Blow-up results for a semilinear parabolic differential inequality in an exterior domain

Abstract

We are concerned with a semilinear parabolic differential inequality posed in an exterior domain of $R^{N}$ , $N ⩾ 3$ . Some blow-up and existence results are established for the considered problem. The novetly of this paper lies in considering a nontrivial Dirichlet boundary condition, which depends both on time and space. Indeed, to the best of our knowledge, in all articles which deal with the blow-up of solutions in exterior domains, the considered boundary condition is trivial or depends only on space.

Keywords

Blow-up exterior domain nontrivial Dirichlet boundary condition

1. Introduction

In this paper, we study the blow-up of solutions to the problem $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u - Δ u ⩾ | u |^{p}, & (t, x) \in (0, \infty) \times D^{c}, \\ u (t, x) ⩾ a (t) f (x), & (t, x) \in (0, \infty) \times \partial D, \\ u (0, x) ⩾ g (x), & x \in D^{c}, \end{matrix} \end{matrix}$ where $D = \overline{B (0, 1)}$ is the closed unit ball in $R^{N}$ , $N ⩾ 3$ , $p > 1$ , $a \in L_{loc}^{1} ([0, \infty))$ , $a ⩾ 0$ , $a ≢ 0$ , $f \in L^{1} (\partial D)$ , $\int_{\partial D} f (x) d S_{x} > 0$ and $g \in L_{loc}^{1} (\overline{D^{c}})$ , $g ⩾ 0$ . Moreover, in some special cases, the existence of global solutions is derived. We list below some motivations for studying problem (1.1).

In [1], the authors studied the Dirichlet exterior problem $\begin{matrix} (1.2) & \{\begin{matrix} \partial_{t} u - Δ u = u^{p}, & (t, x) \in (0, \infty) \times D^{c}, \\ u (t, x) = 0, & (t, x) \in (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x) ⩾ 0, & x \in D^{c} . \end{matrix} \end{matrix}$ They showed that

(1.2) admits no global nontrivial non-negative solutions if $1 < p < 1 + \frac{2}{N}$ ;

(1.2) has global positive solutions if $p > 1 + \frac{2}{N}$ .

This study was motivated by the famous work of Fujita [4], who considered the Cauchy problem

\begin{matrix} (1.3) & \{\begin{matrix} \partial_{t} u - Δ u = u^{p}, & (t, x) \in (0, \infty) \times R^{N}, \\ u (0, x) = u_{0} (x) ⩾ 0, & x \in R^{N}, \end{matrix} \end{matrix}

and proved that:

If $1 < p < 1 + \frac{2}{N}$ and $u_{0}$ is non-negative and nontrivial, then (1.3) admits no global non-negative solutions.

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

The number

1 + \frac{2}{N}

is called the critical exponent in the sense of Fujita for problem (1.3).

For other blow-up results in exterior domains with a zero Dirichlet boundary condition, see, for example [3,7,8,10].

In [2], the authors considered the N-dimensional ( $N ⩾ 3$ ) exterior problem $\begin{matrix} (1.4) & \{\begin{matrix} \partial_{t} u - Δ u = | u |^{p}, & (t, x) \in (0, \infty) \times D^{c}, \\ u (t, x) = f (x), & (t, x) \in (0, \infty) \times \partial D, \\ u (0, x) = u_{0} (x), & x \in D^{c} . \end{matrix} \end{matrix}$ They proved the following results. Suppose that $u_{0} (x) = f (x)$ for $x \in \partial D$ . Then:

If $1 < p ⩽ \frac{N}{N - 2}$ and $\int_{\partial D} f (x) d S_{x} > 0$ , then (1.4) has no global solutions.

If $p > \frac{N}{N - 2}$ , then (1.4) has global solutions for all f with $‖ f ‖_{L^{\infty} (\partial D)}$ sufficiently small and all $u_{0}$ such that $| u_{0} (x) | ⩽ ε {(1 + | x |^{N + δ})}^{- 1}$ for any $δ > 0$ and some sufficiently small $ε > 0$ .

The main idea of the proof is to show that the

L^{p}

-norm of u blows up in a certain selected compact region. Note that the choice of this region depends on the considered problem. Moreover, their proof is based also on the maximum principle and a certain lemma (Lemma 3.2, [2], p. 636), which is valid only if the boundary condition is independent on time.

In [9], using the approach introduced in [2], the author studied the problem $\begin{matrix} (1.5) & \{\begin{matrix} \partial_{t} u - Δ u ⩾ h (t) | x |^{τ} u^{p}, & (t, x) \in (0, \infty) \times D^{c}, u ⩾ 0, \\ u (t, x) ⩾ f (x), & (t, x) \in (0, \infty) \times \partial D, \\ u (0, x) ⩾ g (x), & x \in D^{c}, \end{matrix} \end{matrix}$ where $τ > - 2$ , $p > 1$ , $f \in C^{1} (\partial D)$ , $f ⩾ 0$ , $f ≢ 0$ , $g \in C^{1} (D^{c})$ , $g ⩾ 0$ and h is continuous and satisfies $h (t) \sim t^{α}$ , as $t \to \infty$ , where $α > - 1$ . It was shown that if $N ⩾ 3$ and $\begin{matrix} 1 < p < 1 + \frac{2 + τ + 4 N α}{N - 2}, \end{matrix}$ then problem (1.5) admits no global positive solutions.

Motivated by the above cited works, problem (1.1) is investigated in this paper. Here, the novetly lies in considering a nontrivial Dirichlet boundary condition, which depends both on time and space. Our technique is based on a test function approach and some ideas introduced recently in [5,6].

Let us mention the definition of solutions to problem (1.1).

Definition 1.1.
Let $N ⩾ 3$ , $p > 1$ , $a \in L_{loc}^{1} ([0, \infty))$ , $g \in L_{loc}^{1} (\overline{D^{c}})$ and $f \in L^{1} (\partial D)$ . We say that $u \in L_{loc}^{p} ([0, \infty) \times \overline{D^{c}})$ is a global weak solution to problem (1.1), if and only if, $\begin{matrix} (1.6) & \begin{matrix} \int_{(0, \infty) \times D^{c}} | u |^{p} φ d x d t + \int_{D^{c}} g (x) φ (0, x) d x - \int_{(0, \infty) \times \partial D} \frac{\partial φ}{\partial n^{+}} a (t) f (x) d S_{x} d t \\ ⩽ - \int_{(0, \infty) \times D^{c}} u \partial_{t} φ d x d t - \int_{(0, \infty) \times D^{c}} u Δ φ d x d t, \end{matrix} \end{matrix}$ 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:
There exists $T > 0$ such that $φ (t, \cdot) \equiv 0$ , $t ⩾ T$ ;

There exists $R > 1$ such that $φ (\cdot, x) \equiv 0$ , $| x | ⩾ R$ ;

$φ_{| \partial D} \equiv 0$ ;

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

$\partial_{t} φ, Δ φ \in C ([0, \infty) \times \overline{D^{c}})$ ,
where $n^{+}$ denotes the outward (relative to $D^{c}$ ) unit normal on $\partial D$ .

Let $\begin{array}{l} A = {a \in L_{loc}^{1} ([0, \infty)) : a ⩾ 0, a ≢ 0}, \\ F = {f \in L^{1} (\partial D) : \int_{\partial D} f (x) d S_{x} > 0} \end{array}$ and $\begin{matrix} G = {g \in L_{loc}^{1} (\overline{D^{c}}) : g ⩾ 0} . \end{matrix}$

We state below the main results that we are going to prove in the present article. Our first result is the following general blow-up criterion for problem (1.1).
Theorem 1.2.
Let $N ⩾ 3$ , $p > 1$ , $a \in A$ , $f \in F$ and $g \in G$ . If $\begin{matrix} (1.7) & \underset{T \to \infty}{lim sup} (\int_{0}^{T} a (t) d t) T^{\frac{1}{p - 1} - \frac{N}{2}} = \infty, \end{matrix}$ then problem ( 1.1 ) admits no global weak solutions.

Further, some special cases of functions $a \in A$ will be discussed. We are first concerned with the set of functions $\begin{matrix} B = {a \in A : \underset{t > 0}{essinf} a (t) > 0} . \end{matrix}$
Theorem 1.3.
Let $N ⩾ 3$ .
If $a \in B$ , $f \in F$ and $g \in G$ , then for all $\begin{matrix} 1 < p < \frac{N}{N - 2}, \end{matrix}$ problem ( 1.1 ) admits no global weak solutions.

If $\begin{matrix} p > \frac{N}{N - 2}, \end{matrix}$ then problem ( 1.1 ) admits global solutions for some $a \in B$ , $f \in F$ and $g \in G$ .

Next, we are concerned with the set of functions $\begin{matrix} C_{σ} = {a \in A : a (t) \sim t^{σ}, as t \to \infty}, σ \in R . \end{matrix}$
Theorem 1.4.
Let $N ⩾ 3$ , $a \in C_{σ}$ , $σ \in R$ , $f \in F$ and $g \in G$ .
If $σ ⩽ - 1$ and $\begin{matrix} 1 < p ⩽ 1 + \frac{2}{N} (if σ = - 1), 1 < p < 1 + \frac{2}{N} (if σ < - 1), \end{matrix}$ then problem ( 1.1 ) admits no global weak solutions.

If $- 1 < σ < \frac{N - 2}{2}$ and $\begin{matrix} 1 < p < \frac{N - 2 σ}{N - 2 - 2 σ}, \end{matrix}$ then problem ( 1.1 ) admits no global weak solutions.

If $σ ⩾ \frac{N - 2}{2}$ , then for all $p > 1$ , problem ( 1.1 ) admits no global weak solutions.

Theorem 1.5.
Let $N ⩾ 3$ .
If $σ ⩽ - \frac{N}{2}$ or $1 - \frac{N}{2} ⩽ σ ⩽ 0$ , and $\begin{matrix} p > \frac{N}{N - 2}, \end{matrix}$ then problem ( 1.1 ) admits global solutions for some $a \in C_{σ}$ , $f \in F$ and $g \in G$ .

If $- \frac{N}{2} < σ < 1 - \frac{N}{2}$ and $\begin{matrix} p ⩾ 1 - \frac{1}{σ}, \end{matrix}$ then problem ( 1.1 ) admits global solutions for some $a \in C_{σ}$ , $f \in F$ and $g \in G$ .

The rest of the paper is organized as follows. In Section 2, we provide some estimates that will be used later in the proofs of our main results. In Section 3, we prove the general blow-up result given by Theorem 1.2. In Section 4, we discuss the special case when $a \in B$ and we prove Theorem 1.3. In Section 5, we prove the blow-up results given by Theorem 1.4 in the case when $a \in C_{σ}$ , $σ \in R$ . Finally, the proofs of the existence results given by Theorem 1.5 are given in Section 6.
2. Preliminary estimates

Let $N ⩾ 3$ and H be the harmonic function in $D^{c}$ given by $\begin{matrix} H (x) = 1 - | x |^{2 - N}, x \in D^{c} . \end{matrix}$ Given $0 < T < \infty$ , let $\begin{matrix} Q_{T} = [0, T] \times D^{c} and Γ_{T} = [0, T] \times \partial D . \end{matrix}$ Moreover, we introduce the functions $\begin{matrix} η_{T} (t) = \{\begin{matrix} T^{- λ} {(T - t)}^{λ} & if 0 ⩽ t ⩽ T, λ ≫ 1, \\ 0 & if t > T \end{matrix} \end{matrix}$ and $\begin{matrix} ψ_{T} (x) = H (x) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}), x \in \overline{D^{c}}, r > 0, ω ≫ 1, \end{matrix}$ where $Φ \in C_{0}^{\infty} ([0, \infty))$ is a function satisfying $\begin{matrix} 0 ⩽ Φ ⩽ 1; Φ (σ) = \{\begin{matrix} 1, & 0 ⩽ σ ⩽ 1, \\ 0, & σ ⩾ 2 . \end{matrix} \end{matrix}$ Further, let $\begin{matrix} φ_{T} (t, x) = η_{T} (t) ψ_{T} (x), (t, x) \in [0, \infty) \times \overline{D^{c}} . \end{matrix}$

Lemma 2.1.
For $T ≫ 1$ , the function $φ_{T}$ satisfies conditions (i)–(v) in Definition 1.1 .
Proof.
Let $T ≫ 1$ . By definition of $φ_{T}$ , we have $\begin{array}{l} φ_{T} ⩾ 0, φ_{T} \in C_{t, x}^{1, 2} ((0, \infty) \times D^{c}) \cap C ([0, \infty) \times \overline{D^{c}}), \\ \partial_{t} φ, Δ φ_{T} \in C ([0, \infty) \times \overline{D^{c}}), \\ {φ_{T}}_{| \partial D} \equiv 0, \\ φ_{T} (t, \cdot) \equiv 0, t ⩾ T \end{array}$ and $\begin{matrix} φ (\cdot, x) \equiv 0, | x | ⩾ \sqrt{2} T^{r} . \end{matrix}$ Therefore, conditions (i)–(iii) and (v) are satisfied. On the other hand, for $(t, x) \in (0, \infty) \times \partial D$ , we have $\begin{array}{rcl} \nabla φ_{T} (t, x) & = & η_{T} (t) \nabla ψ_{T} (x) \\ = & η_{T} (t) \nabla (H (x) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})) \\ = & η_{T} (t) (\nabla H (x) Φ^{ω} (\frac{| x |^{2}}{T^{2 r}}) + H (x) \nabla Φ^{ω} (\frac{| x |^{2}}{T^{2 r}})) \\ = & η_{T} (t) Φ^{ω} (\frac{1}{T^{2 r}}) \nabla H (x) \\ = & η_{T} (t) \nabla H (x) \\ = & η_{T} (t) (N - 2) x, \end{array}$ which yields $\begin{matrix} (2.1) & \frac{\partial φ_{T}}{\partial n^{+}} (t, x) = - (N - 2) η_{T} (t), (t, x) \in (0, \infty) \times \partial D . \end{matrix}$ Hence, by definition of $η_{T}$ , condition (iv) is satisfied. □
Lemma 2.2.
Let $N ⩾ 3$ , $p > 1$ , $a \in L_{loc}^{1} ([0, \infty))$ , $a ⩾ 0$ , $f \in F$ and $g \in G$ . If $u \in L_{loc}^{p} ([0, \infty) \times \overline{D^{c}})$ is a global weak solution to problem ( 1.1 ), then for $T ≫ 1$ , we have $\begin{matrix} I_{T} (u) + C \int_{0}^{\frac{T}{2}} a (t) d t ⩽ \int_{Q_{T}} | u | | \partial_{t} φ_{T} | d x d t + \int_{Q_{T}} | u | | Δ φ_{T} | d x d t, \end{matrix}$ where $C > 0$ is a constant (independent of T) and $\begin{matrix} I_{T} (u) = \int_{Q_{T}} | u |^{p} φ_{T} d x d t . \end{matrix}$
Proof.
Let $T ≫ 1$ . Using Lemma 2.1 and taking $φ = φ_{T}$ in (1.6), we get $\begin{matrix} \int_{(0, \infty) \times D^{c}} | u |^{p} φ_{T} d x d t + \int_{D^{c}} g (x) φ_{T} (0, x) d x - \int_{(0, \infty) \times \partial D} \frac{\partial φ_{T}}{\partial n^{+}} a (t) f (x) d S_{x} d t \\ ⩽ \int_{(0, \infty) \times D^{c}} | u | | \partial_{t} φ_{T} | d x d t + \int_{(0, \infty) \times D^{c}} | u | | Δ φ_{T} | d x d t . \end{matrix}$ Since $g ⩾ 0$ , $φ_{T} ⩾ 0$ and $φ_{T} (t, \cdot) \equiv 0$ , $t ⩾ T$ , it follows that $\begin{matrix} (2.2) & I_{T} (u) - \int_{Γ_{T}} \frac{\partial φ_{T}}{\partial n^{+}} a (t) f (x) d S_{x} d t ⩽ \int_{Q_{T}} | u | | \partial_{t} φ_{T} | d x d t + \int_{Q_{T}} | u | | Δ φ_{T} | d x d t . \end{matrix}$ On the other hand, using (2.1), we obtain $\begin{matrix} - \int_{Γ_{T}} \frac{\partial φ_{T}}{\partial n^{+}} a (t) f (x) d S_{x} d t = (N - 2) (\int_{\partial D} f (x) d S_{x}) (\int_{0}^{T} η_{T} (t) a (t) d t) . \end{matrix}$ Since $a, η_{T} ⩾ 0$ , we have $\begin{array}{rcl} \int_{0}^{T} η_{T} (t) a (t) d t & ⩾ & \int_{0}^{\frac{T}{2}} η_{T} (t) a (t) d t \\ = & T^{- λ} \int_{0}^{\frac{T}{2}} {(T - t)}^{λ} a (t) d t \\ ⩾ & \frac{1}{2^{λ}} \int_{0}^{\frac{T}{2}} a (t) d t . \end{array}$ Hence, since $N ⩾ 3$ and $\int_{\partial D} f (x) d S_{x} > 0$ , we get $\begin{matrix} (2.3) & - \int_{Γ_{T}} \frac{\partial φ_{T}}{\partial n^{+}} a (t) f (x) d S_{x} d t ⩾ C \int_{0}^{\frac{T}{2}} a (t) d t, \end{matrix}$ where $\begin{matrix} C = \frac{N - 2}{2^{λ}} (\int_{\partial D} f (x) d S_{x}) > 0 . \end{matrix}$ Combining (2.2) with (2.3), the desired estimate follows. □

We have the following estimates. Lemma 2.3 (see [6]).

Let $m > 1$ and $m^{'} = \frac{m}{m - 1}$ . Then $\begin{matrix} \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{m - 1}} | Δ φ_{T} |^{m^{'}} d x d t = O (T^{1 + N r - 2 m^{'} r}), as T \to \infty . \end{matrix}$

Lemma 2.4 (see [6]).

Let $m > 1$ and $m^{'} = \frac{m}{m - 1}$ . Then $\begin{matrix} \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{m - 1}} | \partial_{t} φ_{T} |^{m^{'}} d x d t = O (T^{1 + N r - m^{'}}), as T \to \infty . \end{matrix}$

3. Proof of Theorem 1.2

Let $N ⩾ 3$ , $p > 1$ , $a \in A$ , $f \in F$ and $g \in G$ . Suppose that (1.7) holds.

We argue by contradiction. Let $u \in L_{loc}^{p} ([0, \infty) \times \overline{D^{c}})$ be a global weak solution to problem (1.1). Using Lemma 2.2, for $T ≫ 1$ , we have $\begin{matrix} (3.1) & I_{T} (u) + C \int_{0}^{\frac{T}{2}} a (t) d t ⩽ \int_{Q_{T}} | u | | \partial_{t} φ_{T} | d x d t + \int_{Q_{T}} | u | | Δ φ_{T} | d x d t . \end{matrix}$ Writing $\begin{matrix} \int_{Q_{T}} | u | | \partial_{t} φ_{T} | d x d t = \int_{Q_{T}} (| u | {[φ_{T}]}^{\frac{1}{p}}) ({[φ_{T}]}^{\frac{- 1}{p}} | \partial_{t} φ_{T} |) d x d t \end{matrix}$ and using the ε-Young inequality, $ε > 0$ , we get $\begin{matrix} (3.2) & \int_{Q_{T}} | u | | \partial_{t} φ_{T} | d x d t ⩽ ε I_{T} (u) + C_{ε, p} \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | \partial_{t} φ_{T} |^{p^{'}} d x d t, \end{matrix}$ where $C_{ε, p} > 0$ is a constant (that depends only on p and ε) and $p^{'} = \frac{p}{p - 1}$ . Similarly, writing $\begin{matrix} \int_{Q_{T}} | u | | Δ φ_{T} | d x d t = \int_{Q_{T}} (| u | {[φ_{T}]}^{\frac{1}{p}}) ({[φ_{T}]}^{\frac{- 1}{p}} | Δ φ_{T} |) d x d t \end{matrix}$ and using the ε-Young inequality, $ε > 0$ , we get $\begin{matrix} (3.3) & \int_{Q_{T}} | u | | Δ φ_{T} | d x d t ⩽ ε I_{T} (u) + C_{ε, p} \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | Δ φ_{T} |^{p^{'}} d x d t . \end{matrix}$ Further, using (3.1), (3.2) and (3.3), for $T ≫ 1$ and $ε > 0$ , we get $\begin{matrix} (1 - 2 ε) I_{T} (u) + C \int_{0}^{\frac{T}{2}} a (t) d t ⩽ C_{ε, p} (\int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | \partial_{t} φ_{T} |^{p^{'}} d x d t + \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | Δ φ_{T} |^{p^{'}} d x d t) . \end{matrix}$ Taking $0 < ε ⩽ \frac{1}{2}$ , it follows that $\begin{matrix} (3.4) & C \int_{0}^{\frac{T}{2}} a (t) d t ⩽ C_{ε, p} (\int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | \partial_{t} φ_{T} |^{p^{'}} d x d t + \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | Δ φ_{T} |^{p^{'}} d x d t), T ≫ 1 . \end{matrix}$ Next, using Lemmas 2.3 and 2.4 with $m = p$ , we obtain $\begin{matrix} \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | \partial_{t} φ_{T} |^{p^{'}} d x d t + \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | Δ φ_{T} |^{p^{'}} d x d t = O (T^{1 + N r - 2 p^{'} r}) + O (T^{1 + N r - p^{'}}), \end{matrix}$ as $T \to \infty$ . Hence, taking $r = \frac{1}{2}$ , we get $\begin{matrix} (3.5) & \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | \partial_{t} φ_{T} |^{p^{'}} d x d t + \int_{Q_{T}} {[φ_{T}]}^{\frac{- 1}{p - 1}} | Δ φ_{T} |^{p^{'}} d x d t = O (T^{- \frac{1}{p - 1} + \frac{N}{2}}), as T \to \infty . \end{matrix}$ Combining (3.4) with (3.5), for $T ≫ 1$ , we get $\begin{matrix} (\int_{0}^{\frac{T}{2}} a (t) d t) T^{\frac{1}{p - 1} - \frac{N}{2}} ⩽ \tilde{C}, \end{matrix}$ where $\tilde{C} > 0$ is a constant (independent of T). Finally, passing to the supremum limit as $T \to \infty$ and using (1.7), a contradiction follows. This proves Theorem 1.2.

4. Proof of Theorem 1.3

Let $N ⩾ 3$ .

4.1. Proof of part (I)

Let $a \in B$ , $f \in F$ , $g \in G$ and $\begin{matrix} (4.1) & 1 < p < \frac{N}{N - 2} . \end{matrix}$ For $T > 0$ , we have $\begin{matrix} (4.2) & (\int_{0}^{T} a (t) d t) T^{\frac{1}{p - 1} - \frac{N}{2}} ⩾ m T^{\frac{p}{p - 1} - \frac{N}{2}}, \end{matrix}$ where $\begin{matrix} m = \underset{s > 0}{essinf} a (s) > 0 . \end{matrix}$ On the other hand, from (4.1), it follows that $\begin{matrix} \frac{p}{p - 1} - \frac{N}{2} > 0 . \end{matrix}$ Hence, passing to the limit as $T \to \infty$ in (4.2), we get (1.7). Using Theorem 1.2, we deduce that problem (1.1) admits no global weak solutions. This proves part (I) of Theorem 1.3.

4.2. Proof of part (II)

Let $\begin{matrix} p > \frac{N}{N - 2} . \end{matrix}$ By [11, Prposition 6.1], for some $f \in F$ with $‖ f ‖_{L^{\infty} (\partial D)}$ is sufficiently small, the problem $\begin{matrix} \{\begin{matrix} - Δ v = v^{p}, & x \in D^{c}, \\ v = f (x), & x \in \partial D \end{matrix} \end{matrix}$ admits a positive solution. Let $\begin{matrix} (4.3) & u (t, x) = v (x), (t, x) \in [0, \infty) \times \overline{D^{c}} . \end{matrix}$ Then u solves problem (1.1) with $a \equiv 1 \in B$ and $g (x) = v (x)$ . This proves part (II) of Theorem 1.3.

5. Proof of Theorem 1.4

Let $N ⩾ 3$ , $a \in C_{σ}$ , $σ \in R$ , $f \in F$ and $g \in G$ .

5.1. Proof of part (I)

Since $a \in C_{σ}$ , there exists $T_{0} > 0$ such that $\begin{matrix} a (t) ⩾ C t^{σ}, t ⩾ T_{0}, \end{matrix}$ where $C > 0$ is a constant. Hence, for $T > T_{0}$ , we have $\begin{matrix} (5.1) & \int_{0}^{T} a (t) d t ⩾ C \int_{T_{0}}^{T} t^{σ} d t = \{\begin{matrix} ln T - ln T_{0} & if σ = - 1, \\ \frac{1}{σ + 1} (T^{σ + 1} - T_{0}^{σ + 1}) & if σ \neq - 1 . \end{matrix} \end{matrix}$ Hence, if $σ = - 1$ and $\begin{matrix} 1 < p ⩽ 1 + \frac{2}{N}, \end{matrix}$ using (5.1), we have $\begin{matrix} (\int_{0}^{T} a (t) d t) T^{\frac{1}{p - 1} - \frac{N}{2}} ⩾ C (ln T - ln T_{0}) T^{\frac{1}{p - 1} - \frac{N}{2}} \to \infty, as T \to \infty . \end{matrix}$ Using Theorem 1.2, we deduce that problem (1.1) admits no global weak solutions. Next, if $σ < - 1$ and $\begin{matrix} 1 < p < 1 + \frac{2}{N}, \end{matrix}$ using (5.1), we get $\begin{matrix} (\int_{0}^{T} a (t) d t) T^{\frac{1}{p - 1} - \frac{N}{2}} ⩾ \frac{C}{σ + 1} (T^{σ + 1} - T_{0}^{σ + 1}) T^{\frac{1}{p - 1} - \frac{N}{2}} \to \infty, as T \to \infty . \end{matrix}$ By Theorem 1.2, it follows that problem (1.1) admits no global weak solutions. This proves part (I) of Theorem 1.4.

5.2. Proof of part (II)

Let $\begin{matrix} - 1 < σ < \frac{N - 2}{2} and 1 < p < \frac{N - 2 σ}{N - 2 - 2 σ} . \end{matrix}$ Using (5.1), we get $\begin{matrix} (5.2) & (\int_{0}^{T} a (t) d t) T^{\frac{1}{p - 1} - \frac{N}{2}} ⩾ \frac{C}{σ + 1} (1 - {(\frac{T_{0}}{T})}^{σ + 1}) T^{σ + \frac{p}{p - 1} - \frac{N}{2}} \to \infty, as T \to \infty . \end{matrix}$ Using Theorem 1.2, we deduce that problem (1.1) admits no global weak solutions. This proves part (II) of Theorem 1.4.

5.3. Proof of part (III)

If $\begin{matrix} σ ⩾ \frac{N - 2}{2}, \end{matrix}$ then for all $p > 1$ , we have $\begin{matrix} σ + \frac{p}{p - 1} - \frac{N}{2} > 0 . \end{matrix}$ Therefore, for all $p > 1$ , (5.2) holds. Hence, the same conclusion as before follows. This proves part (III) of Theorem 1.4.

6. Proof of Theorem 1.5

Let $N ⩾ 3$ .

6.1. Proof of part (I)

Let $σ ⩽ - \frac{N}{2}$ or $1 - \frac{N}{2} ⩽ σ ⩽ 0$ , and $\begin{matrix} p > \frac{N}{N - 2} . \end{matrix}$ Let u be the function given by (4.3), where $f \in F$ and $‖ f ‖_{L^{\infty} (\partial D)}$ is sufficiently small. For $(t, x) \in (0, \infty) \times \partial D$ , we have $\begin{matrix} u (t, x) = f (x) ⩾ {(t + 1)}^{σ} e^{\frac{- 1}{t + 1}} f (x) . \end{matrix}$ Hence, u solves problem (1.1) with $\begin{matrix} a (t) = {(t + 1)}^{σ} e^{\frac{- 1}{t + 1}}, t > 0 \end{matrix}$ and $g (x) = v (x)$ . Note that $a \in C_{σ}$ , for all $σ ⩽ 0$ . This proves part (I) of Theorem 1.5.

6.2. Proof of part (II)

$- \frac{N}{2} < σ < 1 - \frac{N}{2}$ and $\begin{matrix} p ⩾ 1 - \frac{1}{σ} . \end{matrix}$ Let $\begin{matrix} u (t, x) = ε {(t + 1)}^{σ} e^{\frac{- ρ^{2}}{4 (t + 1)}}, ρ = | x |, (t, x) \in [0, \infty) \times \overline{D^{c}}, \end{matrix}$ where $\begin{matrix} 0 < ε ⩽ {(σ + \frac{N}{2})}^{\frac{1}{p - 1}} . \end{matrix}$ For $(t, x) \in (0, \infty) \times D^{c}$ , after calculations, one obtains $\begin{array}{rcl} \partial_{t} u - Δ u & = & ε {(t + 1)}^{σ - 1} (σ + \frac{N}{2}) e^{\frac{- ρ^{2}}{4 (t + 1)}} \\ ⩾ & ε^{p} {(t + 1)}^{σ p} e^{\frac{- p ρ^{2}}{4 (t + 1)}} \\ = & u^{p} (t, x) . \end{array}$ Hence, u solves problem (1.1) with $f \equiv ε$ , $a (t) = {(t + 1)}^{σ} e^{\frac{- 1}{4 (t + 1)}}$ , $t > 0$ , and $g (x) = ε e^{\frac{- ρ^{2}}{4}}$ , $x \in D^{c}$ . This proves part (II) of Theorem 1.5.

Footnotes

Acknowledgements

The authors would like to thank the referee for his valuable comments which helped to improve the manuscript. The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP-1435-034.

References

Bandle and

H.A.

Levine, On the existence and non-existence of global solutions of reaction–diffusion equations in sectorial domains, Trans. Am. Math. Soc. 655 (1989), 595–624. doi:10.1090/S0002-9947-1989-0937878-9.

Bandle,

H.A.

Levine and

Q.S.

Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl. 251 (2000), 624–648. doi:10.1006/jmaa.2000.7035.

A.Z.

Fino,

Ibrahim and

Wehbe, A blow-up result for a nonlinear damped wave equation in exterior domain: The critical case, Comput. Math. Appl. 73 (2017), 2415–2420. doi:10.1016/j.camwa.2017.03.030.

Fujita, On the blowing up of solutions of the Cauchy problem for

u_{t} = Δ u + u^{1 + α}

, J. Fac. Sci. Univ. Tokyo. Sect. I 13 (1966), 109–124.

Jleli and

Samet, On the critical exponent for nonlinear Schrödinger equations without gauge invariance in exterior domains, J. Math. Anal. Appl. 469(1) (2019), 188–201. doi:10.1016/j.jmaa.2018.09.009.

Jleli and

Samet, New blow-up results for nonlinear boundary value problems in exterior domains, Nonlinear Analysis. 178 (2019), 348–365. doi:10.1016/j.na.2018.09.003.

Lai and

Yin, Finite time blow-up for a kind of initial-boundary value problem of semilinear damped wave equation, Math. Methods Appl. Sci. 40 (2017), 1223–1230. doi:10.1002/mma.4046.

Ogawa and

Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (2009), 3696–3701. doi:10.1016/j.na.2008.07.025.

Sun, The absence of global positive solutions to semilinear parabolic differential inequalities in exterior domain, Proc. Amer. Math. Soc. 8(145) (2017), 3455–3464.

10.

Tsutsumi, Global solutions of the nonlinear Schrödinger equation in exterior domains, Comm. Partial Differential Equations 8 (1983), 1337–1374. doi:10.1080/03605308308820306.

11.

Q.S.

Zhang, A general blow-up result on nonlinear boundary-value problems on exterior domains, Proc. R. Soc. Edinburgh. Sect. A 131(2) (2001), 451–475. doi:10.1017/S0308210500000950.