As in [9], throughout the paper, we denote the spaces \(\mathbb{X}=H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\), \(\mathbb{Y}=L^{2}(\Omega )\). We also denote the complexification of the linear space \(\mathbb{X}_{c}:=\mathbb{X}\oplus i\mathbb{X}=\{a+ib\mid a,b\in \mathbb{X}\}\), the domain of a linear operator L by \(\mathfrak{D}(L)\), the kernel of L by \(\mathfrak{N}(L)\), and the range of L by \(\mathfrak{R}(L)\). Moreover, we take \(\langle u,v\rangle =\int _{\Omega }\bar{u}(x)v(x)\,dx\) as the inner product of Hilbert space \(\mathbb{Y}_{c}\). The Banach space of continuous and differentiable mappings from \([-\tau ,0]\) into \(\mathbb{Y}\) is denoted by \(C=C([-\tau ,0],\mathbb{Y})\) and \(C^{1}=C^{1}([-\tau ,0],\mathbb{Y})\), respectively.
For the following analysis, we decompose the spaces \(\mathbb{X}\) and \(\mathbb{Y}\) as follows:
$$ \mathbb{X}=K\oplus \mathbb{X}_{1},\qquad \mathbb{Y}=K\oplus \mathbb{Y}_{1}, $$
where
$$ K=\operatorname{span}\{1\}, \quad\quad \mathbb{X}_{1}= \biggl\{ y\in \mathbb{X} \Bigm| \int _{\Omega }y(x)\,dx=0 \biggr\} , \quad\quad \mathbb{Y}_{1}= \biggl\{ y\in \mathbb{Y}\Bigm| \int _{\Omega }y(x)\,dx=0 \biggr\} . $$
The linearization of (1.8) at \(e^{-\frac{\alpha }{d}P(x)}m(x)\) is given by
$$ \textstyle\begin{cases} \frac{\partial \omega }{\partial t}=e^{-\frac{\alpha }{d}P(x)}\nabla [e^{ \frac{\alpha }{d}P(x)}\nabla \omega ]- \lambda m(x)\omega (t-\tau ),&(t,x) \in (0,+\infty )\times \Omega , \\ \nabla \omega \cdot \vec{n}=0,&t>0,x\in \partial \Omega . \end{cases} $$
(2.1)
It follows that the solution semigroup of the problem (2.1) has the infinitesimal generator satisfying
$$ A_{\tau }(\lambda )\varphi =\dot{\varphi }, $$
with
$$ \mathfrak{D} \bigl(A_{\tau }(\lambda ) \bigr)= \bigl\{ \varphi \in C_{c}\cap C^{1}_{c}: \dot{\varphi }(0)=e^{-\frac{\alpha }{d}P(x)}\nabla \bigl[e^{\frac{\alpha }{d}P(x)} \nabla \varphi (0) \bigr]- \lambda m(x)\varphi (-\tau ) \bigr\} , $$
where \(C^{1}_{c}=C^{1}([-\tau ,0],\mathbb{Y}_{c})\). Then the spectrum of \(A_{\tau }(\lambda )\) is
$$ \sigma \bigl(A_{\tau }(\lambda ) \bigr)= \bigl\{ \mu \in \mathcal{C}, \Lambda (\lambda , \mu ,\tau )\varphi =0, \text{ for some } \varphi \in \mathbb{X}_{c}\setminus \{0\} \bigr\} $$
with
$$ \begin{aligned} \Lambda (\lambda ,\mu ,\tau )\varphi =e^{-\frac{\alpha }{d}P(x)} \nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \varphi \bigr]- \lambda m(x) \varphi e^{-\mu \tau }-\mu \varphi . \end{aligned} $$
(2.2)
Lemma 2.1
For any \(\lambda >0\), the steady state \(e^{-\frac{\alpha }{d}P(x)}m(x)\) of (1.8) is locally asymptotically stable when \(\tau =0\). Moreover, 0 is not the spectrum of \(A_{\tau }(\lambda )\), for any \(\tau >0\).
Proof
When \(\tau =0\), the spectrum of \(A_{\tau }(\lambda )\) becomes
$$ \begin{aligned} \Lambda (\lambda ,\mu ,0)\varphi =e^{-\frac{\alpha }{d}P(x)} \nabla \bigl[e^{ \frac{\alpha }{d}P(x)}\nabla \varphi \bigr]- \lambda m(x)\varphi -\mu \varphi . \end{aligned} $$
(2.3)
This, in turn, leads to the study of the linear eigenvalue problem
$$ \textstyle\begin{cases} e^{-\frac{\alpha }{d}P(x)}\nabla [e^{\frac{\alpha }{d}P(x)}\nabla \varphi ]- \lambda m(x)\varphi =\mu \varphi ,&x \in \Omega , \\ \nabla \varphi \cdot \vec{n}=0,&x\in \partial \Omega . \end{cases} $$
(2.4)
Moreover, \(\mu _{1}\) has the following variational characterization:
$$ \mu _{1}=\max_{\varphi \in \mathbb{X}_{c},\varphi \neq 0} \biggl[ \frac{-\int _{\Omega }e^{\frac{\alpha }{d}P(x)}[(\nabla \varphi )^{2}+ \lambda m(x)\varphi ^{2}]\,dx}{\int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}\,dx} \biggr], $$
(2.5)
which yields \(\mu _{1}\leq -\lambda \min_{x\in \Omega } m(x)<0\). Thus we conclude that the steady state \(e^{-\frac{\alpha }{d}P(x)}m(x)\) of (1.8) is locally asymptotically stable when \(\tau =0\).
Similarly, we can prove \(0\notin \sigma (A_{\tau }(\lambda ))\) for any \(\tau >0\). □
Next, we will show that the eigenvalues of \(A_{\tau }(\lambda )\) could pass through the imaginary axis for some \(\tau >0\). And, this is a necessary condition for hopf bifurcation to occur. Actually, \(A_{\tau }(\lambda )\) has a purely imaginary eigenvalue \(\mu =i\omega \) (\(\omega >0\)) for some \(\tau >0\), if and only if
$$ e^{-\frac{\alpha }{d}P(x)}\nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \varphi \bigr]- \lambda m(x)\varphi e^{-i\theta }-i\omega \varphi =0 $$
(2.6)
is solvable for some value of \(\omega >0\), \(\theta \in [0,2\pi )\), \(\omega \tau =\theta \), \(\varphi \in \mathbb{X}_{c}\), \(\varphi \neq 0\).
First, we give the following lemmas.
Lemma 2.2
For some \(\lambda ^{*}>0\), when \(\lambda \in (0,\lambda ^{*})\), if there exist some \((\omega _{\lambda },\theta _{\lambda },\varphi _{\lambda })\in \mathbb{R}^{+}\times \mathbb{R}\times \mathbb{X}_{c} \backslash \{0\}\) solving system (2.6), then \(\frac{\omega _{\lambda }}{\lambda }\) is uniformly bounded.
Proof
Substituting \((\omega _{\lambda },\theta _{\lambda },\varphi _{\lambda })\) into system (2.6) and multiplying \(e^{\frac{\alpha }{d}P(x)}\overline{\varphi }_{\lambda }\), integrating the result over Ω, then we get
$$ \begin{aligned} \bigl\langle \varphi _{\lambda },\nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \varphi _{\lambda } \bigr] \bigr\rangle -\lambda \int _{\Omega }e^{\frac{\alpha }{d}P(x)} m(x)\varphi ^{2}_{\lambda } e^{-i\theta _{\lambda }}-i\omega _{\lambda } \int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }=0. \end{aligned} $$
(2.7)
Separating the real and imaginary parts of the above equality, one can get
$$ \lambda \sin \theta _{\lambda } \int _{\Omega }e^{\frac{\alpha }{d}P(x)} m(x) \varphi ^{2}_{\lambda }= \omega _{\lambda } \int _{\Omega }e^{ \frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }. $$
Thus,
$$ \frac{\omega _{\lambda }}{\lambda }= \frac{\sin \theta _{\lambda }\int _{\Omega }e^{\frac{\alpha }{d}P(x)} m(x)\varphi ^{2}_{\lambda }}{\int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }} \leq \max_{\Omega }m(x) . $$
□
Lemma 2.3
Let \(\mathbf{L}:=\nabla [e^{\frac{\alpha }{d}P(x)}\nabla ]\). If \(\nu \in \mathbb{X}_{c}\) and \(\langle \nu , 1\rangle =0\), then
$$ \bigl\vert \langle \mathbf{L}\nu ,\nu \rangle \bigr\vert \geq \gamma _{2} \Vert \nu \Vert ^{2}_{ \mathbb{Y}_{c}}, $$
where \(\gamma _{2}\) is the second eigenvalue of operator −L.
Proof
It is well known that the operator −L on the domain Ω with zero-Neumann boundary conditions has a sequence of eigenvalues \(\{\gamma _{n}\}^{\infty }_{n=1}\) satisfying
$$ 0=\gamma _{1}< \gamma _{2}\leq \gamma _{3}\leq \cdots , \quad\quad \lim_{n \rightarrow \infty }\gamma _{n}=\infty $$
and the corresponding eigenfunctions \(\{\phi _{n}\}^{\infty }_{n=1}\), construct an orthogonal basis of \(\mathbb{Y}_{c}\), moreover \(\phi _{1}=1\). In particular, for each \(\nu \in \mathbb{X}_{c}\) satisfying that \(\langle \nu , 1\rangle =0\), there is a sequence of real numbers \(\{c_{n}\}^{\infty }_{n=2}\) such that \(\nu =\sum^{\infty }_{n=2}c_{n}\phi _{n}\) and therefore \(\mathbf{L}\nu =\sum^{\infty }_{n=2}c_{n}\mathbf{L}\phi _{n}= \sum^{\infty }_{n=2}c_{n}\gamma _{n}\phi _{n}\). It follows from the above equality that
$$ \bigl\vert \langle \mathbf{L}\nu , \nu \rangle \bigr\vert =\gamma _{n}\sum^{\infty }_{n=2}c^{2}_{n} \Vert \phi _{n} \Vert ^{2}_{L^{2}} \geq \gamma _{2}\sum^{\infty }_{n=2}c^{2}_{n} \Vert \phi _{n} \Vert ^{2}_{Y_{\mathbb{C}}} =\gamma _{2} \Vert \nu \Vert ^{2}_{Y_{ \mathbb{C}}}. $$
□
For \(\lambda \in (0, \lambda ^{*})\), if \((\omega ,\theta ,\varphi )\) satisfies system (2.6), then, ignoring a scalar factor, we have \(\varphi =\beta +\lambda \nu \), with \(\beta >0\) and \(\Vert \varphi \Vert _{\mathbb{Y}_{c}}= \vert \Omega \vert \). Letting \(\omega =\lambda h\), and substituting these into Eq. (2.6), we obtain
$$ \textstyle\begin{cases} f_{1}(\nu ,\beta ,\theta ,h,\lambda ):=\nabla [e^{\frac{\alpha }{d}P(x)} \nabla \nu ]- e^{\frac{\alpha }{d}P(x)}m(x)(\beta +\lambda \nu ) e^{-i \theta } \\ \hphantom{f_{1}(\nu ,\beta ,\theta ,h,\lambda ):=}{}-ihe^{\frac{\alpha }{d}P(x)}(\beta +\lambda \nu )=0, \\ f_{2}(\nu ,\beta ,\theta ,h,\lambda ):=(\beta ^{2}-1) \vert \Omega \vert + \lambda ^{2} \Vert \nu \Vert _{\mathbb{Y}_{c}}=0. \end{cases} $$
(2.8)
Define \(F=(f_{1},f_{2})\) from \((\mathbb{X}_{1})_{c}\times \mathbb{R}^{4}\rightarrow \mathbb{Y}_{c} \times \mathbb{R}\). The following lemma confirms that \(F(\nu ,\beta ,\theta ,h,\lambda )=0\) is uniquely solvable when \(\lambda \rightarrow 0\).
Lemma 2.4
Consider the following equation:
$$ \textstyle\begin{cases} F(\nu ,\beta ,\theta ,h,0)=0, \\ \nu \in \mathbb{X}_{c},\quad\quad \beta \geq 0,\quad\quad \theta \in [0,\frac{\pi }{2}], \quad\quad h \geq 0. \end{cases} $$
(2.9)
Equation (2.9) has a unique solution \((\nu _{0},\beta _{0},\theta _{0},h_{0})\) as \(\lambda \rightarrow 0\). Here \(\beta _{0}=1\), \(\theta _{0}=\frac{\pi }{2}\), \(h_{0}=\frac{\int _{\Omega }m^{2}(x)}{\int _{\Omega }m(x)}\), and \(\nu _{0}\) satisfies the following equation:
$$ \nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \nu _{0} \bigr]+ie^{\frac{\alpha }{d}P(x)}m(x) -ie^{\frac{\alpha }{d}P(x)}h_{0}=0. $$
(2.10)
Proof
From the second equation of (2.8), we see that \(f_{2}(\nu ,\beta ,\theta ,h,0)\) if and only if \(\beta _{0}=1\). Substituting \(\beta _{0}=1\) into the first equation of (2.8), we can see \(\nu _{0}\) satisfies
$$ \begin{aligned} \nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \nu _{0} \bigr]- m(x)e^{ \frac{\alpha }{d}P(x)} e^{-i\theta _{0}}-ie^{\frac{\alpha }{d}P(x)}h_{0}=0. \end{aligned} $$
(2.11)
Next, integrating Eq. (2.11) over Ω, and separating the real and imaginary parts, we obtain
$$ \textstyle\begin{cases} \cos \theta _{0}\int _{\Omega }e^{\frac{\alpha }{d}P(x)}m(x)=\cos \theta _{0}e^{\frac{\alpha C}{d}}\int _{\Omega }m^{2}(x)=0, \\ \sin \theta _{0}e^{\frac{\alpha C}{d}}\int _{\Omega }m^{2}(x)=h_{0}e^{ \frac{\alpha C}{d}}\int _{\Omega }m(x). \end{cases} $$
(2.12)
By the periodicity of \(\theta _{0}\), we can set \(\theta _{0}=\frac{\pi }{2}\). Then we have \(h_{0}=\frac{\int _{\Omega }m^{2}(x)}{\int _{\Omega }m(x)}\). □
We next prove that there exists a \(\lambda _{*}>0\) such that we can solve \(F(\nu ,\beta ,\theta ,h,\lambda )=0\) for \(\lambda \in (0,\lambda _{*})\).
Lemma 2.5
There exists a \(\lambda _{*}>0\) and a unique continuously differentiable mapping \(\lambda \rightarrow (\nu _{\lambda },\beta _{\lambda },\theta _{ \lambda },h_{\lambda })\) from \((0,\lambda _{*})\) to \((\mathbb{X}_{1})_{c}\times \mathbb{R}^{3}\) such that \(F(\nu _{\lambda },\beta _{\lambda },\theta _{\lambda },h_{\lambda }, \lambda )=0\).
Proof
We define \(T=(T_{1},T_{2})\) \((\mathbb{X}_{1})_{c}\times \mathbb{R}^{3}\rightarrow \mathbb{Y}_{c} \times \mathbb{R}\) be the Fréchet derivative of F with respect to \((\nu ,\beta ,\theta ,h)\) at the point \((\nu _{0},\beta _{0},\theta _{0},h_{0})\). Therefore, we have
$$ \textstyle\begin{cases} T_{1}(\nu _{\epsilon },\beta _{\epsilon },\theta _{\epsilon },h_{ \epsilon }) =\nabla [e^{\frac{\alpha }{d}P(x)}\nabla \nu _{\epsilon }]+ ie^{ \frac{\alpha }{d}P(x)} (m-\frac{\int _{\Omega }m(x)}{ \vert \Omega \vert } )\beta _{\epsilon } \\ \hphantom{T_{1}(\nu _{\epsilon },\beta _{\epsilon },\theta _{\epsilon },h_{ \epsilon })=} {}+e^{\frac{\alpha }{d}P(x)}m(x)\theta _{ \epsilon }-ie^{\frac{\alpha }{d}P(x)}h_{\epsilon }, \\ T_{2}(\nu _{\epsilon },\beta _{\epsilon },\theta _{\epsilon },h_{ \epsilon }) =2\beta _{\epsilon }. \end{cases} $$
(2.13)
One can easily check that T is bijective from \((\mathbb{X}_{1})_{c}\times \mathbb{R}^{3}\) to \(\mathbb{Y}_{c}\times \mathbb{R}\). Thus by the implicit function theorem, there exists a \(\lambda _{*}>0\) and a continuously differentiable mapping \(\lambda \rightarrow (\nu _{\lambda },\beta _{\lambda },\theta _{ \lambda },h_{\lambda })\) from \((0,\lambda _{*})\) to \((\mathbb{X}_{1})_{c}\times \mathbb{R}^{3}\) such that \(F(\nu _{\lambda },\beta _{\lambda },\theta _{\lambda },h_{\lambda }, \lambda )=0\).
We next claim that the uniqueness, by virtue of the uniqueness of the implicit function theorem, it is sufficient to show that, if \(\nu ^{\lambda }\in (\mathbb{X}_{1})_{c}\), \(\beta ^{\lambda }, h^{\lambda }>0\), \(\theta ^{\lambda }\in [0,2\pi )\), and \(F(\nu ^{\lambda },\beta ^{\lambda },\theta ^{\lambda },h^{\lambda }, \lambda )=0\), then \((\nu ^{\lambda },\beta ^{\lambda },\theta ^{\lambda },h^{\lambda }) \rightarrow (\nu _{0},\beta _{0},\theta _{0},h_{0})\) as \(\lambda \rightarrow 0\) in the norm of \((\mathbb{X}_{1})_{c}\times \mathbb{R}^{3}\). First, the boundedness of the sequences \(\{\beta ^{\lambda }\}\), \(\{\theta ^{\lambda }\}\) and \(\{h^{\lambda }\}\) can be easily obtained from the definition, hypothesis and Lemma 2.2, respectively. By Lemma 2.3, and the first equation of Eq. (2.8), we obtain
$$ \bigl\Vert \nu ^{\lambda } \bigr\Vert ^{2}_{\mathbb{Y}_{c}}\leq \frac{1}{\gamma _{2}} \bigl\vert \bigl\langle \mathbf{L}\nu ^{\lambda },\nu ^{\lambda } \bigr\rangle \bigr\vert = \frac{1}{\gamma _{2}} \bigl\vert \bigl\langle \bigl(m(x) e^{-i\theta ^{\lambda }}+ih^{ \lambda } \bigr) \bigl(\beta ^{\lambda }+\lambda \nu ^{\lambda } \bigr),\nu ^{\lambda } \bigr\rangle \bigr\vert . $$
The boundedness of \(m(x)\) and \(\{h^{\lambda }\}\) implies that there exists a constant \(M_{1}\) such that
$$ \frac{1}{\gamma _{2}} \bigl\Vert m(x) e^{-i\theta ^{\lambda }}+ih^{\lambda } \bigr\Vert _{ \infty }\leq M_{1}. $$
Hence, we obtain \(\Vert \nu ^{\lambda } \Vert ^{2}_{\mathbb{Y}_{c}}\leq M_{1} \vert \beta ^{\lambda } \vert \Vert \nu ^{\lambda } \Vert _{\mathbb{Y}_{c}}+\lambda M_{1} \Vert \nu ^{\lambda } \Vert ^{2}_{ \mathbb{Y}_{c}}\). Accordingly, if \(\lambda _{*}\) is sufficiently small, then we have \(\lambda M_{1}\leq \frac{1}{2}\), and \(\Vert \nu ^{\lambda } \Vert _{\mathbb{Y}_{c}}\leq 2M_{1} \vert \beta ^{\lambda } \vert \Vert \nu ^{\lambda } \Vert _{\mathbb{Y}_{c}}\). As a result, \(\{\nu ^{\lambda }\}\) is bounded in \(\mathbb{Y}_{c}\) for \(\lambda \in (0,\lambda _{*})\). Since the operator \(\mathbf{L}^{-1}\) is bounded, we see that \(\{\nu ^{\lambda }\}\) is bounded in \(\mathbb{X}_{c}\), which implies that \((\nu ^{\lambda },\beta ^{\lambda },\theta ^{\lambda },h^{\lambda })\) is precompact in \(\mathbb{Y}_{c}\times \mathbb{R}^{3}\) for \(\lambda \in (0,\lambda _{*})\). Therefore, there is a subsequence \((\nu ^{\lambda _{i}},\beta ^{\lambda _{i}},\theta ^{\lambda _{i}},h^{ \lambda _{i}})\) such that
$$ \bigl(\nu ^{\lambda _{i}},\beta ^{\lambda _{i}},\theta ^{\lambda _{i}},h^{ \lambda _{i}} \bigr)\rightarrow \bigl(\nu ^{0},\alpha ^{0},\theta ^{0},h^{0} \bigr) \quad \text{in } \mathbb{Y}_{c} \times \mathbb{R}^{3}\ \lambda _{i}\rightarrow 0,\text{ as }i \rightarrow \infty . $$
Taking the limit of the equation \(\mathbf{L}^{-1}f_{1}(\nu ^{\lambda _{i}},\beta ^{\lambda _{i}}, \theta ^{\lambda _{i}},h^{\lambda _{i}},\lambda _{i})=0\) as \(i\rightarrow \infty \), we see that
$$ \bigl(\nu ^{\lambda _{i}},\beta ^{\lambda _{i}},\theta ^{\lambda _{i}},h^{ \lambda _{i}} \bigr)\rightarrow \bigl(\nu ^{0},\alpha ^{0},\theta ^{0},h^{0} \bigr) \quad \text{in } \mathbb{X}_{c} \times \mathbb{R}^{3}\ \lambda _{i}\rightarrow 0,\text{ as } i \rightarrow \infty . $$
Moreover, \(F(\nu ,\beta ,\theta ,h,0)=0\) has a unique solution, which implies that \((\nu ^{0},\beta ^{0},\theta ^{0},h^{0})=(\nu _{0},\beta _{0},\theta _{0},h_{0})\). This completes the proof. □
Remark 2.1
From Lemma 2.5, we derive that, for each \(\lambda \in (0,\lambda _{*})\), the eigenvalue problem \(\Lambda (\lambda ,i\omega ,\tau )\varphi =0\), \(\omega >0\), \(\tau \geq 0\), \(\varphi (\neq 0)\in \mathbb{X}_{c}\), has a solution \((\omega ,\tau ,\varphi )\) if and only if
$$ \omega =\omega _{\lambda }=\lambda h_{\lambda }, \quad\quad \varphi =c \varphi _{ \lambda }, \quad \text{and} \quad \tau =\tau _{n}= \frac{\theta _{\lambda }+2n\pi }{\omega _{\lambda }}, \quad n=0,1,2,\ldots, $$
where \(\varphi _{\lambda }=\beta _{\lambda }+\lambda \nu _{\lambda }\).
In the following section, we will always assume \(\lambda \in (0,\lambda _{*})\) for convenience. Actually, the interval of λ might be smaller, since further perturbation arguments are used.
Lemma 2.6
Assume that \(0<\lambda <\lambda _{*}\). Then
$$ S_{n}(\lambda )= \int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{ \lambda } \bigl[ \bigl(1-\tau _{n}\lambda m(x)e^{-i\theta _{\lambda }} \bigr) \bigr]\neq 0. $$
Proof
Since
$$ \lim_{\lambda \rightarrow 0}\theta _{\lambda }=\frac{\pi }{2}, \quad \quad \lim_{ \lambda \rightarrow 0}\varphi _{\lambda }=1, \quad\quad \lim _{\lambda \rightarrow 0}\lambda \tau _{n}= \frac{(\frac{\pi }{2}+2n\pi )\int _{\Omega }m(x)}{\int _{\Omega }m^{2}(x)}, $$
we have
$$ \lim_{\lambda \rightarrow 0}S_{n}(\lambda )=e^{\frac{\alpha C}{d}} \biggl(1+i \biggl( \frac{\pi }{2}+2n\pi \biggr) \biggr) \int _{\Omega }m(x)\neq 0. $$
□
Theorem 2.1
For \(\lambda \in (0,\lambda _{*})\), there is a neighborhood of \((\tau _{n},i\omega _{\lambda },\varphi _{\lambda })\) such that \(A_{\tau _{n}}(\lambda )\) has a simple eigenvalue \(\mu (\tau _{n})=a(\tau _{n})+ib(\tau _{n})\). Moreover, \(a(\tau _{n})=0\), and \(b(\tau _{n})=\omega _{\lambda }\).
Proof
We know that \(\mathcal{N}[A_{\tau _{n}}(\lambda )-i\omega _{\lambda }]=\operatorname{Span}[e^{i \omega _{\lambda }\theta }\varphi _{\lambda }]\), where \(\theta \in [-\tau _{n},0]\). If \(\phi _{1}\in \mathcal{N}[A_{\tau _{n}}(\lambda )-i\omega _{\lambda }]^{2}\), then
$$ \mathcal{N} \bigl[A_{\tau _{n}}(\lambda )-i\omega _{\lambda } \bigr]\phi _{1}\in \mathcal{N} \bigl[A_{\tau _{n}}(\lambda )-i\omega _{\lambda } \bigr]=\operatorname{Span} \bigl[e^{i \omega _{\lambda }\theta }\varphi _{\lambda } \bigr]. $$
Therefore, there exists a constant a such that
$$ \mathcal{N} \bigl[A_{\tau _{n}}(\lambda )-i\omega _{\lambda } \bigr]\phi _{1}=ae^{i \omega _{\lambda }\theta }\varphi _{\lambda }. $$
Hence,
$$ \textstyle\begin{cases} \dot{\phi }_{1}(\theta )=i\omega _{\lambda }\phi _{1}(\theta )+ae^{i \omega _{\lambda }\theta }\varphi _{\lambda }, \quad \theta \in [-\tau _{n},0], \\ \dot{\phi }_{1}(0)=de^{-\frac{\alpha }{d}P(x)}\nabla [e^{ \frac{\alpha }{d}P(x)}\nabla \phi _{1}(0)]- \lambda m(x)\phi _{1}(- \tau _{n}). \end{cases} $$
(2.14)
The first equation of Eq. (2.14) yields
$$ \textstyle\begin{cases} \phi _{1}(\theta )=\phi _{1}(0)e^{i\omega _{\lambda }\theta }+a\theta e^{i \omega _{\lambda }\theta }\varphi _{\lambda }, \\ \dot{\phi }_{1}(0)=i\omega _{\lambda }\phi _{1}(0)+a\varphi _{\lambda }. \end{cases} $$
(2.15)
From Eqs. (2.14) and (2.15), we have
$$ \begin{aligned} e^{\frac{\alpha }{d}P(x)}\Lambda (\lambda ,i\omega _{\lambda },\tau _{n}) \phi _{1}(0) &=\nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \phi _{1}(0)\bigr]- \lambda e^{\frac{\alpha }{d}P(x)}m(x)\phi _{1}(0) e^{-i\theta _{ \lambda }} \\ & \quad{} -ie^{\frac{\alpha }{d}P(x)}\omega _{\lambda }\phi _{1}(0) \\ &=ae^{\frac{\alpha }{d}P(x)}\bigl(\varphi _{\lambda }-\tau _{n}\lambda m(x)e^{-i \theta _{\lambda }}\varphi _{\lambda }\bigr). \end{aligned} $$
(2.16)
Hence
$$ \begin{aligned} \int _{\Omega }\phi _{1}(0)\bigl[e^{\frac{\alpha }{d}P(x)}\Lambda (\lambda ,i \omega _{\lambda },\tau _{n})\varphi _{\lambda } \bigr]&= \int _{\Omega } \varphi _{\lambda }\bigl[e^{\frac{\alpha }{d}P(x)}\Lambda (\lambda ,i\omega _{ \lambda },\tau _{n})\phi _{1}(0) \bigr] \\ &=a \int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda } \bigl[\bigl(1- \tau _{n}\lambda m(x)e^{-i\theta _{\lambda }}\bigr)\bigr]=0, \end{aligned} $$
(2.17)
which implies that \(a=0\). And it leads to that \(\mu =i\omega _{\lambda }\) is a simple eigenvalue of \(A_{\tau _{n}}(\lambda )\). It follows from the implicit function theorem that there is a neighborhood of \((\tau _{n},i\omega _{\lambda },\varphi _{\lambda })\) such that \(A_{\tau }(\lambda )\) has a simple eigenvalue \(\mu (\tau )=a(\tau )+ib(\tau )\), for \(\lambda \in (0,\lambda _{*})\). □
Theorem 2.2
Assume that \(\lambda \in (0,\lambda _{*})\), we have \(\mathcal{R}e\frac{d\mu (\tau _{n}\lambda )}{d\tau }>0\).
Proof
Since
$$ \begin{aligned} e^{\frac{\alpha }{d}P(x)}\Lambda \bigl(\lambda ,\mu (\tau ), \tau \bigr)\varphi ( \tau ) &=\nabla \bigl[e^{\frac{\alpha }{d}P(x)}\nabla \varphi (\tau ) \bigr]- \lambda e^{\frac{\alpha }{d}P(x)} m(x)\varphi (\tau ) e^{-\mu (\tau ) \tau } \\ & \quad{} -\mu (\tau )e^{\frac{\alpha }{d}P(x)}\varphi (\tau )=0. \end{aligned} $$
(2.18)
Differentiating above equation with respect to τ at \(\tau =\tau _{n}\) yields
$$\begin{aligned} \begin{aligned} & e^{\frac{\alpha }{d}P(x)}\Lambda (\lambda ,i\omega _{\lambda },\tau _{n}) \frac{d\varphi (\tau _{n})}{d\tau } +\frac{d\mu (\tau _{n})}{d\tau }e^{ \frac{\alpha }{d}P(x)} \bigl(\lambda \tau _{n}m(x)\varphi _{\lambda }e^{-i \theta _{\lambda }}- \varphi _{\lambda }\bigr) \\ &\quad{} +i\omega _{\lambda }\lambda m(x)e^{\frac{\alpha }{d}P(x)}\varphi _{ \lambda }e^{-i\theta _{\lambda }}=0. \end{aligned} \end{aligned}$$
(2.19)
Then, multiplying the above equation by \(\varphi _{\lambda }\) and integrating the result over Ω, we have
$$ \begin{aligned} \frac{d\mu (\tau _{n})}{d\tau }&= \frac{\int _{\Omega }i\omega _{\lambda }\lambda m(x)e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }e^{-i\theta _{\lambda }}}{\int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }[(1-\tau _{n}\lambda m(x)e^{-i\theta _{\lambda }})]} \\ &= \frac{\int _{\Omega }i\omega _{\lambda }\lambda m(x)e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }e^{-i\theta _{\lambda }}\int _{\Omega }e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }-i\omega _{\lambda }\lambda ^{2}\tau _{n}(\int _{\Omega }m(x)e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda })^{2}}{ \vert S_{n}(\lambda ) \vert ^{2}}. \end{aligned} $$
(2.20)
Since \(\lim_{\lambda \rightarrow 0}\sin \theta _{\lambda }=1\),
$$ \begin{aligned} \mathcal{R}e\frac{d\mu (\tau _{n}\lambda )}{d\tau }= \frac{\lambda \sin \theta _{\lambda }\omega _{\lambda }\int _{\Omega } m(x)e^{\frac{\alpha }{d}P(x)}\varphi ^{2}_{\lambda }\int _{\Omega } e^{\frac{\alpha }{d}P(x)} \varphi ^{2}_{\lambda }}{ \vert S_{n}(\lambda ) \vert ^{2}}>0 \quad \text{for } \lambda \in (0,\lambda _{*}). \end{aligned} $$
(2.21)
□
From the above lemmas and theorems, we immediately have the following result.
Theorem 2.3
For \(\lambda \in (0,\lambda _{*})\), the infinitesimal generator \(A_{\tau _{n}}(\lambda )\) has exactly \(2(n+1)\) eigenvalues with positive real parts when \(\tau \in (\tau _{n},\tau _{n+1}]\), where \(n =0, 1, 2,\ldots \) .
Moreover, by virtue of [22], we have the local Hopf bifurcation theorem for partial functional differential equations as follows.
Theorem 2.4
For each fixed \(\lambda \in (0,\lambda _{*})\), the positive steady state \(e^{-\frac{\alpha }{d}P(x)}m(x)\) of (1.8) is locally asymptotically stable when \(\tau \in [0,\tau _{0})\), and is unstable when \(\tau \in [\tau _{0},\infty )\). Furthermore, in system (1.8) there occurs a Hopf bifurcation at the positive steady state \(e^{-\frac{\alpha }{d}P(x)}m(x)\), when \(\tau =\tau _{n}\) (\(n=0,1,2,\ldots \)).