Nonexistence results for systems of parabolic differential inequalities in 2D exterior domains

Abstract

In this paper, we study, for the first time, the nonexistence of solutions to systems of parabolic differential inequalities in 2D exterior domains with Dirichlet and Neumann boundary conditions. Our obtained results complete those derived recently by Yuhua Sun [Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of $R^{N}$ , Pacific Journal of Mathematics. 293(1) (2018) 245–256] in the N-dimensional case, $N ⩾ 3$ , under Dirichlet boundary condition.

Keywords

Noexistence systems of parabolic differential inequalities 2D exterior domain

1. Introduction

In this paper, we consider the systems of parabolic differential inequalities $\begin{matrix} (1.1) & \begin{matrix} Δ u - \partial_{t} u + | v |^{p} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ Δ v - \partial_{t} v + | u |^{q} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ u (t, x) ⩾ f (x), v (t, x) ⩾ g (x), & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), & in & D^{c} \end{matrix} \end{matrix}$ and $\begin{matrix} (1.2) & \begin{matrix} Δ u - \partial_{t} u + | v |^{p} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ Δ v - \partial_{t} v + | u |^{q} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ \frac{\partial u}{\partial n^{+}} (t, x) ⩾ f (x), \frac{\partial v}{\partial n^{+}} (t, x) ⩾ g (x), & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}$ where $D = \overline{B (0, 1)}$ is the closed unit ball in $R^{2}$ , $D^{c} = R^{2} ∖ D$ and $n^{+}$ is the outward (relative to $D^{c}$ ) unit normal on $\partial D$ . Problems (1.1) and (1.2) are investigated under the assumptions: $\begin{matrix} p, q > 1, u_{0}, v_{0} \in L_{loc}^{1} (\overline{D^{c}}), u_{0}, v_{0} ⩾ 0 and f, g \in L^{1} (\partial D), f, g ⩾ 0 . \end{matrix}$ We are concerned with the nonexistence of global weak solutions to the considered systems. To the best of our knowledge, this work is the first in which the nonexistence of solutions is discussed for systems (1.1) and (1.2) in the two dimensional case.

Let us provide some motivations for studying problems of the form (1.1) and (1.2). In his famous paper [4], Fujita studied the problem $\begin{matrix} (1.3) & \begin{matrix} \partial_{t} u - Δ u = u^{p}, & in & (0, \infty) \times R^{N}, \\ u (0, x) = u_{0} (x), & in & R^{N} . \end{matrix} \end{matrix}$ He proved that

If $1 < p < 1 + \frac{2}{N}$ and $u_{0} > 0$ , then (1.3) admits no global positive solution.

If $p > 1 + \frac{2}{N}$ and $u_{0}$ is smaller than a small Gaussian, then (1.3) admits global solutions.

Later, it was shown in [1,5] that (i) holds true for

p = 1 + \frac{2}{N}

. The real number

1 + \frac{2}{N}

is said to be critical in the sense of Fujita. In [2], Bandle and Levine studied the exterior Dirichlet problem

\begin{matrix} (1.4) & \begin{matrix} \partial_{t} u - Δ u = u^{p}, & in & (0, \infty) \times D^{c}, \\ u (t, x) = 0, & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}

where D is a bounded nonempty domain in

R^{N}

N ⩾ 3

. They proved that the critical exponent for (1.4) is still

1 + \frac{2}{N}

. In [9], Zhang studied the Dirichlet exterior problem (1.4) under the Dirichlet boundary condition

\begin{matrix} u (t, x) = f (x), (t, x) \in (0, \infty) \times \partial D, \end{matrix}

where

f ≢ 0

. He proved that the critical exponent in this case passes from

1 + \frac{2}{N}

to a bigger value

1 + \frac{2}{N - 2}

, where

N ⩾ 3

. In [6], using the test function method [7], Laptev studied the problem

\begin{matrix} (1.5) & \begin{matrix} \partial_{t} u - Δ u ⩾ | x |^{σ} | u |^{p}, & in & (0, \infty) \times D^{c}, \\ u (t, x) ⩾ 0, & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}

where

D = \overline{B (0, R)}

R > 0

σ > - 2

and

N ⩾ 3

. To be more precise, a more general problem was considered in [6], which includes (1.5) as a special case. It was shown that, if

\begin{matrix} 1 < p < 1 + \frac{2 + σ}{N}, \end{matrix}

then (1.5) has no nontrivial global weak solution. Very recently, Sun [8] studied the system of parabolic differential inequalities

\begin{matrix} (1.6) & \begin{matrix} Δ u - \partial_{t} u + | x |^{a} v^{p} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ Δ v - \partial_{t} v + | x |^{b} u^{q} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ u (t, x) ⩾ f (x), v (t, x) ⩾ g (x), & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}

where D is a bounded Lipschitz domain in

R^{N}

N ⩾ 3

, containing the origin, and

f, g ⩾ 0

f ≢ 0

g ≢ 0

. He proved the following interesting result: Assume that

p ⩾ q > 1

. If

\begin{matrix} N < max {\frac{2 p (1 + q) + b p + a}{p q - 1}, \frac{2 q (1 + p) + a q + b}{p q - 1}}, \end{matrix}

then (1.6) admits no positive solution. The used approach in [8] is based on an idea of Zhang [9], which consists to show that the

L^{p}

-norm and

L^{q}

-norm of the solutions blow-up in a certain selected and fixed region. Note that this method, as it was mentioned in [3], is limited to the N-dimensional case, where

N ⩾ 3

. Moreover, its application to the Dirichlet case needs the use of the maximum principle.

Motivated by the above cited works, the nonexistence of global weak solutions to (1.1) and (1.2), in the case $N = 2$ , is investigated in this paper. Our method is based on a test function approach with a judicious choice of the test function. Moreover, our technique doesn’t require the use of the maximum principle.

Before stating the main results of this paper, let us give the definitions of solutions to (1.1) and (1.2).

Definition 1.1.
We say that $(u, v) \in L_{loc}^{q} ([0, \infty) \times \overline{D^{c}}) \times L_{loc}^{p} ([0, \infty) \times \overline{D^{c}})$ is a global weak solution to (1.1), if $\begin{matrix} (1.7) & \begin{matrix} \int_{(0, \infty) \times D^{c}} | v |^{p} φ d x d t - \int_{(0, \infty) \times \partial D} \frac{\partial φ}{\partial n^{+}} f d S_{x} d t + \int_{D^{c}} u_{0} (x) φ (0, x) d x \\ ⩽ - \int_{(0, \infty) \times D^{c}} u \partial_{t} φ d x d t - \int_{(0, \infty) \times D^{c}} u Δ φ d x d t \end{matrix} \end{matrix}$ and $\begin{matrix} (1.8) & \begin{matrix} \int_{(0, \infty) \times D^{c}} | u |^{q} φ d x d t - \int_{(0, \infty) \times \partial D} \frac{\partial φ}{\partial n^{+}} g d S_{x} d t + \int_{D^{c}} v_{0} (x) φ (0, x) d x \\ ⩽ - \int_{(0, \infty) \times D^{c}} v \partial_{t} φ d x d t - \int_{(0, \infty) \times D^{c}} v Δ φ d x d t, \end{matrix} \end{matrix}$ for every non-negative function $φ \in C_{t, x}^{1, 2} ((0, \infty) \times D^{c}) \cap C ([0, \infty) \times \overline{D^{c}})$ satisfying
$\partial_{t} φ, Δ φ \in C ([0, \infty) \times \overline{D^{c}})$ ;

$φ_{| \partial D} \equiv 0$ , $\frac{\partial φ}{\partial n^{+}} \in C ([0, \infty) \times \partial D)$ , $\frac{\partial φ}{\partial n^{+}} ⩽ 0$ in $(0, \infty) \times \partial D$ ;

$\exists T > 0 : φ (t, \cdot) \equiv 0$ , $t ⩾ T$ ;

$\exists R > 1 : φ (\cdot, x) \equiv 0$ , $| x | ⩾ R$ .

Definition 1.2.
We say that $(u, v) \in L_{loc}^{q} ([0, \infty) \times \overline{D^{c}}) \times L_{loc}^{p} ([0, \infty) \times \overline{D^{c}})$ is a global weak solution to (1.2), if $\begin{matrix} (1.9) & \begin{matrix} \int_{(0, \infty) \times D^{c}} | v |^{p} φ d x d t + \int_{(0, \infty) \times \partial D} f φ d S_{x} d t + \int_{D^{c}} u_{0} (x) φ (0, x) d x \\ ⩽ - \int_{(0, \infty) \times D^{c}} u \partial_{t} φ d x d t - \int_{(0, \infty) \times D^{c}} u Δ φ d x d t \end{matrix} \end{matrix}$ and $\begin{matrix} (1.10) & \begin{matrix} \int_{(0, \infty) \times D^{c}} | u |^{q} φ d x d t + \int_{(0, \infty) \times \partial D} g φ d S_{x} d t + \int_{D^{c}} v_{0} (x) φ (0, x) d x \\ ⩽ - \int_{(0, \infty) \times D^{c}} v \partial_{t} φ d x d t - \int_{(0, \infty) \times D^{c}} v Δ φ d x d t, \end{matrix} \end{matrix}$ for every non-negative function $φ \in C_{t, x}^{1, 2} ((0, \infty) \times D^{c}) \cap C ([0, \infty) \times \overline{D^{c}})$ satisfying
$\partial_{t} φ, Δ φ \in C ([0, \infty) \times \overline{D^{c}})$ ;

$\frac{\partial φ}{\partial n^{+}} = 0$ in $(0, \infty) \times \partial D$ ;

$\exists T > 0 : φ (t, \cdot) \equiv 0$ , $t ⩾ T$ ;

$\exists R > 1 : φ (\cdot, x) \equiv 0$ , $| x | ⩾ R$ .

Our first main result is given by the following theorem.
Theorem 1.3.
Let $u_{0}, v_{0} \in L_{loc}^{1} (\overline{D^{c}})$ , $u_{0}, v_{0} ⩾ 0$ and $f, g \in L^{1} (\partial D)$ . Suppose that $\begin{matrix} max {\int_{\partial D} f (x) d S_{x}, \int_{\partial D} g (x) d S_{x}} > 0 . \end{matrix}$ Then for every $p > 1$ and $q > 1$ , ( 1.1 ) admits no global weak solution.
Remark 1.4.
Let us consider the Dirichlet exterior problem $\begin{matrix} (1.11) & \begin{matrix} Δ u - \partial_{t} u + | u |^{p} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ u (t, x) ⩾ f (x), & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}$ where $u_{0} \in L_{loc}^{1} (\overline{D^{c}})$ , $u_{0} ⩾ 0$ , $f \in L^{1} (\partial D)$ and $\int_{\partial D} f (x) d S_{x} > 0$ . From Theorem 1.3, we deduce that for all $p > 1$ , (1.11) admits no global weak solution.

Our next main result is given by the following theorem.
Theorem 1.5.
Let $u_{0}, v_{0} \in L_{loc}^{1} (\overline{D^{c}})$ , $u_{0}, v_{0} ⩾ 0$ and $f, g \in L^{1} (\partial D)$ . Suppose that $\begin{matrix} max {\int_{\partial D} f (x) d S_{x}, \int_{\partial D} g (x) d S_{x}} > 0 . \end{matrix}$ Then for every $p > 1$ and $q > 1$ , ( 1.2 ) admits no global weak solution.
Remark 1.6.
Let us consider the Neumann exterior problem $\begin{matrix} (1.12) & \begin{matrix} Δ u - \partial_{t} u + | u |^{p} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ \frac{\partial u}{\partial n^{+}} (t, x) ⩾ f (x), & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}$ where $u_{0} \in L_{loc}^{1} (\overline{D^{c}})$ , $u_{0} ⩾ 0$ , $f \in L^{1} (\partial D)$ and $\int_{\partial D} f (x) d S_{x} > 0$ . From Theorem 1.5, we deduce that for all $p > 1$ , (1.12) admits no global weak solution.

The rest of the paper is organized as follows. In Section 2, we provide some preliminary estimates. In Section 3, we prove Theorem 1.3. In Section 4, we prove Theorem 1.5. Finally, some open questions are proposed in Section 5.
2. Preliminary estimates

Let us denote by H the harmonic function in $D^{c}$ given by $\begin{matrix} (2.1) & H (x) = ln (| x |), x \in D^{c} . \end{matrix}$ The following result is immediate.

Lemma 2.1.
We have $\begin{matrix} \nabla H (x) = \frac{x}{| x |^{2}}, x \in D^{c} . \end{matrix}$

Next, let $Φ \in C_{0}^{\infty} ([0, \infty))$ be a function satisfying $\begin{matrix} 0 ⩽ Φ ⩽ 1; Φ (σ) = \{\begin{matrix} 1 & if 0 ⩽ σ ⩽ 1, \\ 0 & if σ ⩾ 2 . \end{matrix} \end{matrix}$ Given $0 < T < \infty$ , we introduce the functions $\begin{matrix} (2.2) & D (t, x) = η_{T} (t) ψ_{T} (x), (t, x) \in (0, \infty) \times D^{c} \end{matrix}$ and $\begin{matrix} (2.3) & N (t, x) = η_{T} (t) χ_{T} (x), (t, x) \in (0, \infty) \times D^{c}, \end{matrix}$ where $\begin{array}{l} (2.4) & η_{T} (t) = \{\begin{matrix} T^{- λ} {(T - t)}^{λ} & if 0 ⩽ t ⩽ T, \\ 0 & if t ⩾ T, \end{matrix} λ ≫ 1, \\ (2.5) & ψ_{T} (x) = H (x) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}), r > 0, ω ≫ 1, x \in D^{c}, \\ (2.6) & χ_{T} (x) = Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}), r > 0, ω ≫ 1, x \in D^{c} . \end{array}$
Lemma 2.2.
For T large enough, the function $D$ given by ( 2.2 ) satisfies conditions (i)-(iv) of Definition 1.1 .
Proof.
It is clear that $D ⩾ 0$ , $D \in C_{t, x}^{1, 2} ((0, \infty) \times D^{c}) \cap C ([0, \infty) \times \overline{D^{c}})$ and satisfies (i). Moreover, by (2.1) and (2.5), we have $\begin{matrix} D_{| \partial D} \equiv 0 . \end{matrix}$ Next, for $(t, x) \in [0, \infty) \times \partial D$ , we have $\begin{matrix} \frac{\partial D}{\partial n^{+}} (t, x) = - ⟨ \nabla D (t, x), x ⟩ = - η_{T} (t) ⟨ \nabla ψ_{T} (x), x ⟩, \end{matrix}$ where $⟨ \cdot, \cdot ⟩$ is the scalar product in $R^{2}$ . On the other hand, for $x \in \partial D$ , using Lemma 2.1, we have $\begin{array}{rcl} \nabla ψ_{T} (x) & = & \nabla (H (x) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})) \\ = & Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) \nabla H (x) + H (x) \nabla Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) \\ = & Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) \nabla H (x) \\ = & Φ^{ω} (\frac{1}{T^{2 r}}) x . \end{array}$ Therefore, $\begin{matrix} (2.7) & \frac{\partial D}{\partial n^{+}} (t, x) = - η_{T} (t) Φ^{ω} (\frac{1}{T^{2 r}}), (t, x) \in [0, \infty) \times \partial D . \end{matrix}$ Observe that $\begin{matrix} \frac{\partial D}{\partial n^{+}} \in C ([0, \infty) \times \partial D) \end{matrix}$ and $\begin{matrix} \frac{\partial D}{\partial n^{+}} (t, x) ⩽ 0, (t, x) \in [0, \infty) \times \partial D . \end{matrix}$ Hence, (ii) is satisfied. Note that by (2.4), we have $\begin{matrix} D (t, \cdot) \equiv 0, t ⩾ T, \end{matrix}$ which proves that (iii) is satisfied. Finally, by the definition of the function Φ, we have $\begin{matrix} D (\cdot, x) \equiv 0, | x | ⩾ \sqrt{2} T^{r}, \end{matrix}$ which proves that (iv) is satisfied with $R = \sqrt{2} T^{r}$ . □

Similarly, we have
Lemma 2.3.
For T large enough, the function $N$ given by ( 2.3 ) satisfies conditions (i)-(iv) of Definition 1.2 .

Using Lemma 2.1 and the fact that $Δ H = 0$ in $D^{c}$ , we obtain the following identity.
Lemma 2.4.
We have $\begin{matrix} Δ ψ_{T} (x) = H (x) Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] + \frac{2}{| x |^{2}} ⟨ x, \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] ⟩, x \in D^{c} . \end{matrix}$

Further, we shall prove some estimates that will be used later. Using Lemma 2.4, the following estimate holds.
Lemma 2.5.
We have $\begin{matrix} | Δ ψ_{T} (x) | ⩽ H (x) | Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] | + \frac{2}{| x |} | \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |, x \in D^{c} . \end{matrix}$

Using Lemma 2.5 and the inequality $\begin{matrix} {(a + b)}^{m} ⩽ 2^{m - 1} (a^{m} + b^{m}), a ⩾ 0, b ⩾ 0, m > 1, \end{matrix}$ we obtain the following estimate.
Lemma 2.6.
Let $m > 1$ . For all $x \in D^{c}$ , we have $\begin{matrix} {| Δ ψ_{T} (x) |}^{m} ⩽ C_{m} ({[ln (| x |)]}^{m} {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m} + \frac{1}{| x |^{m}} {| \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m}), \end{matrix}$ where $C_{m} = 2^{2 m - 1}$ .

Further, let $\begin{matrix} Q_{T} = [0, T] \times D^{c} . \end{matrix}$ Lemma 2.7.
Let $m > 1$ and $m^{'} = \frac{m}{m - 1}$ . We have $\begin{matrix} (2.8) & \int_{Q_{T}} {[D (t, x)]}^{\frac{- 1}{m - 1}} {| \partial_{t} D (t, x) |}^{m^{'}} d x d t = O (T^{1 + 2 r - m^{'}} ln T), as T \to \infty \end{matrix}$ and $\begin{matrix} (2.9) & \int_{Q_{T}} {[N (t, x)]}^{\frac{- 1}{m - 1}} {| \partial_{t} N (t, x) |}^{m^{'}} d x d t = O (T^{1 + 2 r - m^{'}}), as T \to \infty . \end{matrix}$
Proof.
Using (2.2), we have $\begin{matrix} {[D (t, x)]}^{\frac{- 1}{m - 1}} {| \partial_{t} D (t, x) |}^{m^{'}} = ψ_{T} (x) {[η_{T} (t)]}^{\frac{- 1}{m - 1}} {| η_{T}^{'} (t) |}^{m^{'}}, (t, x) \in Q_{T}, \end{matrix}$ which yields $\begin{matrix} (2.10) & \begin{matrix} \int_{Q_{T}} {[D (t, x)]}^{\frac{- 1}{m - 1}} | \partial_{t} D |^{m^{'}} d x d t \\ = (\int_{D^{c}} ψ_{T} (x) d x) (\int_{0}^{T} {[η_{T} (t)]}^{\frac{- 1}{m - 1}} {| η_{T}^{'} (t) |}^{m^{'}} d t) . \end{matrix} \end{matrix}$ On the other hand, using (2.5) and the definition of Φ, for T large enough, we obtain $\begin{array}{rcl} \int_{D^{c}} ψ_{T} (x) d x & = & \int_{D^{c}} H (x) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) d x \\ = & \int_{D^{c}} ln (| x |) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) d x \\ = & T^{2 r} \int_{R^{2} ∖ \overline{B (0, T^{- r})}} ln (T^{r} | y |) Φ^{ω} (| y |^{2}) d y \\ = & T^{2 r} \int_{B (0, \sqrt{2}) ∖ \overline{B (0, T^{- r})}} ln (T^{r} | y |) Φ^{ω} (| y |^{2}) d y \\ ⩽ & T^{2 r} ln (\sqrt{2} T^{r}) \int_{B (0, \sqrt{2}) ∖ \overline{B (0, T^{- r})}} Φ^{ω} (| y |^{2}) d y \\ ⩽ & meas (B (0, \sqrt{2})) T^{2 r} ln (\sqrt{2} T^{r}), \end{array}$ which yields $\begin{matrix} (2.11) & \int_{D^{c}} ψ_{T} (x) d x = O (T^{2 r} ln T), as T \to \infty . \end{matrix}$ Further, using (2.4), after simplification, we get $\begin{matrix} (2.12) & \int_{0}^{T} {[η_{T} (t)]}^{\frac{- 1}{m - 1}} {| η_{T}^{'} (t) |}^{m^{'}} d t = O (T^{1 - \frac{m}{m - 1}}), as T \to \infty . \end{matrix}$ Next, combining (2.10), (2.11) and (2.12), (2.8) follows.

Now, we shall prove (2.9). Using (2.3), we have $\begin{matrix} {[N (t, x)]}^{\frac{- 1}{m - 1}} {| \partial_{t} N (t, x) |}^{m^{'}} = χ_{T} (x) {[η_{T} (t)]}^{\frac{- 1}{m - 1}} {| η_{T}^{'} (t) |}^{m^{'}}, (t, x) \in Q_{T}, \end{matrix}$ which yields $\begin{matrix} (2.13) & \begin{matrix} \int_{Q_{T}} {[N (t, x)]}^{\frac{- 1}{m - 1}} | \partial_{t} N |^{m^{'}} d x d t \\ = (\int_{D^{c}} χ_{T} (x) d x) (\int_{0}^{T} {[η_{T} (t)]}^{\frac{- 1}{m - 1}} {| η_{T}^{'} (t) |}^{m^{'}} d t) . \end{matrix} \end{matrix}$ On the other hand, using (2.6) and the definition of Φ, for T large enough, we obtain $\begin{array}{l} \begin{matrix} \int_{D^{c}} χ_{T} (x) d x & = \int_{D^{c}} Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) d x \\ = T^{2 r} \int_{R^{2} ∖ \overline{B (0, T^{- r})}} Φ^{ω} (| y |^{2}) d y \\ = T^{2 r} \int_{B (0, \sqrt{2}) ∖ \overline{B (0, T^{- r})}} Φ^{ω} (| y |^{2}) d y \\ ⩽ meas (B (0, \sqrt{2})) T^{2 r}, \end{matrix} \\ (2.14) & \int_{D^{c}} χ_{T} (x) d x = O (T^{2 r}), as T \to \infty . \end{array}$ Next, combining (2.12), (2.13) and (2.14), (2.9) follows. □
Lemma 2.8.
Let $m > 1$ and $m^{'} = \frac{m}{m - 1}$ . We have $\begin{matrix} (2.15) & \int_{Q_{T}} {[D (t, x)]}^{\frac{- 1}{m - 1}} {| Δ D (t, x) |}^{m^{'}} d x d t = O (T^{1 + 2 r (1 - m^{'})} ln T), as T \to \infty \end{matrix}$ and $\begin{matrix} (2.16) & \int_{Q_{T}} {[N (t, x)]}^{\frac{- 1}{m - 1}} {| Δ N (t, x) |}^{m^{'}} d x d t = O (T^{1 + 2 r (1 - m^{'})}), as T \to \infty . \end{matrix}$
Proof.
Using (2.2), we have $\begin{matrix} {[D (t, x)]}^{\frac{- 1}{m - 1}} {| Δ D (t, x) |}^{m^{'}} = η_{T} (t) {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ ψ_{T} (x) |}^{m^{'}}, (t, x) \in Q_{T}, \end{matrix}$ which yields $\begin{matrix} (2.17) & \begin{matrix} \int_{Q_{T}} {[D (t, x)]}^{\frac{- 1}{m - 1}} {| Δ D (t, x) |}^{m^{'}} d x d t \\ = (\int_{0}^{T} η_{T} (t) d t) (\int_{D^{c}} {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ ψ_{T} (x) |}^{m^{'}} d x) . \end{matrix} \end{matrix}$ On the other hand, it can be easily seen that $\begin{matrix} (2.18) & \int_{0}^{T} η_{T} (t) d t = O (T), as T \to \infty . \end{matrix}$ Further, using Lemma 2.6 and (2.5), we obtain $\begin{array}{l} {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ ψ_{T} (x) |}^{m^{'}} \\ ⩽ C_{m^{'}} {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} ({[ln (| x |)]}^{m^{'}} {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} + \frac{1}{| x |^{m^{'}}} {| \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}}) \\ = C_{m^{'}} {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} {[ln (| x |)]}^{m^{'}} {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} + C_{m^{'}} {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} \frac{1}{| x |^{m^{'}}} {| \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} \\ = C_{m^{'}} ln (| x |) Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} \\ + C_{m^{'}} {(| x |^{m} ln (| x |))}^{\frac{- 1}{m - 1}} Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}}, \end{array}$ for all $x \in D^{c}$ . Hence, we obtain $\begin{matrix} (2.19) & \int_{D^{c}} {[ψ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ ψ_{T} (x) |}^{m^{'}} d x ⩽ I_{1} + I_{2}, \end{matrix}$ where $\begin{matrix} I_{1} = C_{m^{'}} \int_{D^{c}} ln (| x |) Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} d x \end{matrix}$ and $\begin{matrix} I_{2} = C_{m^{'}} \int_{D^{c}} {(| x |^{m} ln (| x |))}^{\frac{- 1}{m - 1}} Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} d x . \end{matrix}$ Next, we have to estimate $I_{1}$ and $I_{2}$ . Using the definition of Φ, for T large enough, we have $\begin{matrix} (2.20) & \begin{matrix} I_{1} & = C_{m^{'}} \int_{\overline{B (0, \sqrt{2} T^{r})} ∖ B (0, T^{r})} ln (| x |) Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} d x \\ = C_{m^{'}} T^{2 r - 2 r m^{'}} \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} ln (T^{r} | y |) Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| Δ [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y \\ = O (T^{2 r (1 - m^{'})} ln T) \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| Δ [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y, \end{matrix} \end{matrix}$ as $T \to \infty$ . On the other hand, for $y \in \overline{B (0, \sqrt{2})} ∖ B (0, 1)$ , we have $\begin{matrix} Δ [Φ^{ω} (| y |^{2})] = Φ^{ω - 2} (| y |^{2}) σ (y), \end{matrix}$ where $\begin{matrix} σ (y) = ω ((ω - 1) {| \nabla [Φ (| y |^{2})] |}^{2} + Φ (| y |^{2}) Δ [Φ (| y |^{2})]) . \end{matrix}$ Therefore, $\begin{array}{l} \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| Δ [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y \\ (2.21) & = \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{ω - 2 m^{'}} (| y |^{2}) {| σ (y) |}^{m^{'}} d y < \infty, \end{array}$ for $ω ⩾ 2 m^{'}$ . Hence, combining (2.20) with (2.21), we obtain $\begin{matrix} (2.22) & I_{1} = O (T^{2 r (1 - m^{'})} ln T), as T \to \infty . \end{matrix}$ Further, we have $\begin{matrix} (2.23) & \begin{matrix} I_{2} & = C_{m^{'}} \int_{\overline{B (0, \sqrt{2} T^{r})} ∖ B (0, T^{r})} {(| x |^{m} ln (| x |))}^{\frac{- 1}{m - 1}} Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| \nabla [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} d x \\ = C_{m^{'}} T^{2 r - 2 r m^{'}} \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} {(ln (T^{r} | y |))}^{\frac{- 1}{m - 1}} Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| \nabla [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y \\ = O (T^{2 r (1 - m^{'})} {(ln T)}^{\frac{- 1}{m - 1}}) \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| \nabla [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y, \end{matrix} \end{matrix}$ as $T \to \infty$ . Note that for $y \in \overline{B (0, \sqrt{2})} ∖ B (0, 1)$ , we have $\begin{matrix} \nabla [Φ^{ω} (| y |^{2})] = ω Φ^{ω - 1} (| y |^{2}) \nabla [Φ (| y |^{2})] . \end{matrix}$ Therefore, $\begin{matrix} (2.24) & \begin{matrix} \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| \nabla [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y \\ = ω^{m^{'}} \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{ω - m^{'}} (| y |^{2}) {| \nabla [Φ (| y |^{2})] |}^{m^{'}} d y \\ < \infty, \end{matrix} \end{matrix}$ for $ω ⩾ m^{'}$ . Then, combining (2.23) with (2.24), we get $\begin{matrix} (2.25) & I_{2} = O (T^{2 r (1 - m^{'})} {(ln T)}^{\frac{- 1}{m - 1}}), as T \to \infty . \end{matrix}$ Next, using (2.17), (2.18), (2.19), (2.22) and (2.25), we obtain (2.15).

Now, we shall prove (2.16). Using (2.3), we have $\begin{matrix} {[N (t, x)]}^{\frac{- 1}{m - 1}} {| Δ N (t, x) |}^{m^{'}} = η_{T} (t) {[χ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ χ_{T} (x) |}^{m^{'}}, (t, x) \in Q_{T}, \end{matrix}$ which yields $\begin{matrix} (2.26) & \begin{matrix} \int_{Q_{T}} {[N (t, x)]}^{\frac{- 1}{m - 1}} {| Δ N (t, x) |}^{m^{'}} d x d t \\ = (\int_{0}^{T} η_{T} (t) d t) (\int_{D^{c}} {[χ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ χ_{T} (x) |}^{m^{'}} d x) . \end{matrix} \end{matrix}$ Using (2.6), for T large enough, we have $\begin{array}{l} \int_{D^{c}} {[χ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ χ_{T} (x) |}^{m^{'}} d x \\ = \int_{D^{c}} Φ^{\frac{- ω}{m - 1}} (\frac{| x |^{2}}{T^{2 r}}) {| Δ [Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})] |}^{m^{'}} d x \\ = T^{2 r (1 - m^{'})} \int_{\overline{B (0, \sqrt{2})} ∖ B (0, 1)} Φ^{\frac{- ω}{m - 1}} (| y |^{2}) {| Δ [Φ^{ω} (| y |^{2})] |}^{m^{'}} d y, \end{array}$ which from (2.21) yields $\begin{matrix} (2.27) & \int_{D^{c}} {[χ_{T} (x)]}^{\frac{- 1}{m - 1}} {| Δ χ_{T} (x) |}^{m^{'}} d x = O (T^{2 r (1 - m^{'})}), as T \to \infty . \end{matrix}$ Combining (2.18), (2.26) and (2.27), (2.16) follows. □

3. Proof of Theorem 1.3

In this section, we prove Theorem 1.3. We argue by contradiction by supposing that $(u, v)$ is a global weak solution to (1.1). Given $T > 0$ large enough, taking $φ = D$ , where $D$ is given by (2.2), using Lemma 2.2, (1.7), (1.8) and condition (iii) of Definition 1.1, we obtain $\begin{array}{l} J_{T} - \int_{(0, T) \times \partial D} \frac{\partial D}{\partial n^{+}} f d S_{x} d t + \int_{D^{c}} u_{0} (x) D (0, x) d x \\ (3.1) & ⩽ \int_{Q_{T}} | u | | \partial_{t} D | d x d t + \int_{Q_{T}} | u | | Δ D | d x d t \end{array}$ and $\begin{array}{l} I_{T} - \int_{(0, T) \times \partial D} \frac{\partial D}{\partial n^{+}} g d S_{x} d t + \int_{D^{c}} v_{0} (x) D (0, x) d x \\ (3.2) & ⩽ \int_{Q_{T}} | v | | \partial_{t} D | d x d t + \int_{Q_{T}} | v | | Δ D | d x d t, \end{array}$ where $\begin{matrix} I_{T} = \int_{Q_{T}} | u |^{q} D d x d t \end{matrix}$ and $\begin{matrix} J_{T} = \int_{Q_{T}} | v |^{p} D d x d t . \end{matrix}$ On the other hand, we have $\begin{matrix} (3.3) & \int_{D^{c}} u_{0} (x) D (0, x) d x ⩾ 0, \int_{D^{c}} v_{0} (x) D (0, x) d x ⩾ 0 . \end{matrix}$ Further, using (2.7) and the definition of Φ, for T large enough, we have $\begin{matrix} \frac{\partial D}{\partial n^{+}} (t, x) = - η_{T} (t), (t, x) \in [0, T] \times \partial D, \end{matrix}$ which yields $\begin{matrix} - \int_{[0, T] \times \partial D} \frac{\partial D}{\partial n^{+}} f d S_{x} d t = (\int_{0}^{T} η_{T} (t) d t) (\int_{\partial D} f (x) d S_{x}), \end{matrix}$ for T large enough. Hence, using (2.4), for T large enough, we get $\begin{matrix} (3.4) & - \int_{[0, T] \times \partial D} \frac{\partial D}{\partial n^{+}} f d S_{x} d t = C_{1} T, \end{matrix}$ where $\begin{matrix} C_{1} = \frac{1}{λ + 1} \int_{\partial D} f (x) d S_{x} . \end{matrix}$ Similarly, we have $\begin{matrix} (3.5) & - \int_{[0, T] \times \partial D} \frac{\partial D}{\partial n^{+}} g d S_{x} d t = C_{2} T, \end{matrix}$ where $\begin{matrix} C_{2} = \frac{1}{λ + 1} \int_{\partial D} g (x) d S_{x} . \end{matrix}$ Next, writing $\begin{matrix} \int_{Q_{T}} | u | | \partial_{t} D | d x d t = \int_{Q_{T}} | u | D^{\frac{1}{q}} D^{\frac{- 1}{q}} | \partial_{t} D | d x d t \end{matrix}$ and using Hölder’s inequality with parameters q and $q^{'} = \frac{q}{q - 1}$ , we obtain $\begin{matrix} \int_{Q_{T}} | u | | \partial_{t} D | d x d t ⩽ I_{T}^{\frac{1}{q}} {(\int_{Q_{T}} D^{\frac{- 1}{q - 1}} | \partial_{t} D |^{q^{'}} d x d t)}^{\frac{q - 1}{q}} . \end{matrix}$ Using Lemma 2.7 (2.8), we get $\begin{matrix} (3.6) & \int_{Q_{T}} | u | | \partial_{t} D | d x d t ⩽ I_{T}^{\frac{1}{q}} O (T^{\frac{(1 + 2 r) (q - 1)}{q} - 1} {(ln T)}^{\frac{q - 1}{q}}), as T \to \infty . \end{matrix}$ Similarly, we have $\begin{matrix} (3.7) & \int_{Q_{T}} | v | | \partial_{t} D | d x d t ⩽ J_{T}^{\frac{1}{p}} O (T^{\frac{(1 + 2 r) (p - 1)}{p} - 1} {(ln T)}^{\frac{p - 1}{p}}), as T \to \infty . \end{matrix}$ Again, writing $\begin{matrix} \int_{Q_{T}} | u | | Δ D | d x d t = \int_{Q_{T}} | u | D^{\frac{1}{q}} D^{\frac{- 1}{q}} | Δ D | d x d t \end{matrix}$ and using Hölder’s inequality, we obtain $\begin{matrix} \int_{Q_{T}} | u | | Δ D | d x d t ⩽ I_{T}^{\frac{1}{q}} {(\int_{Q_{T}} D^{\frac{- 1}{q - 1}} | Δ D |^{q^{'}} d x d t)}^{\frac{q - 1}{q}} . \end{matrix}$ Using Lemma 2.8 (2.15), we obtain $\begin{matrix} (3.8) & \int_{Q_{T}} | u | | Δ D | d x d t ⩽ I_{T}^{\frac{1}{q}} O (T^{\frac{q - 1 - 2 r}{q}} {(ln T)}^{\frac{q - 1}{q}}), as T \to \infty . \end{matrix}$ Similarly, we have $\begin{matrix} (3.9) & \int_{Q_{T}} | v | | Δ D | d x d t ⩽ J_{T}^{\frac{1}{p}} O (T^{\frac{p - 1 - 2 r}{p}} {(ln T)}^{\frac{p - 1}{p}}), as T \to \infty . \end{matrix}$ Further, using (3.1), (3.3), (3.4), (3.6) and (3.8), we obtain $\begin{matrix} (3.10) & J_{T} + C_{1} T ⩽ I_{T}^{\frac{1}{q}} [O (T^{\frac{(1 + 2 r) (q - 1)}{q} - 1} {(ln T)}^{\frac{q - 1}{q}}) + O (T^{\frac{q - 1 - 2 r}{q}} {(ln T)}^{\frac{q - 1}{q}})], as T \to \infty . \end{matrix}$ Similarly, using (3.2), (3.3), (3.5), (3.7) and (3.9), we obtain $\begin{matrix} (3.11) & I_{T} + C_{2} T ⩽ J_{T}^{\frac{1}{p}} [O (T^{\frac{(1 + 2 r) (p - 1)}{p} - 1} {(ln T)}^{\frac{p - 1}{p}}) + O (T^{\frac{p - 1 - 2 r}{p}} {(ln T)}^{\frac{p - 1}{p}})], as T \to \infty . \end{matrix}$ Observe that for $r = \frac{1}{2}$ , we have $\begin{matrix} \frac{(1 + 2 r) (q - 1)}{q} - 1 = \frac{q - 1 - 2 r}{q} = \frac{q - 2}{q} \end{matrix}$ and $\begin{matrix} \frac{(1 + 2 r) (p - 1)}{p} - 1 = \frac{p - 1 - 2 r}{p} = \frac{p - 2}{p} . \end{matrix}$ Therefore, taking $r = \frac{1}{2}$ in (3.10) and (3.11), we obtain $\begin{matrix} (3.12) & J_{T} + C_{1} T ⩽ C_{1}^{'} T^{\frac{q - 2}{q}} {(ln T)}^{\frac{q - 1}{q}} I_{T}^{\frac{1}{q}} \end{matrix}$ and $\begin{matrix} (3.13) & I_{T} + C_{2} T ⩽ C_{2}^{'} T^{\frac{p - 2}{p}} {(ln T)}^{\frac{p - 1}{p}} J_{T}^{\frac{1}{p}}, \end{matrix}$ where $C_{1}^{'} > 0$ and $C_{2}^{'} > 0$ are constants (independent on T). Further, we discuss two cases.

Case 1.
If $\begin{matrix} max {\int_{\partial D} f (x) d S_{x}, \int_{\partial D} g (x) d S_{x}} = \int_{\partial D} f (x) d S_{x} > 0 . \end{matrix}$ In this case, we have $C_{1} > 0$ . Using (3.12) and (3.13), we obtain $\begin{matrix} (3.14) & J_{T} + C_{1} T ⩽ C_{3} T^{μ} {(ln T)}^{ν} J_{T}^{\frac{1}{p q}}, \end{matrix}$ where $\begin{matrix} (3.15) & \begin{matrix} C_{3} = C_{1}^{'} C_{2}^{' \frac{1}{q}}, \\ μ = \frac{q - 2}{q} + \frac{p - 2}{p q} \end{matrix} \end{matrix}$ and $\begin{matrix} (3.16) & ν = \frac{q - 1}{q} + \frac{p - 1}{p q} . \end{matrix}$ Using (3.14), we obtain $\begin{matrix} (3.17) & J_{T} ⩾ \frac{C_{1}^{p q}}{C_{3}^{p q}} T^{- μ p q + p q} {(ln T)}^{- ν p q} . \end{matrix}$ Substituting (3.17) into the left-hand side of (3.14) and simplifying, we get $\begin{matrix} J_{T} ⩾ \frac{C_{1}^{{(p q)}^{2}}}{C_{3}^{p q + {(p q)}^{2}}} T^{- μ (p q + {(p q)}^{2}) + {(p q)}^{2}} {(ln T)}^{- ν (p q + {(p q)}^{2})} . \end{matrix}$ Continuing with this process, by iteration, for every integer $j > 1$ , we obtain $\begin{matrix} (3.18) & J_{T} ⩾ \frac{C_{1}^{{(p q)}^{j}}}{C_{3}^{p q + {(p q)}^{2} + \dots + {(p q)}^{j}}} T^{- μ (p q + {(p q)}^{2} + \dots + {(p q)}^{j}) + {(p q)}^{j}} {(ln T)}^{- ν (p q + {(p q)}^{2} + \dots + {(p q)}^{j})} . \end{matrix}$ Next, observe that $\begin{array}{l} p q + {(p q)}^{2} + \dots + {(p q)}^{j} = p q \frac{1 - {(p q)}^{j}}{1 - p q}, \\ \frac{C_{1}^{{(p q)}^{j}}}{C_{3}^{p q + {(p q)}^{2} + \dots + {(p q)}^{j}}} = {[C_{1} C_{3}^{\frac{p q}{1 - p q}}]}^{{(p q)}^{j}} C_{3}^{\frac{p q}{p q - 1}}, \\ T^{- μ (p q + {(p q)}^{2} + \dots + {(p q)}^{j}) + {(p q)}^{j}} = {[T^{1 - \frac{μ p q}{p q - 1}}]}^{{(p q)}^{j}} T^{\frac{μ p q}{p q - 1}}, \\ {(ln T)}^{- ν (p q + {(p q)}^{2} + \dots + {(p q)}^{j})} = {[{(ln T)}^{\frac{- ν p q}{p q - 1}}]}^{{(p q)}^{j}} {(ln T)}^{\frac{ν p q}{p q - 1}} . \end{array}$ Hence, using (3.18), we get $\begin{matrix} (3.19) & J_{T} ⩾ {[C_{1} C_{3}^{\frac{p q}{1 - p q}} T^{1 - \frac{μ p q}{p q - 1}} {(ln T)}^{\frac{- ν p q}{p q - 1}}]}^{{(p q)}^{j}} C_{3}^{\frac{p q}{p q - 1}} T^{\frac{μ p q}{p q - 1}} {(ln T)}^{\frac{ν p q}{p q - 1}}, j > 1 . \end{matrix}$ On the other hand, using (3.15) and (3.16), we have $\begin{matrix} 1 - \frac{μ p q}{p q - 1} = \frac{p + 1}{p q - 1} > 0 \end{matrix}$ and $\begin{matrix} \frac{- ν p q}{p q - 1} < 0 . \end{matrix}$ Therefore, we can fix $T > 0$ large enough, so that $\begin{matrix} C_{1} C_{3}^{\frac{p q}{1 - p q}} T^{1 - \frac{μ p q}{p q - 1}} {(ln T)}^{\frac{- ν p q}{p q - 1}} > 1 . \end{matrix}$ Next, passing to the limit as $j \to \infty$ in (3.19), we obtain $\begin{matrix} J_{T} = + \infty, \end{matrix}$ which contradicts (3.14), since (3.14) implies that $\begin{matrix} J_{T} ⩽ {[C_{3} T^{μ} {(ln T)}^{ν}]}^{\frac{p q}{p q - 1}} . \end{matrix}$
Case 2.
If $\begin{matrix} max {\int_{\partial D} f (x) d S_{x}, \int_{\partial D} g (x) d S_{x}} = \int_{\partial D} g (x) d S_{x} > 0 . \end{matrix}$ In this case, we have $C_{2} > 0$ . Therefore, by arguing with a similar manner as with $J_{T}$ , we obtain a contradiction. This proves Theorem 1.3.
4. Proof of Theorem 1.5

In this section, we prove Theorem 1.5. Suppose that $(u, v)$ is a global weak solution to (1.2). Given $T > 0$ large enough, taking $φ = N$ , where $N$ is given by (2.3), using Lemma 2.3, (1.9), (1.10) and condition (iii) of Definition 1.2, we obtain $\begin{array}{l} L_{T} + \int_{(0, T) \times \partial D} f N d S_{x} d t + \int_{D^{c}} u_{0} (x) N (0, x) d x \\ (4.1) & ⩽ \int_{Q_{T}} | u | | \partial_{t} N | d x d t + \int_{Q_{T}} | u | | Δ N | d x d t \end{array}$ and $\begin{array}{l} K_{T} + \int_{(0, T) \times \partial D} g N d S_{x} d t + \int_{D^{c}} v_{0} (x) N (0, x) d x \\ (4.2) & ⩽ \int_{Q_{T}} | v | | \partial_{t} N d x d t | + \int_{Q_{T}} | v | | Δ N | d x d t, \end{array}$ where $\begin{matrix} K_{T} = \int_{Q_{T}} | u |^{q} N d x d t \end{matrix}$ and $\begin{matrix} L_{T} = \int_{Q_{T}} | v |^{p} N d x d t . \end{matrix}$ Note that we have $\begin{matrix} (4.3) & \int_{D^{c}} u_{0} (x) N (0, x) d x ⩾ 0, \int_{D^{c}} v_{0} (x) N (0, x) d x ⩾ 0 . \end{matrix}$ Next, for $(t, x) \in [0, T] \times \partial D$ , using (2.3) and the definition of Φ, for T large enough, we have $\begin{matrix} \int_{(0, T) \times \partial D} f N d S_{x} d t = (\int_{0}^{T} η_{T} (t) d t) (\int_{\partial D} f (x) d S_{x}), \end{matrix}$ which yields $\begin{matrix} (4.4) & \int_{(0, T) \times \partial D} f N d S_{x} d t = C_{1} T, \end{matrix}$ where $\begin{matrix} C_{1} = \frac{1}{λ + 1} \int_{\partial D} f (x) d S_{x} . \end{matrix}$ Similarly, we have $\begin{matrix} (4.5) & \int_{(0, T) \times \partial D} g N d S_{x} d t = C_{2} T, \end{matrix}$ where $\begin{matrix} C_{2} = \frac{1}{λ + 1} \int_{\partial D} g (x) d S_{x} . \end{matrix}$ Further, using Lemma 2.7 (2.9), Lemma 2.8 (2.16), and a similar argument as that used in the proof of Theorem 1.3, we obtain $\begin{matrix} (4.6) & \begin{matrix} \int_{Q_{T}} | u | | \partial_{t} N | d x d t ⩽ K_{T}^{\frac{1}{q}} O (T^{\frac{(1 + 2 r) (q - 1)}{q} - 1}), as T \to \infty, \\ \int_{Q_{T}} | v | | \partial_{t} N | d x d t ⩽ L_{T}^{\frac{1}{p}} O (T^{\frac{(1 + 2 r) (p - 1)}{p} - 1}), as T \to \infty, \\ \int_{Q_{T}} | u | | Δ N | d x d t ⩽ K_{T}^{\frac{1}{q}} O (T^{\frac{q - 1 - 2 r}{q}}), as T \to \infty, \\ \int_{Q_{T}} | v | | Δ N | d x d t ⩽ L_{T}^{\frac{1}{p}} O (T^{\frac{p - 1 - 2 r}{p}}), as T \to \infty . \end{matrix} \end{matrix}$ Combining (4.1), (4.2), (4.3), (4.4), (4.5) and (4.6), we obtain $\begin{matrix} (4.7) & L_{T} + C_{1} T ⩽ K_{T}^{\frac{1}{q}} [O (T^{\frac{(1 + 2 r) (q - 1)}{q} - 1}) + O (T^{\frac{q - 1 - 2 r}{q}})], as T \to \infty \end{matrix}$ and $\begin{matrix} (4.8) & K_{T} + C_{2} T ⩽ L_{T}^{\frac{1}{p}} [O (T^{\frac{(1 + 2 r) (p - 1)}{p} - 1}) + O (T^{\frac{p - 1 - 2 r}{p}})], as T \to \infty . \end{matrix}$ Taking $r = \frac{1}{2}$ in (4.7) and (4.8), we get $\begin{matrix} (4.9) & L_{T} + C_{1} T ⩽ C_{1}^{'} T^{\frac{q - 2}{q}} K_{T}^{\frac{1}{q}} \end{matrix}$ and $\begin{matrix} (4.10) & K_{T} + C_{2} T ⩽ C_{2}^{'} T^{\frac{p - 2}{p}} L_{T}^{\frac{1}{p}}, \end{matrix}$ where $C_{1}^{'} > 0$ and $C_{2}^{'} > 0$ are constants (independent on T). Next, we discuss two cases.

Case 1.
If $\begin{matrix} max {\int_{\partial D} f (x) d S_{x}, \int_{\partial D} g (x) d S_{x}} = \int_{\partial D} f (x) d S_{x} > 0 . \end{matrix}$ In this case, we have $C_{1} > 0$ . So, using (4.9), (4.10) and arguing as in the proof of Theorem 1.3, we obtain $\begin{matrix} (4.11) & L_{T} ⩾ {[A T^{1 - \frac{μ p q}{p q - 1}}]}^{{(p q)}^{j}} B T^{\frac{μ p q}{p q - 1}}, j > 1, \end{matrix}$ where $A, B > 0$ are constants (independent on j) and μ is given by (3.15). Since for any $p > 1$ , $q > 1$ , we have $\begin{matrix} 1 - \frac{μ p q}{p q - 1} > 0, \end{matrix}$ fixing $T > 0$ so that $\begin{matrix} A T^{1 - \frac{μ p q}{p q - 1}} > 0 \end{matrix}$ and passing to the limit as $j \to \infty$ in (4.11), we obtain $\begin{matrix} L_{T} = + \infty, \end{matrix}$ which leads to a contradiction.
Case 2.
If $\begin{matrix} max {\int_{\partial D} f (x) d S_{x}, \int_{\partial D} g (x) d S_{x}} = \int_{\partial D} g (x) d S_{x} > 0 . \end{matrix}$ The proof is similar to that of case 1. Hence, Theorem 1.5 is proved.
5. Open questions

In this section, two open questions are proposed.

5.1. Question 1

It would be interesting to study the critical behavior for the mixed exterior problem $\begin{matrix} (5.1) & \begin{matrix} Δ u - \partial_{t} u + | v |^{p} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ Δ v - \partial_{t} v + | u |^{q} ⩽ 0, & in & (0, \infty) \times D^{c}, \\ u (t, x) ⩾ f (x), \frac{\partial v}{\partial n^{+}} (t, x) ⩾ g (x), & in & (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), & in & D^{c}, \end{matrix} \end{matrix}$ where D is the closed unit ball in $R^{N}$ , $N ⩾ 2$ . Note that a judicious choice of the test function is required for studying (5.1).

5.2. Question 2

It would be interesting to generalize the obtained results in this paper to general external domains with compact boundary.

Footnotes

Acknowledgements

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No RGP-237.

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