In this section, we first state some assumptions and definitions needed in the proof of our main result and then prove the existence of solutions.

Throughout the paper, we assume that the exponents

${p}_{1}(x),{p}_{2}(x):\mathrm{\Omega}\to (1,+\mathrm{\infty})$ and the continuous functions

${a}_{1}(x),{a}_{2}(x):\mathrm{\Omega}\to \mathbb{R}$ satisfy the following conditions:

**Definition 2.1** We say that the solution

$(u(x,t),v(x,t))$ for problem (1.1) blows up in finite time if there exists an instant

${T}^{\ast}<\mathrm{\infty}$ such that

$\u2980(u,v)\u2980\to \mathrm{\infty}\phantom{\rule{1em}{0ex}}\text{as}t\to {T}^{\ast},$

where

$\u2980(u,v)\u2980=\underset{t\in [0,T)}{sup}\{{\parallel u(\cdot ,t)\parallel}_{\mathrm{\infty}}+{\parallel v(\cdot ,t)\parallel}_{\mathrm{\infty}}\}.$

Our first result here is the following.

**Theorem 2.1** *Let* $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$ *be a bounded smooth domain*, ${p}_{1}(x)$, ${p}_{2}(x)$, ${a}_{1}(x)$, ${a}_{2}(x)$ *satisfy the conditions in* (2.1), *and assume that* ${u}_{0}(x)$ *and* ${v}_{0}(x)$ *are nonnegative*, *continuous and bounded*. *Then there exists a* ${T}^{0}$, $0<{T}^{0}\le \mathrm{\infty}$, *such that problem* (1.1) *has a nonnegative and bounded solution* $(u,v)$ *in* ${Q}_{{T}^{0}}$.

*Proof* We only prove the case when ${f}_{1}(u,v)={a}_{1}(x){v}^{{p}_{1}(x)}$ and ${f}_{2}(u,v)={a}_{2}(x){u}^{{p}_{2}(x)}$, and the proofs to the cases ${f}_{1}(u,v)={a}_{1}(x){\int}_{\mathrm{\Omega}}{v}^{{p}_{1}(y)}(y,t)\phantom{\rule{0.2em}{0ex}}dy$ and ${f}_{2}(u,v)={a}_{2}(x){\int}_{\mathrm{\Omega}}{u}^{{p}_{2}(y)}(y,t)\phantom{\rule{0.2em}{0ex}}dy$ are similar.

Let us consider the equivalent systems of (1.1)

$\{\begin{array}{c}u(x,t)={\int}_{\mathrm{\Omega}}g(x,y,t){u}_{0}(y)\phantom{\rule{0.2em}{0ex}}dy+{\int}_{0}^{t}{\int}_{\mathrm{\Omega}}g(x,y,t-s){a}_{1}(y){v}^{{p}_{1}(y)}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds,\hfill \\ v(x,t)={\int}_{\mathrm{\Omega}}g(x,y,t){v}_{0}(y)\phantom{\rule{0.2em}{0ex}}dy+{\int}_{0}^{t}{\int}_{\mathrm{\Omega}}g(x,y,t-s){a}_{2}(y){u}^{{p}_{2}(y)}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds,\hfill \end{array}$

where $g(x,y,t)$ is the corresponding Green function. Then the existence and uniqueness of solutions for a given $({u}_{0}(x),{v}_{0}(x))$ could be obtained by a fixed point argument.

We introduce the following iteration scheme:

and the convergence of the sequence

$\{({u}_{n},{v}_{n})\}$ follows by showing that

$\{\begin{array}{c}{\mathrm{\Phi}}_{1}(v)={\int}_{0}^{t}{\int}_{\mathrm{\Omega}}g(x,y,t-s){a}_{1}(y){v}_{n}^{{p}_{1}(y)}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds,\hfill \\ {\mathrm{\Phi}}_{2}(u)={\int}_{0}^{t}{\int}_{\mathrm{\Omega}}g(x,y,t-s){a}_{2}(y){u}_{n}^{{p}_{2}(y)}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds\hfill \end{array}$

is a contraction in the set ${E}_{T}$ to be defined below.

Now, we define

$\mathrm{\Psi}(u,v)=({\mathrm{\Phi}}_{1}(v),{\mathrm{\Phi}}_{2}(u)),$

We denote

$\mathrm{\Psi}(u,v)-\mathrm{\Psi}(w,z)=({\mathrm{\Phi}}_{1}(v)-{\mathrm{\Phi}}_{1}(z),{\mathrm{\Phi}}_{2}(u)-{\mathrm{\Phi}}_{2}(w)),$

and for arbitrary

$T>0$, define the set

${E}_{T}=\{{C}^{1,2}({\mathrm{\Omega}}_{T})\cap C(\overline{{\mathrm{\Omega}}_{T}})|\u2980(u,v)\u2980\le M\},$

where ${\mathrm{\Omega}}_{T}=\mathrm{\Omega}\times [0,T]$, $M>\u2980({u}_{0}(x),{v}_{0}(x))\u2980$ is a fixed positive constant.

We claim that Ψ is a contraction on

${E}_{T}$. In fact, for any

$x\in \mathrm{\Omega}$ fixed, we have

and we always have

$\u2980{p}_{i}(x){w}_{i}^{{p}_{i}(x)-1}({\xi}_{i}-{\eta}_{i})\u2980\le {p}_{i}^{+}{(2M)}^{{p}_{i}^{+}-1}{\parallel {\xi}_{i}-{\eta}_{i}\parallel}_{\mathrm{\infty}},\phantom{\rule{1em}{0ex}}i=1,2.$

(2.2)

Now, we define

$\mu (t)=\underset{x\in \overline{\mathrm{\Omega}},0\le \tau <t}{sup}{\int}_{0}^{\tau}{\int}_{\mathrm{\Omega}}g(x,y,t-s)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds.$

It is obvious that $\mu (t)\to 0$ when $t\to {0}^{+}$.

Then, by using inequality (2.2), we get

Hence, for sufficiently small

*t*, we have

where $\theta <1$ is a constant. Then Ψ is a strict contraction. □