Dirichlet Problem of a Delayed Reaction–Diffusion Equation on a Semi-infinite Interval

Abstract

We consider a nonlocal delayed reaction–diffusion equation in a semi-infinite interval that describes mature population of a single species with two age stages (immature and mature) and a fixed maturation period living in a spatially semi-infinite environment. Homogeneous Dirichlet condition is imposed at the finite end, accounting for a scenario that boundary is hostile to the species. Due to the lack of compactness and symmetry of the spatial domain, the global dynamics of the equation turns out to be a very challenging problem. We first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. Using the estimate, we are able to show the repellency of the trivial equilibrium and the existence of a positive heterogeneous steady state under the Dirichlet boundary condition. We then employ the dynamical system arguments to establish the global attractivity of the heterogeneous steady state. As a byproduct, we also obtain the existence and global attractivity of the heterogeneous steady state for the bistable evolution equation in the whole space.

Appendix: Derivation of Eq. (1.3)

Consider a species living in the semi-infinite interval ${\mathbb {R}}_+=[0, \infty )$. Let $W(t,a,x)$ be the population density (w.r.t. age $a$) of the species at time $t\ge 0$, age $a\ge 0$ and location $x \ge 0$, and $D(a)$ and $d(a)$ be the diffusion rate and death rate, respectively, at age $a$. Then by Metz-Diekmann [19], $W(t,a,x)$ satisfies the following evolution equation

$$\begin{aligned} \frac{\partial W}{\partial t}+\frac{\partial W}{\partial a}=D(a)\frac{\partial ^2 W}{\partial x^2}-d(a)W,\quad t> 0,\quad a >0,\quad x \in (0, \infty ). \end{aligned}$$

(5.1)

In this paper, we consider the homogeneous Dirichlet boundary condition at $x=0$

$$\begin{aligned} W(t,a,0) = 0, \quad (t,a)\in {\mathbb {R}}_+^2, \end{aligned}$$

(5.2)

which accounts for the scenario that the location $x=0$ is hostile to the species. Assume that the species has a fixed maturation time $\tau >0$. Then, the total mature population is given by

$$\begin{aligned} w(t,x)=\int _{\tau }^\infty W(t,a,x)\mathrm {d}a. \end{aligned}$$

(5.3)

By the biological meaning of $W(t,a,x)$, one should have the following conditions with respect to the age variable $a$:

$$\begin{aligned} \left\{ \begin{array}{ll} W(t,\infty ,x) &{}= 0, \\ W(t,0,x)&{}=b(w(t,x)) \end{array} \right. \qquad (t,x)\in {\mathbb {R}}_+^2, \end{aligned}$$

(5.4)

where $b$ is the birth function.

Differentiating (5.3), making use of (5.4), yields

$$\begin{aligned} \frac{\partial w}{\partial t}=W(t,\tau ,x) +D_m \frac{\partial ^2 w}{\partial x^2}-d_m w, \quad t >0,\quad x \in (0, \infty ) \end{aligned}$$

(5.5)

Here, for simplicity of presentation, we have assumed that the diffusion and death rates for the mature population are age independent, that is, $D(a)=D_m$ and $d(a)=d_m$ for $a \in [\tau , \infty )$. We need to determine $W(t,\tau ,x)$ in terms of $w(t,x)$. To this end, for any $s \in {\mathbb {R}}_+$, let $V^s(t,x)=W(t,t-s,x)$ for $(t,x)\in [s,s+\tau ] \times {\mathbb {R}}_+$. Then by (5.1), (5.2) and (5.4), we have

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{\partial V^s}{\partial t}(t,x) &{} = &{} D(t-s)\frac{\partial ^2 V^s}{\partial x^2}(t,x)-d(t-s)V^s(t,x), \quad (t,x)\in (s,s+\tau ]\times (0,\infty ), \\ V^s(t,0) &{} = &{} 0, \quad t\in [s,s+\tau ], \\ \ V^s(s,x) &{} = &{} b(w(s,x)), \quad x\in {\mathbb {R}}_+. \end{array} \right. \end{aligned}$$

(5.6)

Applying the Fourier sine transform (see, e.g., [11], pp. 471–473]) to (5.6), we have

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{d \hat{V}^s(t,\xi )}{d t}&{}=&{} -[D(t-s)\xi ^2+d(t-s)] \hat{V}^s(t,\xi ), \quad (t,\xi )\in (s,s+\tau ]\times (0,\infty ), \\ \hat{V}^s (s,\xi ) &{} = &{} c(s,\xi ), \quad \xi \in {\mathbb {R}}_+, \end{array} \right. \end{aligned}$$

(5.7)

where $\text{ for } \text{ all } (t,\xi )\in (s,s+\tau ]\times [0,\infty )$,

$$\begin{aligned} \hat{V}^s(t,\xi )=\frac{2}{\pi }\int _0^{\infty } V^s(t,x)sin(\xi x) \mathrm {d} x \end{aligned}$$

and

$$\begin{aligned} c(s,\xi )=\frac{2}{\pi }\int _0^{\infty } b(w(s,x))sin(\xi x) \mathrm {d} x. \end{aligned}$$

Solving (5.7) leads to

$$\begin{aligned} \begin{array}{rcl} \hat{V}^s(t,\xi )= & {} c(s,\xi )e^{-\int _s^t[D(t-s)\xi ^2+d(t-s)] \mathrm {d} t}. \end{array}. \end{aligned}$$

Hence,

$$\begin{aligned} \begin{array}{rcl} \hat{V}^{t-\tau } (t,\xi )= & {} \varepsilon \cdot c(t-\tau ,\xi ) \cdot e^{-\alpha \xi ^2}. \end{array} \end{aligned}$$

(5.8)

where $\varepsilon = \exp (-\int _0^\tau d(s) \mathrm {d}s)$ and $\alpha =\int _0^\tau D(s) \mathrm {d}s$. Taking the inverse Fourier sine transform in (5.8) gives

$$\begin{aligned} V^{t-\tau } (t,x)= \int _0^{\infty } \varepsilon \cdot c(t-\tau ,\xi ) \cdot e^{-\alpha \xi ^2} \sin (x \xi ) \mathrm {d} \xi . \end{aligned}$$

In order to obtain a concrete formula for $V^{t-\tau } (t,x)$, one only needs to follow (almost repeat) the steps on P477 in [11] for deriving the formula Eq. (10.5.39) (involving an odd extension of $b(w(s,x))$ from $[0, \infty )$ to the whole space $(-\infty , \infty )$, formulas of Fourier transforms of Gaussian functions, and the Convolution Theorem), and this will lead to

$$\begin{aligned} \begin{aligned} V^{t-\tau } (t,x)&=\frac{\varepsilon }{\sqrt{4 \pi \alpha }} \int _0^\infty b(w(t-\tau ,y))\left[ \exp \left( -\frac{(x-y)^2}{4\alpha }\right) -\exp \left( -\frac{(x+y)^2}{4\alpha }\right) \right] \mathrm {d}y \\&= \varepsilon \int _0^{\infty } b(w(t-\tau , y) [\Gamma _{\alpha }(x-y) - \Gamma _{\alpha }(x+y)] \mathrm {d}y. \end{aligned} \end{aligned}$$

(5.9)

Note that $W(t, \tau , x)=V^{t-\tau } (t,x)$. Plugging (5.9) into (5.5) gives the PDE in (1.3), where we have used $D_I(a)$ and $d_I(a)$ to denote the diffusion and death rates of the immature respectively, that is, $D_I=D|_{[0,\tau ]}$ and $d_I=d|_{[0,\tau ]}$.