonexistence of a local in time solution of (1.1). In the proof we combine methods of Gal–Warma [9] and Quittner–Souplet [15, 16].

To define a mild solution, we recall the Wright type function [10] defined by

$$\begin{aligned} \Phi _{\alpha }(z):=\sum _{n=0}^{\infty } \frac{(-z)^n}{n!\Gamma (-\alpha n+1-\alpha )} \,\, {\text {for}}\,\, 0<\alpha <1 \,\, {\text {and}} \,\, z\in \mathbb {C}. \end{aligned}$$

This is also sometimes called the Mainardi function and studied in [14, 18, 22]. It follows from [2, 10] that

$$\begin{aligned} \Phi _{\alpha }(t)\ge 0 \,\, {\text {for}} \,\, t\ge 0 \,\, {\text {and}}\, \, \int _0^{\infty } \Phi _{\alpha }(t) dt=1. \end{aligned}$$

(1.4)

We prove our nonexistence result using (1.4). It is a key point that the function $\Phi _{\alpha }(t)$ is nonnegative and integrable. Moreover, it is well known ([10]) that

$$\begin{aligned} \displaystyle \int _0^{\infty } t^p\Phi _{\alpha }(t) dt=\frac{\Gamma (p+1)}{\Gamma (\alpha p+1)} \,\, {\text {for}} \,\, p>-1 \,\,{\text {and}}\,\, 0<\alpha <1. \end{aligned}$$

(1.5)

Next, we consider functional spaces. Let $1<r<\infty$. We denote by $W^{k,r}(\Omega )$ the Sobolev space (resp. the Sobolev–Slobodeckii space) if k is an integer (resp. if k is not an integer). We also denote by $\mathcal {D}(\Omega )$ the space of $C^\infty$-functions with compact support in $\Omega$. Put $X_0(r):=L^r(\Omega )$ and $X_1(r):=W^{2,r}(\Omega )\cap W^{1,r}_0(\Omega )$, where $W^{1,r}_0(\Omega )$ is the closure of $\mathcal {D}(\Omega )$ in $W^{1,r}(\Omega )$. Let $\Delta$ be the Laplace operator with the domain $D(\Delta )=X_1(r)$. Then $\Delta$ generates a $C^0$ analytic semigroup in $X_0$ by [16, Examples 51.4 (i)]. Let $X_{-1}(r)$ be the completion of $X_0(r)$ endowed with the norm $|x|_{X_{-1}(r)}:=|(\omega +\Delta )^{-1}x|_{X_0(r)}$, where $\omega \in \mathbb {R}$ satisfies that $\omega +\Delta : X_1(r)\rightarrow X_0(r)$ is an isomorphism ([16, p.466, 467]). For $0<\theta <1$, set $X_{\theta }(r):= (X_0(r),X_1(r))_{\theta }$ and $X_{-1+\theta }(r):= (X_{-1}(r),X_0(r))_{\theta }$, where $(\cdot ,\cdot )_{\theta }$ is the complex interpolation functor if $\theta =\dfrac{1}{2}$ and the real interpolation functor $(\cdot ,\cdot )_{\theta ,r}$ otherwise. Due to [16, Theorem 51. 1 (i) and Examples 51.4 (i)], we have $X_{\theta _1}(r)\hookrightarrow X_{\theta _2}(r)$ if $-1\le \theta _2\le \theta _1\le 1$,

$$\begin{aligned} X_{\theta }(r)\hookrightarrow W^{2\theta ,r}(\Omega ) \ \ {\text {if}}\,\, \theta \ge 0, \,\, {\text {and}}\,\, X_{\theta }(r) \,\, {\doteq}\,\, (X_{-\theta }(r'))' \,\, {\text {if}} \,\,\theta <0, \end{aligned}$$

(1.6)

where $r'$ is the conjugate exponent of r, i.e., $\dfrac{1}{r}+\dfrac{1}{r'}=1$. We denote $X'$ by the (topological) dual space if X is a Banach space. We write $X\hookrightarrow Y$ if X is continuously embedded in Y. Moreover, $X\doteq Y$ means that $X\hookrightarrow Y$ and $Y\hookrightarrow X$.

We are ready to introduce some operators related to fractional derivatives. Let $0\le \theta \le 1$ and $1<r<\infty$. We observe from [16, Theorem 51.1 (iv)] that the operator $\Delta$ also generates a $C^0$ analytic semigroup in $X_{\theta }(r)$, which is denoted by S(t). For $0<\alpha <1$ and $t>0$, we define

$$\begin{aligned} S_{\alpha }(t): X_{\theta }(r)\rightarrow X_{\theta }(r), \ P_{\alpha }(t): X_{\theta }(r)\rightarrow X_{\theta }(r) \end{aligned}$$

by

$${\left\{ \begin{array}{ll} S_{\alpha }(t)w:= \int _0^{\infty } \Phi _{\alpha }(\tau )S(\tau t^{\alpha })w d\tau ,\\ P_{\alpha }(t)w:=\alpha t^{\alpha -1} \int _0^{\infty } \tau \Phi _{\alpha }(\tau )S(\tau t^{\alpha })w d\tau \end{array}\right. }$$

(1.7)

for $w\in X_{\theta }(r)$. Note that if $w\in L^r(\Omega )$ for some $1\le r<\infty$, then we can also define in the same way as (1.7) (cf.[9]). Moreover, by definition the operator $S_\alpha$ is strongly continuous, i.e.,

$$\begin{aligned} \displaystyle \lim _{t\rightarrow 0} \Vert S_\alpha (t)w-w\Vert _{X}=0 \, \, {\text {for}}\,\, w\in X, \end{aligned}$$

(1.8)

where $X=X_{\theta }(r)$, $0\le \theta \le 1$ and $1<r<\infty$, or $X=L^r(\Omega )$, $1\le r<\infty$.

Definition 1.1

(Mild solution) By a mild solution of (1.1) on [0, T) we mean that the measurable functions (u, v) have the following properties:

• (a) $u(t)=u(\cdot ,t)\in L^1(\Omega )$ and $v(t)=v(\cdot ,t)\in L^1(\Omega )$ for $t\in (0,T)$,

• (b) $f_1(t,v(t))=f_1(\cdot ,t,v(\cdot ,t))\in L^1(\Omega )$ and $f_2(t,u(t))=f_2(\cdot ,t,u(\cdot ,t))\in L^1(\Omega )$ for almost all $t\in (0,T)$,

• (c) $\int _0^t \left\| f_1(s,v(s))\right\| _{L^1(\Omega )}ds<\infty$ and $\int _0^t \left\| f_2(s,u(s))\right\| _{L^1(\Omega )} ds<\infty$ for $t\in (0,T)$,

• (d) the functions (u, v) satisfy

  $$\begin{aligned} u(t)&= S_{\alpha _1}(t)u_0 +\displaystyle \int _0^t P_{\alpha _1}(t-s) f_1(s,v(s)) ds \, \, {\text {in}} \, \,\Omega \times (0,T),\\ v(t)&= S_{\alpha _2}(t)v_0 +\displaystyle \int _0^t P_{\alpha _2}(t-s) f_2(s,u(s)) ds \, \, {\text {in}}\, \, \Omega \times (0,T), \end{aligned}$$

  where the integral terms are absolutely converging Bochner integrals in $L^1(\Omega )$,

• (e) the initial functions $(u_0,v_0)$ satisfy

  $$\begin{aligned} \Vert u(t)-u_0\Vert _{L^{r_1}(\Omega )} \rightarrow 0 \ \ (t\rightarrow 0) \, \, {\text {and}}\, \, \Vert v(t)-v_0\Vert _{L^{r_2}(\Omega )} \rightarrow 0 \ \ (t\rightarrow 0) \end{aligned}$$

  for $(u_0,v_0)\in L^{r_1}(\Omega )\times L^{r_2}(\Omega )$, if $1\le r_1,r_2<\infty$.

It follows from (1.8) and [9, Remark 3.1.2] that the property of Definition 1.1 (e) holds if and only if

$$\begin{aligned} \lim _{t\rightarrow 0}\Vert u(t)-S_{\alpha _1}(t)u_0\Vert _{L^{r_1}(\Omega )}=0 \ \ \, \,{\text {and}} \, \, \lim _{t\rightarrow 0}\Vert v(t)-S_{\alpha _2}(t)v_0\Vert _{L^{r_2}(\Omega )}=0, \end{aligned}$$

(1.9)

which are equivalent to the convergence in the norms of the integral terms in Definition 1.1 (d) to 0.

We are ready to state our main results.

Theorem 1.2

(Local in time existence) Let $N \ge 1$, $0<\alpha _1\le \alpha _2<1$, $1<p_1, p_2<\infty$, $q_1, q_2\in (1,\infty ]\cap \left( \dfrac{N}{2},\infty \right]$, $1<r_1, r_2<\infty$, $m_1\in [0,\alpha _1)$ and $m_2\in [0,\alpha _2).$ Suppose that Assumption A holds. Put

$$\begin{aligned} \mathcal {P}&:=\frac{N}{2}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right) +\frac{m_1}{\alpha _1},\ \ \mathcal {Q}:=\frac{N}{2}\left( \frac{p_2}{r_1}-\frac{1}{r_2}+\frac{1}{q_2}\right) +\frac{m_2}{\alpha _2}\, \, {\text {and}} \, \, \mathcal {R}&:=\frac{N}{2}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right) +\frac{m_1}{\alpha _2} +\left( 1-\frac{\alpha _1}{\alpha _2}\right) \left\{ 1-\frac{N}{2}\left( 1-\frac{1}{r_1}\right) \right\} . \end{aligned}$$

Suppose that one of the following holds:

• (a) $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}<1$,

• (b) $\alpha _1=\alpha _2$ and $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}=1$,

• (c) $\alpha _1<\alpha _2$ and $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}=1,$ where we also suppose that $m_1\in (0,\alpha _1)$ when $\mathcal {P}=1$.

Then for any $(u_0,v_0)\in L^{r_1}(\Omega )\times L^{r_2}(\Omega )$, there exist $T>0$ and a unique local in time mild solution (u, v) of (1.1) in the sense of Definition 1.1on the interval [0, T).

We also obtain the following nonexistence result.

Theorem 1.3

(Local in time nonexistence) Let $N \ge 1$, $0<\alpha _1\le \alpha _2<1$, $0<p_1, p_2<\infty$, $q_1, q_2\in [1,\infty ]$, $1\le r_1, r_2<\infty$, $m_1\in [0,\alpha _1)$ and $m_2\in [0,\alpha _2).$ Put

$$\begin{aligned} \mathcal {Q}_1:=\frac{N\alpha _1}{2\alpha _2}\left( \frac{p_2}{r_1}-\frac{1}{r_2}+\frac{1}{q_2}\right) +\frac{m_2}{\alpha _2}\, \, {\text {and}} \, \, \mathcal {Q}_2:=\frac{N}{2}\left( \frac{\alpha _1p_2}{\alpha _2r_1}-\frac{1}{r_2}+\frac{1}{q_2}\right) +\frac{m_2}{\alpha _2}. \end{aligned}$$

Put $\mathcal {P}$ and $\mathcal {R}$ in the same way as in Theorem 1.2. Suppose that $\max \{\mathcal {P},\mathcal {Q}_1,\mathcal {Q}_2,\mathcal {R}\}>1$. Then there exist nonnegative functions $(c_1,c_2,u_0,v_0)\in L^{q_1}(\Omega )\times L^{q_2}(\Omega )\times L^{r_1}(\Omega )\times L^{r_2}(\Omega )$ such that, for every $T>0,$ the problem (1.1) with $f_1(x,t,v)=c_1(x)\cdot t^{-m_1}v^{p_1}$ and $f_2(x,t,u)=c_2(x)\cdot t^{-m_2}u^{p_2}$ admits no local in time nonnegative mild solution (u, v) in the sense of Definition 1.1on the interval [0, T).

If $1<p_1, p_2<\infty$, then the nonlinear terms $f_1(x,t,v)=c_1(x)\cdot t^{-m_1}v^{p_1}$ and $f_2(x,t,u)=c_2(x)\cdot t^{-m_2}u^{p_2}$ mentioned in Theorem 1.3 satisfy Assumption A with $\mathbb {R}$ replaced by $[0,\infty )$.

We deduce the following corollary from Theorems 1.2 and 1.3.

Corollary 1.4

Let $N \ge 1$ and $0<\alpha _1=\alpha _2<1.$ Then the following are true:

• (i) Let $1<p_1, p_2<\infty$, $q_1, q_2\in (1,\infty ]\cap \left( \dfrac{N}{2},\infty \right]$, $1<r_1, r_2<\infty$, $m_1\in [0,\alpha _1)$ and $m_2\in [0,\alpha _2).$ Suppose that Assumption A holds. Put $\mathcal {P}$ and $\mathcal {Q}$ in the same way as in Theorem 1.2. If $\max \{\mathcal {P},\mathcal {Q}\}\le 1,$ then for any $(u_0,v_0)\in L^{r_1}(\Omega )\times L^{r_2}(\Omega ),$ there exist $T>0$ and a unique local in time mild solution (u, v) of (1.1).

• (ii) Let $0<p_1, p_2<\infty$, $q_1, q_2\in [1,\infty ]$, $1\le r_1, r_2<\infty$, $m_1\in [0,\alpha _1)$ and $m_2\in [0,\alpha _2).$ If $\max \{\mathcal {P},\mathcal {Q}\}>1,$ then there exist nonnegative functions $(c_1,c_2,u_0,v_0)\in L^{q_1}(\Omega )\times L^{q_2}(\Omega )\times L^{r_1}(\Omega )\times L^{r_2}(\Omega )$ such that, for every $T>0,$ the problem (1.1) with $f_1(x,t,v)=c_1(x)\cdot t^{-m_1}v^{p_1}$ and $f_2(x,t,u)=c_2(x)\cdot t^{-m_2}u^{p_2}$ admits no local in time nonnegative mild solution (u, v).

Corollary 1.4 implies that when $\alpha _1=\alpha _2$, we can explicitly determine the existence/nonexistence of a solution. Our conditions cover all the cases $(p_1,p_2,q_1,q_2,m_1,m_2)$ in Assumption A. Moreover, Corollary 1.4 leads to the following pure power case result, which corresponds to [15].

Corollary 1.5

Let $N \ge 1$ and $0<\alpha _1=\alpha _2<1.$ Then the following are true:

• (i) Let $1<p_1, p_2<\infty$ and $1<r_1, r_2<\infty$. Put

  $$\begin{aligned} \widetilde{\mathcal {P}}:=\frac{N}{2}\left( \frac{p_1}{r_2}-\frac{1}{r_1}\right) \, \, {\text {and}} \, \, \widetilde{\mathcal {Q}}:=\frac{N}{2}\left( \frac{p_2}{r_1}-\frac{1}{r_2}\right) . \end{aligned}$$

  If $\max \{\widetilde{\mathcal {P}},\widetilde{\mathcal {Q}}\}\le 1,$ then for any $(u_0,v_0)\in L^{r_1}(\Omega )\times L^{r_2}(\Omega )$, there exist $T>0$ and a unique local in time mild solution (u, v) of (1.1) with $f_1(x,t,v)=|v|^{p_1-1}v$ and $f_2(x,t,u)=|u|^{p_2-1}u$.

• (ii) Let $0<p_1, p_2<\infty$ and $1\le r_1, r_2<\infty$. If $\max \{\widetilde{\mathcal {P}},\widetilde{\mathcal {Q}}\}>1,$ then there are nonnegative functions $(u_0,v_0)\in L^{r_1}(\Omega )\times L^{r_2}(\Omega )$ such that, the problem (1.1) with $f_1(x,t,v)=v^{p_1}$ and $f_2(x,t,u)=u^{p_2}$ has no local in time nonnegative mild solution on any time interval.

Let us recall fundamental properties of scalar problems. Fractional in time parabolic equations with nonlinear terms have not been well studied until recently. Gal–Warma [9] has studied the fractional in time scalar problem

$${\left\{ \begin{array}{ll} \partial ^{\alpha }_t u=Au+f(x,t,u) &{} {\text {in}} \, \,\Omega \times (0,T), \\ u(x,0)=u_0 (x) &{} {\text {in}} \, \, \Omega , \end{array}\right. }$$

(1.10)

where A is a differential operator which generates a strongly continuous semigroup on $L^2 (\Omega )$. Detailed results can be found in [1, 3, 8, 12]. Let $1\le p<\infty$ and $q_1,q_2\in [1,\infty ]$ be given constants. In [9] the authors assumed that there exists a nonnegative function $c\in L_{q_1,q_2}$ such that the following hold:

(F1’):

$f(x,t,\cdot ):\mathbb {R}\rightarrow \mathbb {R}$ is a measurable function such that

$$\begin{aligned} |f(x,t,\xi )|\le c(x,t)\ (1+|\xi |)^{p} \, \, {\text {for}}\, \, \xi \in \mathbb {R}, \, \, {\text {a.e.}} (x,t)\in \Omega \times (0,\infty ), \end{aligned}$$

(F2’):

f satisfies the local Lipschitz condition

$$\begin{aligned} |f(x,t,\xi )-f(x,t,\eta )|\le c(x,t)\ (1+|\xi |+|\eta |)^{p-1}|\xi -\eta | \, \, {\text {for}}\, \, \xi ,\eta \in \mathbb {R}, \, \,{\text {a.e.}}\, \, (x,t)\in \Omega \times (0,\infty ). \end{aligned}$$

Here $L_{q_1,q_2}$ denotes the Banach space defined by

$$\begin{aligned} L_{q_1,q_2}:=\left\{ c:\Omega \times (0,\infty )\rightarrow \mathbb {R}\ \, \,{\text {measurable}}, \ \ \Vert c\Vert _{L_{q_1,q_2}}:=\displaystyle \sup _{\begin{subarray}{c} t_1,t_2\in (0,\infty ),\\ 0\le t_2-t_1\le 1 \end{subarray}} \left( \displaystyle \int _{t_1}^{t_2} \Vert c(\cdot ,s)\Vert ^{q_2}_{L^{q_1}(\Omega )} ds \right) ^{\frac{1}{q_2}}<\infty \right\} \end{aligned}$$

for $q_1\in [1,\infty ]$ and $q_2\in [1,\infty )$ with the obvious modifications when $q_2=\infty$. Proposition 2.2.2 of [9, p.21, 22] states the following existence result. Let $0<\alpha <1$, $1\le p<\infty$, $q_1\in [1,\infty ]\cap (\beta _A,\infty ]$ and $q_2\in \left( \dfrac{1}{\alpha },\infty \right]$. The constant $\beta _A$ is related to the $L^p$-$L^q$ estimate of the semigroup generated by A. Assume (F1’) and (F2’). If

$$\begin{aligned}&\beta _A\left( \frac{p-1}{r}+\frac{1}{q_1}\right) +\frac{1}{\alpha q_2}<1 \, \, {\text {and}}\, \, 1\le p,r<\infty , \, \, {\text {or}} \\&\beta _A\left( \frac{p-1}{r}+\frac{1}{q_1}\right) +\frac{1}{\alpha q_2}=1 \, \, {\text {and}} \, \, 1<p,r<\infty , \end{aligned}$$

then for any $u_0\in L^{r}(\Omega )$, there exist $T>0$ and a unique local in time solution of (1.10). Note that if $\Omega$ is a bounded domain in ${\mathbb {R}^N}$, $N\ge 1$, with $C^2$ boundary, $A=\Delta$ and $f(x,t,u)=|u|^{p-1}u$, $p>1$, then $\beta _A=\dfrac{N}{2}$, $q_1=q_2=\infty$ and hence this result corresponds to the existence part in [20].

In [9, Remark 5.0.2], based on [20, 21], they conjectured the nonexistence of a local in time solution of (1.10) in the super-critical case

$$\begin{aligned} \beta _A\left( \frac{p-1}{r}+\frac{1}{q_1}\right) +\frac{1}{\alpha q_2}>1. \end{aligned}$$

(1.11)

We give an affirmative answer to the conjecture when $A=\Delta$. Sect. 5 is devoted to this nonexistence result.

Let us explain a sketch of the proofs. The main points of the proofs are Cauchy sequences for the existence part, including

$$\begin{aligned} u_n (t) &= S_{\alpha _1}(t) u_0 +\displaystyle \int _0^t P_{\alpha _1}(t-s) f_1 (s, v_{n-1} (s)) ds, \end{aligned}$$

(1.12)

$$\begin{aligned} v_n (t) &= S_{\alpha _2}(t) v_0 +\displaystyle \int _0^t P_{\alpha _2}(t-s) f_2 (s, u_{n-1} (s)) ds \end{aligned}$$

(1.13)

for $n\ge 2$, $u_1=v_1=0$, and the contradiction argument for the nonexistence part.

For the existence part by induction method we can show that if $T>0$ is sufficiently small, then $\{u_n\}^{\infty }_{n=1}$ and $\{v_n\}^{\infty }_{n=1}$ are Cauchy sequences. Then limits u and v of the sequences exist and (u, v) is a mild solution of (1.1) in the sense of Definition 1.1. In order to show that these are indeed Cauchy sequences, it is crucial to find various exponents including $\theta _1$, $\theta _2$ and $\theta _3$. However, it is not obvious how to find these exponents. Section 3 addresses this aspect.

For the nonexistence part we construct initial data $(u_0,v_0)$. Assume that (1.1) has a local in time nonnegative mild solution (u, v). Since the singularity of the constructed functions are strong, the norm of at least one integral term in Definition 1.1 (d) diverges as $t\rightarrow 0$, which follows from estimates of $S_{\alpha }(t)$ and $P_{\alpha }(t)$. This is a contradiction. It is known (cf. [16, p.440]) that there exists a positive $C^\infty$-function $G_{\Omega } : \Omega \times \Omega \times (0,\infty )\rightarrow \mathbb {R}$ (Dirichlet heat kernel) such that

$$\begin{aligned} \left( S(t)\phi \right) (x)=\displaystyle \int _\Omega G_{\Omega } (x,y,t)\phi (y) dy \end{aligned}$$

for $\phi \in L^r (\Omega )$, $1\le r\le \infty$. After that, we abbreviate $G_{\Omega }$ as G. Since G has a lower bound with respect to t, we obtain estimates of $S_{\alpha }(t)$ and $P_{\alpha }(t)$ from (1.7).

This paper is organized as follows. In Sect. 2 we give and recall some properties of $S_{\alpha }(t)$, $P_{\alpha }(t)$ and the Dirichlet heat kernel. In Sects. 3 and 4 we use these properties and prove Theorems 1.2 and 1.3, respectively. In Sect. 5 we give a nonexistence result for scalar problems. In Sect. 6 we discuss our results and explain possible future problems ensuing from the current analysis.

2 Preliminaries

For any set $\mathcal {X}$ and the mappings $a=a(x)$ and $b=b(x)$ from $\mathcal {X}$ to $[0,\infty )$, we say

$$\begin{aligned} a(x) \,\, \lesssim \,\, b(x) \, \, {\text {for all}}\, \, x\in \mathcal {X} \end{aligned}$$

if there exists a positive constant C such that $a(x) \le Cb(x)$ for all $x\in \mathcal {X}$.

Proposition 2.1

Let $0<\alpha <1,$ $1<r<\infty$ and $-1\le \theta _1\le \theta _2\le 1.$ Then the following are true:

• (i) If $\theta _2-\theta _1<1,$ then there exists $C>0$ such that

  $$\begin{aligned} |S_{\alpha }(t)w|_{X_{\theta _2}(r)}\le Ct^{\alpha (\theta _1-\theta _2)}|w|_{X_{\theta _1}(r)} \end{aligned}$$

  for $t>0$ and $w\in X_{\theta _1}(r).$

• (ii) If $\theta _1>-1$ or $\theta _2<1,$ then there exists $C>0$ such that

  $$\begin{aligned} |t^{1-\alpha }P_{\alpha }(t)w|_{X_{\theta _2}(r)}\le Ct^{\alpha (\theta _1-\theta _2)}|w|_{X_{\theta _1}(r)} \end{aligned}$$

  for $t>0$ and $w\in X_{\theta _1}(r)$.

Proof

We prove (i) in a similar manner to [9, Proposition 2.2.2]. Using (1.5), (1.7) and [15, Theorem 51.1 (iv)], we have

$$\begin{aligned} |S_{\alpha }(t)w|_{X_{\theta _2}(r)}&\le \displaystyle \int _0^{\infty } \Phi _{\alpha }(\tau )|S(\tau t^{\alpha })w|_{X_{\theta _2}(r)} d\tau \\&\lesssim \displaystyle \int _0^{\infty } \Phi _{\alpha }(\tau ) \tau ^{\theta _1-\theta _2} t^{\alpha (\theta _1-\theta _2)}|w|_{X_{\theta _1}(r)} d\tau \\&=t^{\alpha (\theta _1-\theta _2)}|w|_{X_{\theta _1}(r)}\displaystyle \int _0^{\infty } \Phi _{\alpha }(\tau ) \tau ^{\theta _1-\theta _2} d\tau \\&=\frac{\Gamma (1+\theta _1-\theta _2)}{\Gamma (1+\alpha (\theta _1-\theta _2))}t^{\alpha (\theta _1-\theta _2)}|w|_{X_{\theta _1}(r)}. \end{aligned}$$

We can obtain the assertion (ii) in the same way as the assertion (i). $\square$

Lemma 2.2

Let $0<\alpha _1\le \alpha _2<1$ and $0<T<\infty .$ For $i=1,2,$ let $0<\theta _i<1$, $1<r_i<\infty$ and $\Pi _i \subset X_0(r_i)$ $(=L^{r_i}(\Omega )).$ Suppose that for $i=1,2$,

$$\begin{aligned} \kappa (\Pi _i):=\left\{ u|u|^{-1}_{X_0(r_i)}: u\in \Pi _i, \ u\ne 0\right\} \end{aligned}$$

is precompact in $X_0(r_i).$ Then there exists a continuous and nondecreasing function $g:(0,T)\rightarrow (0,\infty ),$ depending on $\alpha _i$, $\theta _i$, $r_i$ and $\Pi _i$ $(i=1,2)$ such that the following are true:

• (i) For $i=1,2,$ the following is true:

  $$\begin{aligned} |S_{\alpha _i}(t)u|_{X_{\theta _i}(r_i)} \,\, \lesssim \,\, g(t)\cdot t^{-\alpha _2\theta _i}|u|_{X_0(r_i)} \, \, {\text {for}}\, \, 0<t<T \,\, and \,\,u\in \Pi _i. \end{aligned}$$

• (ii) We have $\lim _{t\rightarrow 0}g(t)=0$. For $i=1,2,$ the function $w_i=w_i (t)$ defined by

  $$\begin{aligned} (w_i (t))^{-\alpha _2\theta _i}=g(t)\cdot t^{-\alpha _2\theta _i} \end{aligned}$$

  has the properties

  $$\begin{aligned} \lim _{t\rightarrow 0}w_i (t)=0 \, \, {\text {and}}\, \, \min \left\{ t,t^{1-\frac{\min \{\theta _1,\theta _2\}}{2\theta _i}}\right\} \le w_i (t)\le t^{1-\frac{\min \{\theta _1,\theta _2\}}{2\max \{\theta _1,\theta _2\}}}. \end{aligned}$$

Proof

Define

$$\begin{aligned} {\left\{ \begin{array}{ll} h_1(t,u):=|S_{\alpha _1}(t)u|_{X_{\theta _1}(r_1)}C_0^{-1}t^{\alpha _1\theta _1}|u|_{X_0(r_1)}^{-1} &{} {\text {for}}\, \, (t,u)\in (0,T)\times \Pi _1\backslash \{0\},\\ \overline{h_1}(t):=\sup \left\{ h_1(s,u): s\in (0,t],\ u\in \Pi _1\backslash \{0\}\right\} &{} {\text {for}}\, \, t\in (0,T), \end{array}\right. } \end{aligned}$$

where $C_0(=C)>0$ is the constant from Proposition 2.1 (i). We observe from [9, Lemma A.0.2] that $0\le \overline{h_1}\le 1$, $\lim _{t\rightarrow 0}\overline{h_1}(t)=0$ and

$$\begin{aligned} |S_{\alpha _1}(t)u|_{X_{\theta _1}(r_1)}& \le \,\, C_0\overline{h_1}(t)\cdot t^{-\alpha _1\theta _1}|u|_{X_0(r_1)}\\& \lesssim \,\, C_0\overline{h_1}(t)\cdot t^{-\alpha _2\theta _1}|u|_{X_0(r_1)} \, \, {\text {for}}\, \, 0<t<T \, \, {\text {and}}\, \, u\in \Pi _1. \end{aligned}$$

Put $\overline{h_2}$ in the same way. We set

$$\begin{aligned} g(t):=\max \left\{ \overline{h_1}(t), \overline{h_2}(t), t^{\delta \alpha _2\theta _1}, t^{\delta \alpha _2\theta _2}\right\} , \end{aligned}$$

where $\delta :=\dfrac{\min \{\theta _1,\theta _2\}}{2\max \{\theta _1,\theta _2\}}$.

It suffices to prove the estimates of $w_i(t)$ for $i=1,2$. Since $g(t)\ge t^{\delta \alpha _2\theta _1}$, we obtain

$$\begin{aligned} w_1(t)=g(t)^{-\frac{1}{\alpha _2\theta _1}}\cdot t\le t^{1-\delta } \, \, {\text {for}} \,\, t>0. \end{aligned}$$

On the other hand, let $0<t<\min \{T,1\}$. Due to $\overline{h_1},\overline{h_2}\le 1$, we have $g(t)\le 1$ and hence $w_1(t)\ge t$. If $t\ge 1$, then $g(t)=t^{\delta \alpha _2\max \{\theta _1,\theta _2\}}$. Thus it follows that

$$\begin{aligned} w_1(t)=t^{1-\frac{\delta \max \{\theta _1,\theta _2\}}{\theta _1}} =t^{1-\frac{\min \{\theta _1,\theta _2\}}{2\theta _1}}. \end{aligned}$$

Therefore, we deduce the desired estimate of $w_1(t)$. We can obtain the estimate of $w_2(t)$ in the same way.$\square$

Proposition 2.3

([16, Proposition 49.10]) Let $N\ge1$ and $\Omega$ be an arbitrary domain in ${\mathbb {R}^N}$. There exist constants $c_1 > 0$ and $c_2 \ge 2$ depending only on N,  such that the Dirichlet heat kernel G(x, y, t) in $\Omega$ satisfies

$$\begin{aligned} G(x,y,t)\ge c_1 t^{-\frac{N}{2}} \end{aligned}$$

for $t>0$ and $x,y\in \Omega$ such that

$$\begin{aligned} dist(x,\partial \Omega )\ge c_2 \sqrt{t} \, \, {\text {and}}\, \, |x-y|\le \sqrt{t}. \end{aligned}$$

3 Existence result

Proposition 3.1

Let $N \ge 1$, $0<\alpha _1\le \alpha _2<1$, $1<p_1, p_2<\infty$, $q_1, q_2\in (1,\infty ]\cap \left( \dfrac{N}{2},\infty \right]$, $1<r_1, r_2<\infty$, $m_1\in [0,\alpha _1)$ and $m_2\in [0,\alpha _2)$. Put $\mathcal {P}$, $\mathcal {Q}$ and $\mathcal {R}$ in the same way as in Theorem 1.2. If $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}<1,$ then there exist $(s_1,\theta _2,z_1,z_2)$, $(s_2,\theta _1,z_3,z_4)\in (0,1)\times (0,1)\times (1,\infty )\times (1,\infty )$ such that

$$\begin{aligned}&\frac{1}{z_1}-\frac{2}{N}(1-s_1)=\frac{1}{r_1}, \ \frac{1}{q_1}+\frac{p_1}{z_2}=\frac{1}{z_1} \, \, {\text {and}}\ \ \frac{1}{r_2}-\frac{2}{N}\theta _2\le \frac{1}{z_2}, \end{aligned}$$

(3.1)

$$\begin{aligned}&\alpha _1 s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1>0,\end{aligned}$$

(3.2)

$$\begin{aligned}&m_1+p_1\alpha _2\theta _2<1, \end{aligned}$$

(3.3)

$$\begin{aligned}&\theta _1<s_1, \end{aligned}$$

(3.4)

$$\begin{aligned}&\frac{1}{z_3}-\frac{2}{N}(1-s_2)=\frac{1}{r_2}, \ \frac{1}{q_2}+\frac{p_2}{z_4}=\frac{1}{z_3} \, \, {\text {and}}\ \ \frac{1}{r_1}-\frac{2}{N}\theta _1\le \frac{1}{z_4}, \end{aligned}$$

(3.5)

$$\begin{aligned}&\alpha _2 s_2-m_2-p_2\alpha _2\theta _1>0, \end{aligned}$$

(3.6)

$$\begin{aligned}&\theta _2<s_2. \end{aligned}$$

(3.7)

Moreover, if $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}=1,$ then there exist $(s_1,\theta _2,z_1,z_2)$, $(s_2,\theta _1,z_3,z_4)\in (0,1)\times (0,1)\times (1,\infty )\times (1,\infty )$ such that (3.1)–(3.7) hold with (3.2) and (3.6) replaced by $\alpha _1 s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1\ge 0$ and $\alpha _2 s_2-m_2-p_2\alpha _2\theta _1\ge 0,$ respectively.

Proof

Suppose that $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}<1$. For $1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right)<s<\min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_1}-\dfrac{1}{r_1}\right) \right\}$, put

$$\begin{aligned} \theta _2(s):= & {} \frac{N}{2p_1}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right) -\frac{1}{p_1}(1-s), \\ \frac{1}{z_1(s)}:= & {} \frac{2}{N}(1-s)+\frac{1}{r_1} \, \, {\text {and}}\\ \frac{1}{z_2(s)}:= & {} \frac{1}{p_1}\left\{ \frac{1}{r_1}-\frac{1}{q_1}+\frac{2}{N}(1-s)\right\} . \end{aligned}$$

We see that

$$\begin{aligned} \max \left\{ \frac{1}{q_1},\frac{1}{r_1}\right\}<\frac{1}{z_1(s)}<1 \, \, {\text {and}}\, \, \max \left\{ \frac{1}{p_1}\left( \frac{1}{r_1}-\frac{1}{q_1}\right) ,0 \right\}<\frac{1}{z_2(s)}<\frac{1}{p_1}\left( 1-\frac{1}{q_1}\right) . \end{aligned}$$

Hence, $z_1(s)\in (1,\infty )$ and $z_2(s)\in (1,\infty )$. Let $(s_1,\theta _2,z_1,z_2)=(s,\theta _2(s),z_1(s),z_2(s))$. By direct calculation we have (3.1). In the same way we derive (3.5) with $(s_2,\theta _1,z_3,z_4)= (\tilde{s},\theta _1(\tilde{s}),z_3(\tilde{s}),z_4(\tilde{s}))$, where

$$\begin{aligned} \theta _1(\tilde{s}):= & {} \dfrac{N}{2p_2}\left( \dfrac{p_2}{r_1}-\dfrac{1}{r_2}+\dfrac{1}{q_2}\right) -\dfrac{1}{p_2}(1-\tilde{s}),\\ \dfrac{1}{z_3(\tilde{s})}:= & {} \dfrac{2}{N}(1-\tilde{s})+\dfrac{1}{r_2} \, \, \text {and}\\ \dfrac{1}{z_4(\tilde{s})}:= & {} \dfrac{1}{p_2}\left\{ \dfrac{1}{r_2}-\dfrac{1}{q_2}+\dfrac{2}{N}(1-\tilde{s})\right\} \end{aligned}$$

for $1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_2}\right)<\tilde{s}<\min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right) \right\}$. Moreover, we see that (3.3) and (3.6) follow from $\mathcal {P}<1$ and $\mathcal {Q}<1$, respectively.

Substituting $(s_1,\theta _2,s_2,\theta _1)=(s,\theta _2(s),\tilde{s},\theta _1(\tilde{s}))$ into (3.4) and (3.7), we obtain

$$\begin{aligned}&\frac{N}{2p_1}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right) -\frac{1}{p_1}(1-s)<\tilde{s}, \end{aligned}$$

(3.8)

$$\begin{aligned}&\frac{N}{2p_2}\left( \frac{p_2}{r_1}-\frac{1}{r_2}+\frac{1}{q_2}\right) -\frac{1}{p_2}(1-\tilde{s})<s. \end{aligned}$$

(3.9)

We show that (3.8) holds with $(s,\tilde{s})=\left( \max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) ,\dfrac{m_1}{\alpha _1}\right\} , \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right) \right\} \right)$. Since $\mathcal {P}<1$, we have

$$\begin{aligned} \frac{N}{2p_1}\left( \frac{p_1}{r_2}-1+\frac{1}{q_1}\right)<\frac{N}{2p_1}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right)<\frac{1}{p_1}\left( 1-\frac{m_1}{\alpha _1}\right) <1. \end{aligned}$$

Moreover, it follows from $q_1, q_2\in (1,\infty ]\cap \left( \dfrac{N}{2},\infty \right]$ that $\dfrac{N}{2p_1}\left( \dfrac{p_1}{r_2}-1+\dfrac{1}{q_1}\right)<\dfrac{N}{2r_2}<1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right)$. Thus (3.8) holds when $s=1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right)$. Since $\mathcal {P}<1$, the left hand side of (3.8) is negative when $s=\dfrac{m_1}{\alpha _1}$. Therefore, (3.8) holds with $(s,\tilde{s})=\left( \max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) ,\dfrac{m_1}{\alpha _1}\right\} , \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right) \right\} \right)$. In the same way (3.9) holds with $(s,\tilde{s})=\left( \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_1}-\dfrac{1}{r_1}\right) \right\} ,\max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_2}\right) ,\dfrac{m_2}{\alpha _2}\right\} \right)$.

We divide the possibilities into two cases: $\alpha _1=\alpha _2$ and $\alpha _1<\alpha _2$.

Let $\alpha _1=\alpha _2$. Since $\mathcal {P}<1$, we obtain (3.2). Put

$$\begin{aligned} I_1&:=\left( \max \left\{ 1-\frac{N}{2}\left( 1-\frac{1}{r_1}\right) , \frac{m_1}{\alpha _1}\right\} ,\min \left\{ 1,1-\frac{N}{2}\left( \frac{1}{q_1}-\frac{1}{r_1}\right) \right\} \right) ,\\ I_2&:=\left( \max \left\{ 1-\frac{N}{2}\left( 1-\frac{1}{r_2}\right) , \frac{m_2}{\alpha _2}\right\} ,\min \left\{ 1,1-\frac{N}{2}\left( \frac{1}{q_2}-\frac{1}{r_2}\right) \right\} \right) \, \, \text {and} \, \, R:={I_1}\times {I_2}. \end{aligned}$$

The area of $(s,\tilde{s})$ satisfying (3.8) and (3.9) is the shaded portion on the graphs. See Fig. 1. Due to the above calculation results, the position of R is as shown in the left figure, but not as shown in the right one. Thus there exists $(s,\tilde{s})\in R$ such that (3.8) and (3.9) hold.

Fig. 1

Possible relative positions of the rectangle R and sector defined by (3.8)–(3.9)

It remains to consider $\theta _1$ and $\theta _2$. We observe from (3.8) and (3.9) that $\theta _2(s)<1$ and $\theta _1(\tilde{s})<1$. If $\theta _2(s)\le 0$, then we replace $\theta _2=\theta _2(s)$ with $\theta _2=\varepsilon >0$, where $\varepsilon$ is sufficiently small such that $\alpha _1 s_1-m_1-p_1\alpha _2\varepsilon >0$ and $\varepsilon <\tilde{s}$. We see that (3.1), (3.2), (3.3) and (3.7) hold. If $\theta _1(\tilde{s})\le 0$, then we can replace $\theta _1(\tilde{s})$ with $\theta _1\in (0,1)$ in the same way.

Let $\alpha _1<\alpha _2$. We recall that

$$\begin{aligned} (s_1,\theta _2,z_1,z_2)=(s,\theta _2(s),z_1(s),z_2(s)) \, \, {\text {and}}\, \, (s_2,\theta _1,z_3,z_4)=(\tilde{s},\theta _1(\tilde{s}),z_3(\tilde{s}),z_4(\tilde{s})). \end{aligned}$$

We divide the possibilities into two cases: (i) $\dfrac{p_2}{r_1}-\dfrac{1}{r_2}+\dfrac{1}{q_2}>0$ and (ii) $\dfrac{p_2}{r_1}-\dfrac{1}{r_2}+\dfrac{1}{q_2}\le 0$.

We prove (i). It follows that (3.1), (3.3), (3.5) and (3.6) hold. The inequality (3.2) is equivalent to

$$\begin{aligned} \frac{N}{2p_2}\left( \frac{p_2}{r_1}-\frac{1}{r_2}+\frac{1}{q_2}\right) -\frac{1}{p_2}(1-\tilde{s})>s+\frac{\frac{N}{2}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right) +\frac{m_1}{\alpha _2}-1}{1-\frac{\alpha _1}{\alpha _2}}, \end{aligned}$$

(3.10)

which holds with $(s,\tilde{s})= \left( \max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) ,\dfrac{m_1}{\alpha _1}\right\} , \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right) \right\} \right)$. Indeed, the right hand side of (3.10) is negative when $s=\dfrac{m_1}{\alpha _1}$ (resp. when $s=1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right)$), since $\mathcal {P}<1$ (resp. since $\mathcal {R}<1$). On the other hand, since $\dfrac{p_2}{r_1}-\dfrac{1}{r_2}+\dfrac{1}{q_2}>0$, the left hand side of (3.10) is positive.

The area of $(s,\tilde{s})$ satisfying (3.8), (3.9) and (3.10) is the shaded portion on the graphs. See Fig. 2. Here, R is defined in the same way as when $\alpha _1=\alpha _2$. Due to the above calculation results, the position of R is as shown in the left figure, but not as shown in the right one. Thus there exists $(s,\tilde{s})\in R$ such that (3.8), (3.9) and (3.10) hold.

Fig. 2

Possible relative positions of the rectangle R and the strip defined by (3.8)–(3.10)

It remains to consider $\theta _1$ and $\theta _2$. We observe from (3.8) and (3.9) that $\theta _2(s)<1$ and $\theta _1(\tilde{s})<1$. If $\theta _2(s)\le 0$ or $\theta _1(\tilde{s})\le 0$, then it is sufficient to replace it/them in the same way as when $\alpha _1=\alpha _2$.

We prove (ii). Let $\varepsilon >0$ be sufficiently small. Put

$$\begin{aligned} s:=\max \left\{ 1-\frac{N}{2}\left( 1-\frac{1}{r_1}\right) , \frac{m_1}{\alpha _1}\right\} +\varepsilon \in I_1 \ \ {\text {and}}\ \ \tilde{s}:=\min \left\{ 1,1-\frac{N}{2}\left( \frac{1}{q_2}-\frac{1}{r_2}\right) \right\} -\varepsilon \in I_2 \end{aligned}$$

Due to $\theta _1(\tilde{s})<0$ in this case, we replace $\theta _1=\theta _1(\tilde{s})$ with $\theta _1=\varepsilon$. It follows that (3.1), (3.3) and (3.5) hold. Since $\theta _1=\varepsilon$ is sufficiently small, $s\ge \varepsilon >0$ and $\tilde{s}>\dfrac{m_2}{\alpha _2}$, we obtain (3.4) and (3.6). The inequality (3.2) is equivalent to (3.10) with the left hand side replaced by $\varepsilon$. Then by $\max \{\mathcal {P},\mathcal {R}\}<1$ we deduce (3.2). Moreover, since (3.8) holds with $(s,\tilde{s})=\left( \max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) ,\dfrac{m_1}{\alpha _1}\right\} , \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right) \right\} \right)$, (3.8) also holds with $(s,\tilde{s})=\left( \max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) ,\dfrac{m_1}{\alpha _1}\right\} +\varepsilon , \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_2}-\dfrac{1}{r_2}\right) \right\} -\varepsilon \right)$.

It remains to consider $\theta _2$. We observe from (3.8) that $\theta _2(s)<1$. If $\theta _2(s)\le 0$, then it is sufficient to replace it in the same way as when $\alpha _1=\alpha _2$.

Since we can prove the case where $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}=1$ in a similar way, we omit the proof.$\square$

Lemma 3.2

Suppose that $\alpha _1<\alpha _2$ and $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}<1$ hold in particular in the assumptions of Proposition 3.1. Let $s_1\in (0,1)$ be chosen in Proposition 3.1. Put

$$\begin{aligned} \mathcal {S}:= \frac{1-\frac{N}{2}\left( \frac{p_1}{r_2}-\frac{1}{r_1}+\frac{1}{q_1}\right) -\frac{m_1}{\alpha _2}}{1-\frac{\alpha _1}{\alpha _2}}. \end{aligned}$$

If $s_1\ge \mathcal {S},$ then the following are true:

• (i) There exists $s_3\in (0,1)$ such that $1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right)<s_3< \mathcal {S}$ and that $\theta _3:=\theta _2(s_3)>0,$ where $\theta _2(s)$ is defined in the proof of Proposition 3.1.

• (ii) $\theta _2(s_1)>0$.

Proof

By direct calculation we have $p_1\alpha _2\theta _2(\mathcal {S})=\alpha _1\mathcal {S}-m_1$. It follows from $\mathcal {P}<1$ that $\mathcal {S}>\dfrac{m_1}{\alpha _1}$. Then $\theta _2(\mathcal {S})>0$ holds. Note that $\mathcal {R}<1$ yields $1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) <\mathcal {S}$. Choosing $\max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right) ,\dfrac{m_1}{\alpha _1}\right\}<s_3<\mathcal {S}$ sufficiently large, we obtain $\theta _2(s_3)>0$. Moreover, we see that

$$\begin{aligned} \max \left\{ 1-\frac{N}{2}\left( 1-\frac{1}{r_1}\right) ,\frac{m_1}{\alpha _1}\right\}<s_3<\mathcal {S}\le s_1<1. \end{aligned}$$

Thus $s_3\in (0,1)$ holds. Since $\theta _2(s)$ is increasing with respect to s, we have $\theta _2(s_1)>\theta _2(s_3)>0$. The proof is complete.$\square$

Proof of Theorem 1.2

Set $u_1:=0$ and $v_1:=0$. For $n\ge 2$, define the functions $u_n$ and $v_n$ by (1.12) and (1.13), respectively. We introduce the Banach space defined by

$$\begin{aligned} Y_{\theta ,r,T}:=\left\{ u\in L^{\infty }_{loc}((0,T],X_{\theta }(r)): \Vert u\Vert _{Y_{\theta ,r,T}}:=\sup _{t\in (0,T)}t^{\alpha _2\theta }|u(t)|_{X_{\theta }(r)}<\infty \right\} \end{aligned}$$

for $0<\theta <1$, $1<r<\infty$ and $T>0$.

We prove the case (a). Choose $(s_1,\theta _2,z_1,z_2)$ and $(s_2,\theta _1,z_3,z_4)$ as in Proposition 3.1.

We consider the existence part. Let $T>0$. Assume that $(u_{n-1},v_{n-1})\in Y_{\theta _1,r_1,T}\times Y_{\theta _2,r_2,T}$. We obtain from Proposition 2.1 that for $0<t<T$,

$$\begin{aligned} t^{\alpha _2\theta _1}|u_{n}|_{X_{\theta _1}(r_1)} & \le \,\, t^{\alpha _2\theta _1} |S_{\alpha _1}(t) u_0|_{X_{\theta _1}(r_1)} +t^{\alpha _2\theta _1}\displaystyle \int _0^t |P_{\alpha _1}(t-s) f_1 (s, v_{n-1} (s))|_{X_{\theta _1}(r_1)}ds\\&\lesssim \,\, |u_0|_{X_0 (r_1)} +t^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} |f_1 (s, v_{n-1} (s))|_{X_{s_1-1}(r_1)}ds. \end{aligned}$$

(3.11)

It follows from (1.6) and the first equality of (3.1) that $X_{1-s_1}(r_1')\hookrightarrow L^{z_1'}(\Omega )$, which implies that $L^{z_1}(\Omega )\hookrightarrow X_{s_1-1}(r_1)$. By the last inequality of (3.1) we have $X_{\theta _2}(r_2)\hookrightarrow L^{z_2}(\Omega )$. Then we can deduce from Assumption A and (3.1) that for $0<s<t$,

$$\begin{aligned} |f_1 (s, v_{n-1} (s))|_{X_{s_1-1}(r_1)}& \lesssim \,\, \Vert f_1 (s, v_{n-1} (s))\Vert _{L^{z_1}(\Omega )} \le \Vert c_1\cdot s^{-m_1}(1+|v_{n-1} (s)|)^{p_1}\Vert _{L^{z_1}(\Omega )}\\&\le s^{-m_1}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert (1+|v_{n-1} (s)|)^{p_1}\Vert _{L^{\frac{z_2}{p_1}}(\Omega )}\\&=s^{-m_1}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1} (s)|\Vert ^{p_1}_{L^{z_2}(\Omega )}\\&\lesssim \, s^{-m_1}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}|1+|v_{n-1} (s)||^{p_1}_{X_{\theta _2}(r_2)}\\&\le s^{-m_1-p_1\alpha _2\theta _2}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1}|\Vert ^{p_1}_{Y_{\theta _2,r_2,T}}, \end{aligned}$$

(3.12)

which yields

$$\begin{aligned} &t^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} |f_1 (s, v_{n-1} (s))|_{X_{s_1-1}(r_1)}ds\\&\lesssim \, t^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} s^{-m_1-p_1\alpha _2\theta _2}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1}|\Vert ^{p_1}_{Y_{\theta _2,r_2,T}}ds\\&\le \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1}|\Vert ^{p_1}_{Y_{\theta _2,r_2,T}}\cdot t^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1}\\&\qquad \times \displaystyle \int _0^1 (1-s)^{\alpha _1(s_1-\theta _1)-1}s^{-m_1-p_1\alpha _2\theta _2}ds \end{aligned}$$

(3.13)

for $0<t<T$. Here we use $\alpha _1(s_1-\theta _1)-1>-1$ and $0<m_1+p_1\alpha _2\theta _2<1$, which follow from (3.4) and (3.3), respectively. Combining (3.2), (3.11) and (3.13), we have $u_n\in Y_{\theta _1,r_1,T}$. We can obtain $v_n\in Y_{\theta _2,r_2,T}$ in the same way. By induction we derive $(u_{n},v_{n})\in Y_{\theta _1,r_1,T}\times Y_{\theta _2,r_2,T}$ for $n\ge 1$.

We divide the possibilities into two cases: (i) $\alpha _1=\alpha _2$, or $\alpha _1<\alpha _2$ and $s_1<\mathcal {S}$ and (ii) $\alpha _1<\alpha _2$ and $s_1\ge \mathcal {S}$. In the case (ii) we also choose $s_3$ and put $\theta _3$ as in Lemma 3.2. Moreover, let us introduce the Banach space defined by

$$\begin{aligned} Y_{\theta _2,\theta _3,r_2,T}:= \bigg \{ u\in L^{\infty }_{loc}((0,T],X_{\theta _2}(r_2) \cap X_{\theta _3}(r_2)): \Vert u\Vert _{Y_{\theta _2,\theta _3,r_2,T}}:= \max \left\{ \Vert u\Vert _{Y_{\theta _2,r_2,T}},\Vert u\Vert _{Y_{\theta _3,r_2,T}}\right\} <\infty \bigg\} . \end{aligned}$$

In a similar way to (3.11), (3.12) and (3.13) we obtain

$$\begin{aligned} &t^{\alpha _2\theta _3}|v_{n}|_{X_{\theta _3}(r_2)}\\& \quad \le \, t^{\alpha _2\theta _3} |S_{\alpha _2}(t) v_0|_{X_{\theta _3}(r_2)} +t^{\alpha _2\theta _3}\displaystyle \int _0^t |P_{\alpha _2}(t-s) f_2 (s, u_{n-1} (s))|_{X_{\theta _3}(r_2)}ds\\&\quad \lesssim \, |v_0|_{X_0 (r_2)} +t^{\alpha _2\theta _3}\displaystyle \int _0^t (t-s)^{\alpha _2(s_2-\theta _3)-1} |f_2 (s, u_{n-1} (s))|_{X_{s_2-1}(r_2)}ds\\&\quad \lesssim \, |v_0|_{X_0 (r_2)}+ t^{\alpha _2\theta _3}\displaystyle \int _0^t (t-s)^{\alpha _2(s_2-\theta _3)-1} s^{-m_2-p_2\alpha _2\theta _1}\cdot \Vert c_2\Vert _{L^{q_2}(\Omega )}\Vert 1+|u_{n-1}|\Vert ^{p_2}_{Y_{\theta _1,r_1,T}}ds\\&\quad \le |v_0|_{X_0 (r_2)}+\Vert c_2\Vert _{L^{q_2}(\Omega )}\Vert 1+|u_{n-1}|\Vert ^{p_2}_{Y_{\theta _1,r_1,T}}\cdot t^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1}\\&\qquad \times \displaystyle \int _0^1 (1-s)^{\alpha _2(s_2-\theta _3)-1}s^{-m_2-p_2\alpha _2\theta _1}ds. \end{aligned}$$

(3.14)

Here we apply $s_2>\theta _2=\theta _2(s_1)>\theta _2(s_3)=\theta _3$, which follows from (3.7) and Lemma 3.2. Then $u_{n-1}\in Y_{\theta _1,r_1,T}$ leads not only to $v_{n}\in Y_{\theta _2,r_2,T}$ but also to $v_{n}\in Y_{\theta _3,r_2,T}$. Thus $v_{n}\in Y_{\theta _2,\theta _3,r_2,T}$ for $n\ge 1$.

We consider the rest of the existence part only in the case (i). In the case (ii) it can be proved by replacing $Y_{\theta _2,r_2,T}$ and $Y_{\theta _2,r_2,T_*}$ with $Y_{\theta _2,\theta _3,r_2,T}$ and $Y_{\theta _2,\theta _3,r_2,T_*}$, respectively.

In a similar way there exist $C_1>0$ and $C_2>0$ such that for $n\ge 2$,

$$\begin{aligned} \begin{aligned} \Vert u_{n+1}-u_n\Vert _{Y_{\theta _1,r_1,T}}&\le C_1 T^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2 +(\alpha _2-\alpha _1)\theta _1} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_n|+|v_{n-1}|\Vert ^{p_1-1}_{Y_{\theta _2,r_2,T}} \Vert v_n-v_{n-1}\Vert _{Y_{\theta _2,r_2,T}},\\ \Vert v_{n+1}-v_n\Vert _{Y_{\theta _2,r_2,T}}&\le C_2 T^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1} \Vert c_2\Vert _{L^{q_2}(\Omega )}\Vert 1+|u_n|+|u_{n-1}|\Vert ^{p_2-1}_{Y_{\theta _1,r_1,T}} \Vert u_n-u_{n-1}\Vert _{Y_{\theta _1,r_1,T}}. \end{aligned} \end{aligned}$$

(3.15)

Put $U:=2\max \left\{ \Vert u_2\Vert _{Y_{\theta _1,r_1,T}}, \Vert v_2\Vert _{Y_{\theta _2,r_2,T}}\right\}$ and choose a sufficiently small time $T_*>0$ such that

$$\begin{aligned} \begin{aligned} C_1 {T_*}^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1} \Vert c_1\Vert _{L^{q_1}(\Omega )}(1+2U)^{p_1-1}&\le \frac{1}{2},\\ C_2 {T_*}^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1} \Vert c_2\Vert _{L^{q_2}(\Omega )}(1+2U)^{p_2-1}&\le \frac{1}{2}. \end{aligned} \end{aligned}$$

(3.16)

Note that $T_*$ is independent of n. By induction together with (3.15) and (3.16) we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert u_n\Vert _{Y_{\theta _1,r_1,T_*}}\le U &{} {\text {for}} \,\,n\ge 1,\\ \Vert v_n\Vert _{Y_{\theta _2,r_2,T_*}}\le U &{} {\text {for}} \,\, n\ge 1,\\ \Vert u_{n+1}-u_n\Vert _{Y_{\theta _1,r_1,T_*}} \le \dfrac{1}{2}\Vert v_n-v_{n-1}\Vert _{Y_{\theta _2,r_2,T_*}} &{} {\text {for}} \,\, n\ge 2,\\ \Vert v_{n+1}-v_n\Vert _{Y_{\theta _2,r_2,T_*}} \le \dfrac{1}{2}\Vert u_n-u_{n-1}\Vert _{Y_{\theta _1,r_1,T_*}} &{} {\text {for}}\,\, n\ge 2. \end{array}\right. } \end{aligned}$$

(3.17)

By iteration in (3.17) the sequences $\{u_n\}^{\infty }_{n=1}$ and $\{v_n\}^{\infty }_{n=1}$ are Cauchy in the Banach spaces $Y_{\theta _1,r_1,T_*}$ and $Y_{\theta _2,r_2,T_*}$, respectively. Consequently, there exist limits $u\in Y_{\theta _1,r_1,T_*}$ and $v\in Y_{\theta _2,r_2,T_*}$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert u_n-u\Vert _{Y_{\theta _1,r_1,T_*}}=0 \,\, {\text {and}}\,\, \lim _{n \rightarrow \infty } \Vert v_n-v\Vert _{Y_{\theta _2,r_2,T_*}}=0. \end{aligned}$$

(3.18)

We show that the limits u and v have all the properties of Definition 1.1 on $[0,T_*)$. Property (a) immediately follows from $u\in Y_{\theta _1,r_1,T_*}$ and $v\in Y_{\theta _2,r_2,T_*}$. We obtain in the same way as we deduce (3.12) that

$$\begin{aligned} \displaystyle \int _0^{T_*} \Vert f_1(s,v(s))\Vert _{L^1(\Omega )} ds&\lesssim \, \displaystyle \int _0^{T_*} \Vert f_1(s,v(s))\Vert _{L^{z_1}(\Omega )} ds\\&\lesssim \, \displaystyle \int _0^{T_*} s^{-m_1-p_1\alpha \theta _2}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v|\Vert ^{p_1}_{Y_{\theta _2,r_2,T_*}}ds\\&=\frac{(T_*)^{1-m_1-p_1\alpha _2\theta _2}}{1-m_1-p_1\alpha _2\theta _2}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v|\Vert ^{p_1}_{Y_{\theta _2,r_2,T_*}}. \end{aligned}$$

Here we use $m_1+p_1\alpha _2\theta _2<1$ by (3.3). Since the same is true for $f_2$, (u, v) satisfies the properties (b) and (c).

We mention the properties (d) and (e). We divide the possibilities into the same two cases as in the existence part.

In the case (i) it follows that

$$\begin{aligned} \alpha _1s_1-m_1-p_1\alpha _2\theta _2>0. \end{aligned}$$

(3.19)

Indeed, when $\theta _2=\theta _2(s_1)$, (3.19) follows from $\mathcal {P}<1$ (resp. $s_1<\mathcal {S}$) if $\alpha _1=\alpha _2$ (resp. if $\alpha _1<\alpha _2$). On the other hand, when $\theta _2=\varepsilon$, i.e., $\theta _2(s_1)\le 0$, (3.19) follows from the choice of $\varepsilon$ in the proof of Proposition 3.1. Then in the same way as (3.15) with $\theta _1$ replaced by 0 we can evaluate

$$\begin{aligned} \begin{aligned}&\left\| \displaystyle \int _0^t P_{\alpha _1}(t-s) \left( f_1(s,v_n(s))-f_1(s,v(s))\right) ds \right\| _{L^{r_1}(\Omega )}\\&\quad \le C_1 (T_*)^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_n|+|v|\Vert ^{p_1-1}_{Y_{\theta _2,r_2,T_*}} \Vert v_n-v\Vert _{Y_{\theta _2,r_2,T_*}}, \end{aligned} \end{aligned}$$

(3.20)

which converges to zero as $n\rightarrow \infty$, by (3.18).

In the case (ii) it follows from Lemma 3.2 (ii) that $\theta _2=\theta _2(s_1)$. Since $s_3<\mathcal {S}$, we have

$$\begin{aligned} \alpha _1s_3-m_1-p_1\alpha _2\theta _3 =\alpha _1s_3-m_1-p_1\alpha _2\theta _2(s_3)>0. \end{aligned}$$

(3.21)

Then we obtain in the same way as (3.11), (3.12) and (3.13) with $(s_1,\theta _2,z_1,z_2)=(s_1,\theta _2(s_1),z_1(s_1),z_2(s_1))$ and $\theta _1$ replaced by $(s_3,\theta _3,z_1(s_3),z_2(s_3))$ and 0, respectively that

$$\begin{aligned}&|u_{n+1}-u_n|_{X_{\theta _1}(r_1)}\\&\quad \le C_1 {(T_*)}^{\alpha _1s_3-m_1-p_1\alpha _2\theta _3} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_n|+|v_{n-1}|\Vert ^{p_1-1}_{Y_{\theta _3,r_2,T_*}} \Vert v_n-v_{n-1}\Vert _{Y_{\theta _3,r_2,T_*}}\\&\quad \le C_1 {(T_*)}^{\alpha _1s_3-m_1-p_1\alpha _2\theta _3} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_n|+|v_{n-1}|\Vert ^{p_1-1}_{Y_{\theta _2,\theta _3,r_2,T_*}} \Vert v_n-v_{n-1}\Vert _{Y_{\theta _2,\theta _3,r_2,T_*}}, \end{aligned}$$

which implies that

$$\begin{aligned} \begin{aligned}&\left\| \displaystyle \int _0^t P_{\alpha _1}(t-s) \left( f_1(s,v_n(s))-f_1(s,v(s))\right) ds \right\| _{L^{r_1}(\Omega )}\\&\quad \le C_1 (T_*)^{\alpha _1s_3-m_1-p_1\alpha _2\theta _3} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_n|+|v|\Vert ^{p_1-1}_{Y_{\theta _2,\theta _3,r_2,T_*}} \Vert v_n-v\Vert _{Y_{\theta _2,\theta _3,r_2,T_*}}. \end{aligned} \end{aligned}$$

(3.22)

Both (3.18) and (3.20) (or (3.22)) enable us to take the limit of (1.12) as $n\rightarrow \infty$ in $L^1(\Omega )$. Since we can take the limit of (1.13) in a similar way, we deduce the integral system in Definition 1.1 (d). For the last property (e), it suffices to prove (1.9). In the case (i) it follows from (3.13) with $\theta _1$ and $v_{n-1}$ replaced by 0 and v, respectively and (3.19) that

$$\begin{aligned} \begin{aligned}&\Vert u(t)-S_{\alpha _1}(t)u_0\Vert _{L^{r_1}(\Omega )}\\&\lesssim \, \displaystyle \int _0^t (t-s)^{\alpha _1 s_1-1} |f_1 (s, v(s))|_{X_{s_1-1}(r_1)}ds\\&\lesssim \, \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v|\Vert ^{p_1}_{Y_{\theta _2,r_2,T_*}}\cdot t^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2} \displaystyle \int _0^1 (1-s)^{\alpha _1 s_1-1}s^{-m_1-p_1\alpha _2\theta _2}ds\\&\rightarrow 0 \ \ {(t\rightarrow 0).} \end{aligned} \end{aligned}$$

(3.23)

In the case (ii) we observe from considering (3.13) and (3.19) that (3.23) also holds with $s_1$ and $\theta _2$ replaced by $s_3$ and $\theta _3$, respectively. Moreover, in both cases (i) and (ii) since we can obtain $\lim _{t\rightarrow 0}\Vert v(t)-S_{\alpha _2}(t)v_0\Vert _{L^{r_2}(\Omega )}=0$ in a similar way, (1.9) holds. Thus (u, v) possesses the desired properties of Definition 1.1.

We consider the uniqueness of the mild solution only in the case (i). In the case (ii) it can be proved by replacing $Y_{\theta _2,r_2,T}$ with $Y_{\theta _2,\theta _3,r_2,T}$. We can derive the uniqueness from calculations similar to (3.15). Indeed, let $T\in (0, T_*]$ and let $(u,v),(\tilde{u},\tilde{v})\in Y_{\theta _1,r_1,T}\times Y_{\theta _2,r_2,T}$ be any two mild solutions of (1.1) with the same initial data $(u_0,v_0)$. In a similar way to deduce (3.15) it follows that

$$\begin{aligned} \begin{aligned} \Vert u-\tilde{u}\Vert _{Y_{\theta _1,r_1,T}}&\le C_1 T^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v|+|\tilde{v}|\Vert ^{p_1-1}_{Y_{\theta _2,r_2,T}} \Vert v-\tilde{v}\Vert _{Y_{\theta _2,r_2,T}},\\ \Vert v-\tilde{v}\Vert _{Y_{\theta _2,r_2,T}}&\le C_2 T^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1} \Vert c_2\Vert _{L^{q_2}(\Omega )}\Vert 1+|u|+|\tilde{u}|\Vert ^{p_2-1}_{Y_{\theta _1,r_1,T}} \Vert u-\tilde{u}\Vert _{Y_{\theta _1,r_1,T}}. \end{aligned} \end{aligned}$$

(3.24)

We see that there exists a sufficiently small time $\widehat{T}\in (0,T]$ such that

$$\begin{aligned} \begin{aligned}&C_1 {\widehat{T}}^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1} \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v|+|\tilde{v}|\Vert ^{p_1-1}_{Y_{\theta _2,r_2,T}} \le \frac{1}{2},\\&C_2 {\widehat{T}}^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1} \Vert c_2\Vert _{L^{q_2}(\Omega )}\Vert 1+|u|+|\tilde{u}|\Vert ^{p_2-1}_{Y_{\theta _1,r_1,T}} \le \frac{1}{2}. \end{aligned} \end{aligned}$$

By this together with (3.24) we have $u(\cdot ,t)\equiv \tilde{u}(\cdot ,t)$ and $v(\cdot ,t)\equiv \tilde{v}(\cdot ,t)$ for $t\in [0,\widehat{T}]$. We obtain from a standard continuation argument (cf.[17]) that the uniqueness over the whole interval $[0,T_*]$ holds.

We prove the cases (b) and (c). Choose $(s_1,\theta _2,z_1,z_2)$ and $(s_2,\theta _1,z_3,z_4)$ as in Proposition 3.1. We divide the possibilities into two cases: (i) $\alpha _1=\alpha _2$, or $\alpha _1<\alpha _2$ and $s_1\le \mathcal {S}$ and (ii) $\alpha _1<\alpha _2$ and $s_1>\mathcal {S}$.

We consider the existence part in the case (i). Set $\Pi _1:=\{u_0\}\subset L^{r_1}(\Omega )$ and $\Pi _2:=\{v_0\}\subset L^{r_2}(\Omega )$. Let $0<T<\infty$. We can apply Lemma 2.2 and consider the constructed functions g and $w_i$ $(i=1,2)$. For $i=1,2$, put

$$\begin{aligned} Y_{w_i,\theta _i,r_i,T}:=\left\{ u\in L^{\infty }_{loc}((0,T],X_{\theta _i}(r_i)): \Vert u\Vert _{Y_{w_i,\theta _i,r_i,T}}:=\sup _{t\in (0,T)} (w_i (t))^{\alpha _2\theta _i}|u(t)|_{X_{\theta _i}(r_i)}<\infty \right\} . \end{aligned}$$

(3.25)

Lemma 2.2 (ii) implies that the spaces $Y_{w_i,\theta _i,r_i,T}$, $i=1,2$, are Banach spaces. Note that for $i=1,2$, $Y_{w_i,\theta _i,r_i,T}\subset Y_{\theta _i,r_i,T}$ follows if $0<T<\infty$. Define the functions $(u_n,v_n)$ in the same way as in the case (a). Assume that $(u_{n-1},v_{n-1})\in Y_{w_1,\theta _1,r_1,T}\times Y_{w_2,\theta _2,r_2,T}$. In the same way as (3.11) and (3.13) we have for $0<t<T$,

$$\begin{aligned} (w_1(t))^{\alpha _2\theta _1}|u_{n}|_{X_{\theta _1}(r_1)}&\le (w_1(t))^{\alpha _2\theta _1} |S_{\alpha _1}(t) u_0|_{X_{\theta _1}(r_1)} +(w_1(t))^{\alpha _2\theta _1}\displaystyle \int _0^t |P_{\alpha _1}(t-s) f_1 (s, v_{n-1} (s))|_{X_{\theta _1}(r_1)}ds\\&\lesssim \, |u_0|_{X_0 (r_1)} +(w_1(t))^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} |f_1 (s, v_{n-1} (s))|_{X_{s_1-1}(r_1)}ds \end{aligned}$$

and

$$\begin{aligned} &(w_1(t))^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} |f_1 (s, v_{n-1} (s))|_{X_{s_1-1}(r_1)}ds\\&\quad \lesssim \, (w_1(t))^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} s^{-m_1} (w_2(s))^{-p_1\alpha _2\theta _2}\cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1}|\Vert ^{p_1}_{Y_{w_2,\theta _2,r_2,T}}ds\\&\quad \le \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1}|\Vert ^{p_1}_{Y_{w_2,\theta _2,r_2,T}}\cdot (g(t))^{p_1-1}\cdot t^{\alpha _2\theta _1}\displaystyle \int _0^t (t-s)^{\alpha _1(s_1-\theta _1)-1} s^{-m_1-p_1\alpha _2\theta _2}ds\\&\quad =\Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_{n-1}|\Vert ^{p_1}_{Y_{w_2,\theta _2,r_2,T}}\\&\qquad \cdot (g(t))^{p_1-1} \cdot t^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1} \displaystyle \int _0^1 (1-s)^{\alpha _1(s_1-\theta _1)-1}s^{-m_1-p_1\alpha _2\theta _2}ds. \end{aligned}$$

(3.26)

We recall that $\lim _{t\rightarrow 0} g(t)=0$ and $\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1\ge 0$. Hence, $u_n\in Y_{w_1,\theta _1,r_1,T}$. We can obtain $v_n\in Y_{w_2,\theta _2,r_2,T}$ in the same way. By induction we derive $(u_{n},v_{n})\in Y_{w_1,\theta _1,r_1,T}\times Y_{w_2,\theta _2,r_2,T}$ for $n\ge 1$. In a similar way there exist $C_3>0$ and $C_4>0$ such that for $n\ge 2$,

$$\begin{aligned} \begin{aligned} \Vert u_{n+1}-u_n\Vert _{Y_{w_1,\theta _1,r_1,T}}&\le C_3 (g(T))^{p_1-1} T^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1}\\&\quad \cdot \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v_n|+|v_{n-1}|\Vert ^{p_1-1}_{Y_{w_2,\theta _2,r_2,T}} \Vert v_n-v_{n-1}\Vert _{Y_{w_2,\theta _2,r_2,T}},\\ \Vert v_{n+1}-v_n\Vert _{Y_{w_2,\theta _2,r_2,T}}&\le C_4 (g(T))^{p_2-1} T^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1}\\&\quad \cdot \Vert c_2\Vert _{L^{q_2}(\Omega )}\Vert 1+|u_n|+|u_{n-1}|\Vert ^{p_2-1}_{Y_{w_1,\theta _1,r_1,T}} \Vert u_n-u_{n-1}\Vert _{Y_{w_1,\theta _1,r_1,T}}. \end{aligned} \end{aligned}$$

Put $V:=2\max \left\{ \Vert u_2\Vert _{Y_{w_1,\theta _1,r_1,T}}, \Vert v_2\Vert _{Y_{w_2,\theta _2,r_2,T}}\right\}$ and choose a sufficiently small time $T_{**}>0$ such that

$$\begin{aligned} \begin{aligned}&C_3 (g(T_{**}))^{p_1-1} {T_{**}}^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2+(\alpha _2-\alpha _1)\theta _1} \Vert c_1\Vert _{L^{q_1}(\Omega )}(1+2V)^{p_1-1} \le \frac{1}{2},\\&C_4 (g(T_{**}))^{p_2-1} {T_{**}}^{\alpha _2s_2-m_2-p_2\alpha _2\theta _1} \Vert c_2\Vert _{L^{q_2}(\Omega )}(1+2V)^{p_2-1} \le \frac{1}{2}. \end{aligned} \end{aligned}$$

We deduce the analogue of (3.17) in $Y_{w_1,\theta _1,r_1,T_{**}}\times Y_{w_2,\theta _2,r_2,T_{**}}$ instead of $Y_{\theta _1,r_1,T_*}\times Y_{\theta _2,r_2,T_*}$. Therefore, there exist limits $u\in Y_{w_1,\theta _1,r_1,T_{**}}$ and $v\in Y_{w_2,\theta _2,r_2,T_{**}}$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert u_n-u\Vert _{Y_{w_1,\theta _1,r_1,T_{**}}}=0 \,\, {\text {and}}\,\, \lim _{n \rightarrow \infty } \Vert v_n-v\Vert _{Y_{w_2,\theta _2,r_2,T_{**}}}=0. \end{aligned}$$

We consider the case (ii). Let $s_3=\mathcal {S}$ and $\theta _3=\theta _2(s_3)$, where $\theta _2(s)$ is defined in the proof of Proposition 3.1. It follows from $\mathcal {P}<1$, or $\mathcal {P}=1$ and $m_1>0$ that $s_3>0$ and $\theta _3\ge 0$ hold. By this together with $\mathcal {S}<s_1<1$ and $\theta (s_1)<1$ we have $s_3\in (0,1)$ and $\theta _3\in [0,1)$. Let $0<T<\infty$. In a similar way to Lemma 2.2 we can construct a continuous and nondecreasing function $g:(0,T)\rightarrow (0,\infty )$ such that

$$\begin{aligned} |S_{\alpha _1}(t)u_0|_{X_{\theta _1}(r_1)} \, & \lesssim \,\, g(t)\cdot t^{-\alpha _2\theta _1}|u_0|_{X_0(r_1)},\ \ |S_{\alpha _2}(t)v_0|_{X_{\theta _2}(r_2)} \lesssim \, g(t)\cdot t^{-\alpha _2\theta _2}|v_0|_{X_0(r_2)} \,\, \text {and}\\ |S_{\alpha _3}(t)v_0|_{X_{\theta _3}(r_2)} \, &\lesssim \,\, g(t)\cdot t^{-\alpha _2\theta _3}|v_0|_{X_0(r_2)} \,\, \text {for} 0<t<T, \end{aligned}$$

$\lim _{t\rightarrow 0}g(t)=0$, and $\lim _{t\rightarrow 0}w_i (t)=0$ for $i=1,2,3$, where $(w_i (t))^{-\alpha _2\theta _i}=g(t)\cdot t^{-\alpha _2\theta _i}$. Put $Y_{w_1,\theta _1,r_1,T}$, $Y_{w_2,\theta _2,r_2,T}$ and $Y_{w_3,\theta _3,r_2,T}$ in the same way as (3.25). Then it follows that $(u_{n},v_{n})\in Y_{w_1,\theta _1,r_1,T}\times Y_{w_2,\theta _2,r_2,T}$ for $n\ge 1$. Let us introduce the Banach space defined by

$$\begin{aligned}&Y_{w_2,\theta _2,w_3,\theta _3,r_2,T}\\&\quad :=\left\{ u\in L^{\infty }_{loc}((0,T],X_{\theta _2}(r_2) \cap X_{\theta _3}(r_2)): \Vert u\Vert _{Y_{w_2,\theta _2,w_3,\theta _3,r_2,T}}:= \max \left\{ \Vert u\Vert _{Y_{w_2,\theta _2,r_2,T}},\Vert u\Vert _{Y_{w_3,\theta _3,r_2,T}}\right\} <\infty \right\} . \end{aligned}$$

We observe from a similar manner to (3.14) that $u_{n-1}\in Y_{w_1,\theta _1,r_1,T}$ leads not only to $v_{n}\in Y_{w_2,\theta _2,r_2,T}$ but also to $v_{n}\in Y_{w_3,\theta _3,r_2,T}$. Thus $v_{n}\in Y_{w_2,\theta _2,w_3,\theta _3,r_2,T}$ for $n\ge 1$. We omit the rest of the existence part, since it can be proved by replacing $Y_{w_2,\theta _2,r_2,T}$ and $Y_{w_2,\theta _2,r_2,T_*}$ with $Y_{w_2,\theta _2,w_3,\theta _3,r_2,T}$ and $Y_{w_2,\theta _2,w_3,\theta _3,r_2,T_*}$, respectively.

We can obtain in a similar way to the case (a) that u and v have the properties (a)–(d) of Definition 1.1 on $[0,T_{**})$. In the case (i) it follows that $\alpha _1s_1-m_1-p_1\alpha _2\theta _2\ge 0$. Then we deduce from (3.26) by replacing $\theta _1$ and $v_{n-1}$ with 0 and v, respectively that

$$\begin{aligned} \begin{aligned}&\Vert u(\cdot ,t)-S_{\alpha _1}(t)u_0\Vert _{L^{r_1}(\Omega )}\\&\lesssim \, \displaystyle \int _0^t (t-s)^{\alpha _1 s_1-1} |f_1 (s, v(s))|_{X_{s_1-1}(r_1)}ds\\& \lesssim \, \Vert c_1\Vert _{L^{q_1}(\Omega )}\Vert 1+|v|\Vert ^{p_1}_{Y_{w_2,\theta _2,r_2,T_{**}}}\cdot (g(t))^{p_1-1}\cdot t^{\alpha _1s_1-m_1-p_1\alpha _2\theta _2} \displaystyle \int _0^1 (1-s)^{\alpha _1 s_1-1}s^{-m_1-p_1\alpha _2\theta _2}ds\\&\rightarrow 0 \ \ {(t\rightarrow 0).} \end{aligned} \end{aligned}$$

(3.27)

In the same way it follows that $\lim _{t\rightarrow 0}\Vert v(t)-S_{\alpha _2}(t)v_0\Vert _{L^{r_2}(\Omega )}=0$. In the case (ii) we observe from considering (3.26) that (3.27) also holds with $s_1$, $\theta _2$ and $Y_{w_2,\theta _2,r_2,T_{**}}$ replaced by $s_3$, $\theta _3$ and $Y_{w_3,\theta _3,r_2,T_{**}}$, respectively. Thus (u, v) is a mild solution of (1.1) in the sense of Definition 1.1 on $[0,T_{**})$.

Note that in the case of (ii) how to derive the limit corresponding to (3.27) differs between when $\mathcal {R}<1$ and when $\mathcal {R}=1$. When $\mathcal {R}<1$, by considering (3.11) and (3.12) we can evaluate

$$\begin{aligned} \Vert u(\cdot ,t)-S_{\alpha _1}(t)u_0\Vert _{L^{r_1}(\Omega )} \, & \lesssim \,\, \displaystyle \int _0^t (t-s)^{\alpha _1 s_3-1}|f_1 (s, v(s))|_{X_{s_3-1}(r_1)}ds\\&\lesssim \,\, \displaystyle \int _0^t (t-s)^{\alpha _1 s_3-1}\Vert f_1 (s, v(s))\Vert _{L^{z_1}(\Omega )}ds, \end{aligned}$$

where $z_1=z_1(s_3)\in (1,\infty )$. On the other hand, when $\mathcal {R}=1$, since $z_1=z_1(s_3)=1$, the inequality $|f_1 (s, v(s))|_{X_{s_3-1}(r_1)}\lesssim \, \Vert f_1 (s, v(s))\Vert _{L^{z_1}(\Omega )}$ does not hold. Then we use the $L^p$-$L^q$ estimate of the heat semigroup ([16, Proposition 48.4 (c)-(e)]) and deduce that

$$\begin{aligned}&\Vert P_{\alpha _1}(t-s)f_1 (s, v(s))\Vert _{L^{r_1}(\Omega )}\\&\le \alpha _1 (t-s)^{\alpha _1-1}\displaystyle \int _0^{\infty } \tau \Phi _{\alpha _1}(\tau ) \Vert S(\tau (t-s)^{\alpha _1})f_1 (s, v(s))\Vert _{L^{r_1}(\Omega )} d\tau \\&\lesssim \, \alpha _1 (t-s)^{\alpha _1-1}\displaystyle \int _0^{\infty } \tau \Phi _{\alpha _1}(\tau ) \cdot \tau ^{-\frac{N}{2}\left( 1-\frac{1}{r_1}\right) } (t-s)^{-\frac{N}{2}\alpha _1\left( 1-\frac{1}{r_1}\right) }\Vert f_1 (s, v(s))\Vert _{L^{1}(\Omega )} d\tau \\&=\alpha _1 (t-s)^{\alpha _1s_3-1}\Vert f_1 (s, v(s))\Vert _{L^{1}(\Omega )} \displaystyle \int _0^{\infty } \tau ^{s_3}\Phi _{\alpha _1}(\tau ) d\tau \\&=\frac{\alpha _1\Gamma (s_3+1)}{\Gamma (\alpha _1s_3+1)} (t-s)^{\alpha _1s_3-1}\Vert f_1 (s, v(s))\Vert _{L^{1}(\Omega )}. \end{aligned}$$

Here we apply (1.5) and $\mathcal {S}=1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_1}\right)$, which follows from $\mathcal {R}=1$. Thus we have

$$\begin{aligned} \Vert u(\cdot ,t)-S_{\alpha _1}(t)u_0\Vert _{L^{r_1}(\Omega )} \,\, \lesssim \, \displaystyle \int _0^t (t-s)^{\alpha _1 s_3-1}\Vert f_1 (s, v(s))\Vert _{L^{1}(\Omega )}ds. \end{aligned}$$

We also obtain the uniqueness of the mild solution in the space $Y_{w_1,\theta _1,r_1,T_{**}}\times Y_{w_2,\theta _2,r_2,T_{**}}$ through an argument similar to (3.24). We omit the details.$\square$

4 Nonexistence result

Proof of Theorem 1.3

Without loss of generality, we suppose that $0\in \Omega$. Choose $\rho >0$ such that $B(0,\rho )\subset \Omega$, where $B(0,\rho )$ denotes the ball of radius $\rho >0$ centered at 0.

We prove the case where $\max \{\mathcal {P},\mathcal {R}\}>1$. Let $0<l<\dfrac{N}{q_1}$ if $1\le q_1<\infty$, $l=0$ if $q_1=\infty$ and $0<k<\dfrac{N}{r_2}$. Note that k is chosen to be sufficiently large and the same is valid for l when $1\le q_1<\infty$. Put

$$\begin{aligned} c_1(x):=|x|^{-l},\ c_2(x):=1,\ u_0(x):=0\,\, \text {and}\ v_0(x):=|x|^{-k}\chi _{B(0,\rho )}(x), \end{aligned}$$

where $\chi _{B(0,\rho )}$ is a characteristic function. Then $(c_1,c_2,u_0,v_0)\in L^{q_1}(\Omega )\times L^{q_2}(\Omega )\times L^{r_1}(\Omega )\times L^{r_2}(\Omega )$ holds.

The proof is by contradiction. Assume that there exists $T>0$ such that the problem (1.1) with $f_1(x,t,v)=c_1(x)\cdot t^{-m_1}v^{p_1}$ and $f_2(x,t,u)=c_2(x)\cdot t^{-m_2}u^{p_2}$ possesses a local in time nonnegative mild solution (u, v) in the sense of Definition 1.1 on [0, T). Let $1<\tau <2$ and let $s>0$ be sufficiently small. Let $|x|\le \dfrac{\sqrt{s^{\tilde{\alpha }}}}{2}$, where $\tilde{\alpha }=\alpha _1$ or $\tilde{\alpha }=\alpha _2$. Then we can apply Proposition 2.3 and obtain

$$\begin{aligned} \left( S(\tau s^{\alpha _2}) v_0\right) (x)&=\displaystyle \int _{\{|y|<\rho \}}G(x,y,\tau s^{\alpha _2})|y|^{-k}dy \ge \displaystyle \int _{\{|y-x|<\sqrt{\tau s^{\alpha _2}}\}}G(x,y,\tau s^{\alpha _2})|y|^{-k}dy\\&\ge c_1 \displaystyle \int _{\{|y-x|<\sqrt{\tau s^{\alpha _2}}\}} \left( \tau s^{\alpha _2}\right) ^{-\frac{N}{2}}|y|^{-k}dy \gtrsim s^{-\frac{N}{2}\alpha _2}\displaystyle \int _{\{|y-x|<\sqrt{\tau s^{\alpha _2}}\}}|y|^{-k}dy\\&\ge s^{-\frac{N}{2}\alpha _2}\left( \sqrt{\tau s^{\alpha _2}}+\frac{\sqrt{s^{\tilde{\alpha }}}}{2}\right) ^{-k} \displaystyle \int _{\{|y-x|<\sqrt{\tau s^{\alpha _2}}\}}dy \gtrsim s^{-\frac{k}{2}\tilde{\alpha }}. \end{aligned}$$

Due to (1.4) and (1.7), we have

$$\begin{aligned} \left( S_{\alpha _2}(s)v_0\right) (x) \ge \displaystyle \int _1^2 \Phi _{\alpha _2}(\tau )\left( S(\tau s^{\alpha _2}) v_0\right) (x)d\tau \gtrsim \displaystyle \int _1^2 \Phi _{\alpha _2}(\tau )d\tau \cdot s^{-\frac{k}{2}\tilde{\alpha }} \end{aligned}$$

and hence

$$\begin{aligned} S_{\alpha _2}(s)v_0 \,\, \gtrsim \,\, s^{-\frac{k}{2}\tilde{\alpha }}\chi _{B\left( 0,\frac{\sqrt{s^{\tilde{\alpha }}}}{2}\right) } \end{aligned}$$

(4.1)

for sufficiently small $s>0$.

Let $t>0$ be sufficiently small and let $\dfrac{t}{3}\le s \le \dfrac{t}{2}$, $|x|<\dfrac{\sqrt{s^{\alpha _1}}}{2}$ and $1<\tau <2$. Then for $|y|<\dfrac{\sqrt{s^{\tilde{\alpha }}}}{2}$,

$$\begin{aligned} |x-y|<\sqrt{s^{\alpha _1}}\le \sqrt{(t-s)^{\alpha _1}}<\sqrt{\tau (t-s)^{\alpha _1}}. \end{aligned}$$

Using Proposition 2.3, we obtain

$$\begin{aligned} G(x,y,\tau (t-s)^{\alpha _1})\gtrsim \tau ^{-\frac{N}{2}}(t-s)^{-\frac{N}{2}\alpha _1} \gtrsim s^{-\frac{N}{2}\alpha _1} \end{aligned}$$

(4.2)

for $|y|<\dfrac{\sqrt{s^{\tilde{\alpha }}}}{2}$. It follows from (4.1) and (4.2) that

$$\begin{aligned} \left( S(\tau (t-s)^{\alpha _1}) f_1(s,S_{\alpha _2}(s)v_0)\right) (x)&=\displaystyle \int _\Omega G(x,y,\tau (t-s)^{\alpha _1})|y|^{-l} s^{-m_1} \left( (S_{\alpha _2}(s)v_0)(y)\right) ^{p_1}dy\\&\gtrsim s^{-m_1-\frac{k}{2}p_1\tilde{\alpha }} \displaystyle \int _{\left\{ |y|<\frac{\sqrt{s^{\tilde{\alpha }}}}{2}\right\} } G(x,y,\tau (t-s)^{\alpha _1})|y|^{-l}dy\\&\gtrsim s^{-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1} \displaystyle \int _{\left\{ |y|<\frac{\sqrt{s^{\tilde{\alpha }}}}{2}\right\} }dy\\&\gtrsim s^{-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }}, \end{aligned}$$

which yields

$$\begin{aligned} \left( P_{\alpha _1}(t-s) f_1(s,S_{\alpha _2}(s)v_0)\right) (x)&\ge \alpha _1 (t-s)^{\alpha _1-1}\displaystyle \int _1^2 \tau \Phi _{\alpha _1}(\tau ) \left( S(\tau (t-s)^{\alpha _1})f_1(s,S_{\alpha _2}(s)v_0)\right) (x)d\tau \\&\gtrsim (t-s)^{\alpha _1-1} s^{-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }}\\&\gtrsim (t-s)^{\alpha _1-1} t^{-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }}. \end{aligned}$$

Here we use $s\le \dfrac{t}{2}$. By direct calculation we have

$$\begin{aligned} \displaystyle \int _0^t \left( P_{\alpha _1}(t-s) f_1(s,S_{\alpha _2}(s)v_0)\right) (x) ds \ge \displaystyle \int _{\frac{t}{3}}^{\frac{t}{2}} \left( P_{\alpha _1}(t-s) f_1(s,S_{\alpha _2}(s)v_0)\right) (x) ds \gtrsim t^{\alpha _1-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }} \end{aligned}$$

and

$$\begin{aligned} \left\| \displaystyle \int _0^t P_{\alpha _1}(t-s) f_1(s,S_{\alpha _2}(s)v_0) ds \right\| ^{r_1}_{L^{r_1}(\Omega )}&\gtrsim \displaystyle \int _{\left\{ |x|<\frac{\sqrt{s^{\alpha _1}}}{2}\right\} } t^{r_1\left( \alpha _1-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }\right) } dx\\&\gtrsim t^{\frac{N}{2}\alpha _1+r_1\left( \alpha _1-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }\right) }. \end{aligned}$$

Taking the limit as $l\rightarrow \dfrac{N}{q_1}$ (if $1\le q_1<\infty$) and $k\rightarrow \dfrac{N}{r_2}$, we deduce that

$$\begin{aligned} \frac{N}{2}\alpha _1+r_1\left( \alpha _1-\frac{l}{2}\tilde{\alpha }-m_1-\frac{k}{2}p_1\tilde{\alpha }-\frac{N}{2}\alpha _1+\frac{N}{2}\tilde{\alpha }\right) \rightarrow {\left\{ \begin{array}{ll} \alpha _1 r_1(1-\mathcal {P}) &{} \,\, {\text {when}} \,\, \tilde{\alpha }=\alpha _1, \\ \alpha _2 r_1(1-\mathcal {R}) &{} \,\, {\text {when}} \,\,\tilde{\alpha }=\alpha _2. \end{array}\right. } \end{aligned}$$

Thus we obtain $\left\| \int _0^t P_{\alpha _1}(t-s) f_1(s,S_{\alpha _2}(s)v_0) ds\right\| ^{r_1}_{L^{r_1}(\Omega )}\rightarrow \infty$ as $t\rightarrow 0$. By Definition 1.1 (d) we have $v(t)\ge S_{\alpha _2}(t)v_0$ in $\Omega \times (0,T)$, which yields

$$\begin{aligned} \left\| \displaystyle \int _0^t P_{\alpha _1}(t-s) f_1(s,v(s)) ds\right\| _{L^{r_1}(\Omega )}\ge \left\| \displaystyle \int _0^t P_{\alpha _1}(t-s) f_1(s,S_{\alpha _2}(s)v_0) ds\right\| _{L^{r_1}(\Omega )} \rightarrow \infty (t\rightarrow 0). \end{aligned}$$

This contradicts the property of Definition 1.1 (e).

We prove the case where $\max \{\mathcal {Q}_1,\mathcal {Q}_2\}>1$. Let $0<\tilde{l}<\dfrac{N}{q_2}$ if $1\le q_2<\infty$, $\tilde{l}=0$ if $q_2=\infty$ and $0<\tilde{k}<\dfrac{N}{r_1}$. Note that $\tilde{k}$ is chosen to be sufficiently large and the same is valid for $\tilde{l}$ when $1\le q_2<\infty$. Put

$$\begin{aligned} c_1(x):=1,\ c_2(x):=|x|^{-\tilde{l}},\ u_0(x):=|x|^{-\tilde{k}}\chi _{B(0,\rho )}(x)\,\, {\text {and}}\ v_0(x):=0. \end{aligned}$$

Then $(c_1,c_2,u_0,v_0)\in L^{q_1}(\Omega )\times L^{q_2}(\Omega )\times L^{r_1}(\Omega )\times L^{r_2}(\Omega )$ holds.

The proof is also by contradiction. As in the case where $\max \{\mathcal {P},\mathcal {R}\}>1$, we assume that there exists a local in time nonnegative mild solution (u, v) in the sense of Definition 1.1. We obtain in the same way as we deduce (4.1) that

$$\begin{aligned} S_{\alpha _1}(s)u_0 \, \gtrsim \, s^{-\frac{\tilde{k}}{2}\alpha _1}\chi _{B\left( 0,\frac{\sqrt{s^{\alpha _1}}}{2}\right) } \end{aligned}$$

(4.3)

for sufficiently small $s>0$.

We prove the case where $\mathcal {Q}_1>1$. Let $t>0$ be sufficiently small and let $\dfrac{t}{3}\le s \le \dfrac{t}{2}$, $|x|<\dfrac{\sqrt{s^{\alpha _1}}}{3}$ and $1<\tau <2$. It follows from Proposition 2.3 and (4.3) that

$$\begin{aligned} \left( S(\tau (t-s)^{\alpha _2}) f_2(s,S_{\alpha _1}(s)u_0)\right) (x)&=\displaystyle \int _\Omega G(x,y,\tau (t-s)^{\alpha _2})|y|^{-\tilde{l}} s^{-m_2} \left( (S_{\alpha _1}(s)u_0)(y)\right) ^{p_2}dy\\&\gtrsim s^{-\frac{N}{2}\alpha _2} \displaystyle \int _{\{|y-x|<\sqrt{\tau s^{\alpha _2}}\}} |y|^{-\tilde{l}} s^{-m_2} \left( (S_{\alpha _1}(s)u_0)(y)\right) ^{p_2}dy\\&\gtrsim s^{-\frac{N}{2}\alpha _2-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1} \displaystyle \int _{\{|y-x|<\sqrt{\tau s^{\alpha _2}}\}} dy\\&\gtrsim s^{-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1}. \end{aligned}$$

Note that we apply $|y|\le |y-x|+|x|<\sqrt{\tau s^{\alpha _2}}+\dfrac{\sqrt{s^{\alpha _1}}}{3}\le \dfrac{\sqrt{s^{\alpha _1}}}{2}$, since s is sufficiently small. Then by $s\le \dfrac{t}{2}$ we have

$$\begin{aligned} \left( P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0)\right) (x)&\ge \alpha _2 (t-s)^{\alpha _2-1}\displaystyle \int _1^2 \tau \Phi _{\alpha _2}(\tau ) \left( S(\tau (t-s)^{\alpha _2})f_1(s,S_{\alpha _1}(s)u_0)\right) (x)d\tau \\&\gtrsim (t-s)^{\alpha _2-1}s^{-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1}\\&\gtrsim (t-s)^{\alpha _2-1}t^{-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1}. \end{aligned}$$

By direct calculation we get

$$\begin{aligned} \displaystyle \int _0^t \left( P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0)\right) (x) ds \ge \displaystyle \int _{\frac{t}{3}}^{\frac{t}{2}} \left( P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0)\right) (x) ds \gtrsim t^{\alpha _2-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1} \end{aligned}$$

and

$$\begin{aligned} \left\| \displaystyle \int _0^t P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0) ds \right\| ^{r_2}_{L^{r_2}(\Omega )}&\gtrsim \displaystyle \int _{\left\{ |x|<\frac{\sqrt{s^{\alpha _1}}}{3}\right\} } t^{r_2\left( \alpha _2-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1\right) } dx\\&\gtrsim t^{\frac{N}{2}\alpha _1+ r_2\left( \alpha _2-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1\right) }. \end{aligned}$$

Taking the limit as $\tilde{l}\rightarrow \dfrac{N}{q_2}$ (if $1\le q_2<\infty$) and $\tilde{k}\rightarrow \dfrac{N}{r_1}$, we deduce that

$$\begin{aligned} \frac{N}{2}\alpha _1+ r_2\left( \alpha _2-\frac{\tilde{l}}{2}\alpha _1-m_2-\frac{\tilde{k}}{2}p_2\alpha _1\right) \rightarrow \alpha _2 r_2(1-\mathcal {Q}_1)<0. \end{aligned}$$

Thus we obtain $\left\| \int _0^t P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0) ds\right\| ^{r_2}_{L^{r_2}(\Omega )}\rightarrow \infty$ as $t\rightarrow 0$. This contradicts the property of Definition 1.1(e) in a similar way to the case where $\max \{\mathcal {P},\mathcal {R}\}>1$.

We prove the case where $\mathcal {Q}_2>1$. Let $t>0$ be sufficiently small and let $\dfrac{t}{3}\le s \le \dfrac{t}{2}$, $|x|<\dfrac{\sqrt{s^{\alpha _2}}}{2}$ and $1<\tau <2$. Then for $|y|<\dfrac{\sqrt{s^{\alpha _2}}}{2}$,

$$\begin{aligned} |x-y|<\sqrt{s^{\alpha _2}}\le \sqrt{(t-s)^{\alpha _2}}<\sqrt{\tau (t-s)^{\alpha _2}}. \end{aligned}$$

Due to Proposition 2.3, we have

$$\begin{aligned} G(x,y,\tau (t-s)^{\alpha _2})\gtrsim \tau ^{-\frac{N}{2}}(t-s)^{-\frac{N}{2}\alpha _2} \gtrsim s^{-\frac{N}{2}\alpha _2} \end{aligned}$$

(4.4)

for $|y|<\dfrac{\sqrt{s^{\alpha _2}}}{2}$. Considering in the same way as in the case where $\max \{\mathcal {P},\mathcal {R}\}>1$, we obtain from (4.3) and (4.4) that

$$\begin{aligned} \left( S(\tau (t-s)^{\alpha _2}) f_2(s,S_{\alpha _1}(s)u_0)\right) (x) \gtrsim s^{-\frac{\tilde{l}}{2}\alpha _2-m_2-\frac{\tilde{k}}{2}p_2\alpha _1}, \end{aligned}$$

which yields

$$\begin{aligned} \left( P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0)\right) (x) \gtrsim (t-s)^{\alpha _2-1}t^{-\frac{\tilde{l}}{2}\alpha _2-m_2-\frac{\tilde{k}}{2}p_2\alpha _1}. \end{aligned}$$

Moreover, this leads to

$$\begin{aligned} \left\| \displaystyle \int _0^t P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0) ds \right\| ^{r_2}_{L^{r_2}(\Omega )}\gtrsim t^{\frac{N}{2}\alpha _2+ r_2\left( \alpha _2-\frac{\tilde{l}}{2}\alpha _2-m_2-\frac{\tilde{k}}{2}p_2\alpha _1\right) }. \end{aligned}$$

Taking the limit as $\tilde{l}\rightarrow \dfrac{N}{q_2}$ (if $1\le q_2<\infty$) and $\tilde{k}\rightarrow \dfrac{N}{r_1}$, we deduce that

$$\begin{aligned} \frac{N}{2}\alpha _2+ r_2\left( \alpha _2-\frac{\tilde{l}}{2}\alpha _2-m_2-\frac{\tilde{k}}{2}p_2\alpha _1\right) \rightarrow \alpha _2 r_2(1-\mathcal {Q}_2)<0. \end{aligned}$$

Thus we obtain $\left\| \int _0^t P_{\alpha _2}(t-s) f_2(s,S_{\alpha _1}(s)u_0) ds\right\| ^{r_2}_{L^{r_2}(\Omega )}\rightarrow \infty$ as $t\rightarrow 0$. This contradicts the property of Definition 1.1 (e). Therefore, the proof is complete.$\square$

5 Nonexistence result for scalar problems

In this section we apply our study to the nonexistence of a local in time solution of the scalar problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial ^{\alpha }_t u=\Delta u+f(x,t,u) &{} {\text {in}}\,\, \Omega \times (0,T), \\ u(x,t)=0 &{} {\text {on}}\,\, \partial \Omega \times (0,T), \\ u(x,0)=u_0 (x) &{} {\text {in}}\,\, \Omega . \end{array}\right. } \end{aligned}$$

(5.1)

We obtain the following nonexistence theorem.

Theorem 5.1

Let $N \ge 1$, $0<\alpha <1$, $0<p<\infty$, $q_1\in [1,\infty ]$, $q_2\in \left( \dfrac{1}{\alpha },\infty \right]$ and $1\le r<\infty$. Suppose that (1.11) holds with $\beta _A=\dfrac{N}{2}$. Then there exist nonnegative functions $c(x,t)\in L_{q_1,q_2}$ and $u_0\in L^r(\Omega )$ such that, for every $T>0,$ the problem (5.1) with $f(x,t,u)=c(x,t)\cdot u^p$ admits no local in time nonnegative mild solution u in the sense of Definition 1.1 (more precisely, in the sense of [9, Definition 3.1.1]) on the interval [0, T).

Since we can prove in the same way as in the proof of Theorem 1.3, we leave the proof to readers. Note that we only solve the nonexistence conjecture in [9] when the nonlinear term f is separable with respect to x, t and u.

Remark that if $1\le r<\dfrac{N}{2}(p-1)$, then there exists a nonnegative initial function $u_0 \in L^r (\Omega )$ such that, the problem (5.1) with $f(x,t,u)=u^p$ has no local in time nonnegative mild solution on any time interval. Hence, our nonexistence result corresponds to [20]. In conclusion, for scalar problems and systems with pure power nonlinear terms, the existence/nonexistence results correspond to [20] and [15], respectively.

6 Discussion

In this paper we consider a local in time solution of a time fractional weakly coupled reaction-diffusion system in two components with possibly distinct fractional orders. In Theorems 1.2 and 1.3 we derive the integrability conditions on the initial state functions for the local in time existence and nonexistence results. The parameters $\mathcal {P}$, $\mathcal {Q}$, $\mathcal {Q}_1$, $\mathcal {Q}_2$ and $\mathcal {R}$ describe the balance between the factors: the growth rates (resp. the singularities) of the nonlinear terms with respect to u or v (resp. x and t), the singularities of the initial data, and the fractional exponents. For instance, the larger the growth rates $p_1$ and $p_2$ become, the larger these five parameters become. Then Theorems 1.2 and 1.3 imply that the existence result is less likely to hold, and that the nonexistence result is more likely to hold. The integrability is determined by $\max \{\mathcal {P},\mathcal {Q},\mathcal {R}\}$ and $\max \{\mathcal {P},\mathcal {Q}_1,\mathcal {Q}_2,\mathcal {R}\}$ in the existence and nonexistence part, respectively.

When $\alpha _1=\alpha _2$, the equalities $\mathcal {P}=\mathcal {R}$ and $\mathcal {Q}=\mathcal {Q}_1=\mathcal {Q}_2$ hold. Therefore, as seen in Corollary 1.4, we can explicitly determine the existence/nonexistence of a solution. The threshold integrability condition on initial data, which is a pair $(r_1,r_2)$, is defined by $\max \{\mathcal {P},\mathcal {Q}\}=1$. When $m_1>0$ and $m_2>0$, the larger $\alpha _1$ and $\alpha _2$ become, the wider the space of initial data for the existence result becomes. On the other hand, when $\alpha _1<\alpha _2$, the inequalities $\max \{\mathcal {Q}_1,\mathcal {Q}_2\}\le 1<\mathcal {Q}$ and $\max \{\mathcal {P},\mathcal {R}\}\le 1$ can occur. In this case since Theorems 1.2 and 1.3 cannot be applied, we cannot determine whether the problem (1.1) possesses a local in time solution or not. We mention the following points:

• 1. In Theorem 1.3 even if we assume $\mathcal {Q}>1$, we cannot obtain the nonexistence result, since the inequality (4.3) does not hold with $\alpha _1$ replaced by $\alpha _2$ on the right hand side. If this is true, we can evaluate $\left( S(\tau (t-s)^{\alpha _2}) f_2(s,S_{\alpha _1}(s)u_0)\right) (x)$ in the same way as in the case where $\max \{\mathcal {P},\mathcal {R}\}>1$. Hence, we can get the nonexistence result even if $\mathcal {Q}>1$.

• 2. If we assume $\max \{\mathcal {Q}_1,\mathcal {Q}_2\}<1$ instead of $\mathcal {Q}<1$, then Proposition 3.1 does not hold. In particular, the inequality $\max \{\mathcal {Q}_1,\mathcal {Q}_2\}<1$ does not lead to (3.6) and (3.9) with $(s,\tilde{s})=\left( \min \left\{ 1,1-\dfrac{N}{2}\left( \dfrac{1}{q_1}-\dfrac{1}{r_1}\right) \right\} ,\max \left\{ 1-\dfrac{N}{2}\left( 1-\dfrac{1}{r_2}\right) ,\dfrac{m_2}{\alpha _2}\right\} \right)$. Thus we cannot obtain the existence result under this assumption.

The author conjectures that the nonexistence result does not hold in the above case, since the former point is a greater reason. In the existence result it seems that the solvability of the problem (1.1) in the above case may hold by using a functional space different from the Banach spaces introduced in the proof of Theorem 1.2.

Possible future problems ensuing from the current analysis are as follows:

• 1. What are the consequences of the problem (1.1) with a different boundary condition or situation, e.g. the Neumann boundary condition, the boundary condition where u and v are non-zero positive bounded functions, and the situation where the boundary is broken into parts with a condition of a different type set on each?

• 2. What happens to a local in time solution of (1.1) as $\alpha \rightarrow 1^-$? Given that (1.3) is the limit of (1.1) as $\alpha \rightarrow 1^-$, is it possible to show that the estimates obtained by this paper approach those known for (1.3)? If not, which is more conservative and why?

• 3. For the problem (1.1), what is the solvability when one equation has an integer in time derivative and the other has a fractional in time derivative?