Regularity of Transition Fronts in Nonlocal Dispersal Evolution Equations

Abstract

It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction–diffusion equations to nonlocal equations, and hence serves as an initial and fundamental step for further studying various important qualitative properties of transition fronts such as stability, uniqueness and asymptotic speeds. We also prove the existence of continuously differentiable and increasing interface location functions, which give a better characterization of the propagation of transition fronts and are of great technical importance.

Appendix: Ignition Traveling Waves

Consider the homogeneous ignition equation

$$\begin{aligned} u_{t}=J*u-u+f_{I}(u),\quad (t,x)\in \mathbb {R}\times \mathbb {R}, \end{aligned}$$

(7.1)

where J is as in (H1), and the $C^{2}$ function $f_{I}:[0,1]\rightarrow \mathbb {R}$ is of standard ignition type, that is, there is $\theta _{I}\in (0,1)$ such that

$$\begin{aligned} f_{I}(u)=0,\,\,u\in [0,\theta _{I}]\cup \{1\},\quad f_{I}(u)>0,\,\,u\in (\theta _{I},1)\quad \text {and}\quad f_{I}'(1)<0. \end{aligned}$$

It was proven in [16] that the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} J*\phi -\phi +c\phi _{x}+f_{I}(\phi )=0,\\ \phi _{x}<0,\,\,\phi (0)=\theta _{I},\,\,\phi (-\infty )=1\,\,\text {and}\,\,\phi (\infty )=0. \end{array}\right. } \end{aligned}$$

for $(c,\phi )$ has a unique classical solution $(c_{I},\phi _{I})$ with $c_{I}>0$.

We used the following result in the previous sections.

Lemma 7.1

Let $u_{0}:\mathbb {R}\rightarrow [0,1]$ be uniformly continuous and satisfy

$$\begin{aligned} \lim _{x\rightarrow -\infty }u_{0}(x)=1\quad \text {and}\quad u_{0}(x)\le e^{-\alpha _{0} (x-x_{0})},\,\,x\in \mathbb {R}\end{aligned}$$

for some $\alpha _{0}>0$ and $x_{0}\in \mathbb {R}$, then there exist $\omega _{I}=\omega _{I}(\alpha _{0})>0$ and $\epsilon _{I}>0$ such that for any $\epsilon \in (0,\epsilon _{I}]$ there exist $\xi ^{\pm }_{I}=\xi ^{\pm }_{I}(\epsilon ,u_{0})\in \mathbb {R}$ such that

$$\begin{aligned} \phi _{I}(x-c_{I}t-\xi ^{-}_{I})-\epsilon e^{-\omega _{I}t}\le u_{I}(t,x;u_{0})\le \phi _{I}(x-c_{I}t-\xi ^{+}_{I})+\epsilon e^{-\omega _{I}t},\quad x\in \mathbb {R}\end{aligned}$$

for all $t\ge 0$, where $u_{I}(t,x;u_{0})$ is the solution of (7.1) with initial data $u_{I}(0,\cdot ;u_{0})=u_{0}$.

Lemma 7.1 can be proven as [51, Theorem 1.4]; we here simply recall it for completeness. To do so, we fix $L_{1}>0$ so large that

$$\begin{aligned} \phi _{I}(-L_{1})\ge \frac{1+\tilde{\theta }_{I}}{2}\quad \text {and}\quad \phi _{I}(L_{1})\le \frac{\theta _{I}}{2}, \end{aligned}$$

where $\tilde{\theta }_{I}\in (\theta _{I},1)$ is such that

$$\begin{aligned} f_{I}'(u)\le -\tilde{\beta }_{I},\quad u\in [\tilde{\theta }_{I},1] \end{aligned}$$

(7.2)

for some $\tilde{\beta }_{I}>0$ (such $\tilde{\theta }_{I}$ and $\tilde{\beta }_{I}$ exist due to $f_{I}'(1)<0$).

For $\alpha >0$, let $\Gamma _{\alpha }:\mathbb {R}\rightarrow [0,1]$ be a smooth nonincreasing function satisfying

$$\begin{aligned} \Gamma _{\alpha }={\left\{ \begin{array}{ll} 1,\quad &{}x\le -L_{1}-1,\\ e^{-\alpha (x-L_{1})},\quad &{}x\ge L_{1}+1. \end{array}\right. } \end{aligned}$$

We have

Lemma 7.2

There exists $\alpha _{*}>0$ such that such that for any $\alpha \in (0,\alpha _{*}]$ there exists $L_{2}=L_{2}(\alpha )>L_{1}+1$ such that

$$\begin{aligned} \big |[J*\Gamma _{\alpha }](x)-e^{-\alpha (x-L_{1})}\big |\le \frac{c_{I}}{4}\alpha e^{-\alpha (x-L_{1})},\quad x\ge L_{2}. \end{aligned}$$

Proof

See [51, Lemma 4.1]. It requires the symmetry of J so that $\int _{\mathbb {R}}J(x)e^{\alpha x}dx-1=O(\alpha ^{2})$. $\square$

Now, we prove Lemma 7.1.

Proof of Lemma 7.1

Let $\alpha =\frac{\alpha _{0}}{2}$ and $L_{2}=L_{2}(\alpha )$ as in Lemma 7.2. For any $\epsilon \in (0,\epsilon _{I}]$, where $\epsilon _{I}>0$ is to be chosen, we can find $\xi ^{\pm }=\xi _{0}^{\pm }(\epsilon ,u_{0})\in \mathbb {R}$ such that

$$\begin{aligned} \phi _{I}(x-\xi ^{-})-\epsilon \Gamma _{\alpha }(x-\xi ^{-})\le u_{0}(x)\le \phi _{I}(x-\xi ^{+})+\epsilon \Gamma _{\alpha }(x-\xi ^{+}),\quad x\in \mathbb {R}. \end{aligned}$$

(7.3)

Setting $\xi ^{\pm }(t)=\xi ^{\pm }\pm \frac{A\epsilon }{\omega }(1-e^{-\omega t})$, where $A>0$ and $\omega >0$ is to be chosen, we define

$$\begin{aligned} u^{\pm }(t,x)=\phi (x-ct-\xi ^{\pm }(t))\pm \epsilon e^{-\omega t}\Gamma (x-ct-\xi ^{\pm }(t)),\quad t\ge 0, \end{aligned}$$

where $\phi =\phi _{I}$, $c=c_{I}$ and $\Gamma =\Gamma _{\alpha }$. Clearly, $u^{-}(0,\cdot )\le u_{0}\le u^{+}(0,\cdot )$. Thus, if we can show that $u^{-}(t,x)$ and $u^{+}(t,x)$ are sub- and super-solutions, respectively, then the lemma follows.

We show that $u^{-}(t,x)$ is a sub-solution; $u^{+}(t,x)$ being a super-solution can be proven along the same line. We compute

$$\begin{aligned}&u^{-}_{t}-[J*u^{-}-u^{-}]-f_{I}(u^{-})\\&\quad =A\epsilon e^{-\omega t}\phi '+\epsilon \omega e^{-\omega t}\Gamma -\epsilon e^{-\omega t}(A\epsilon e^{-\omega t}-c)\Gamma '\\&\qquad +\,\epsilon e^{-\omega t}[J*\Gamma -\Gamma ]+f_{I}(\phi )-f_{I}(u^{-}), \end{aligned}$$

where $\phi$, $\phi '$, $\Gamma$ and $\Gamma '$ are computed at $x-ct-\xi ^{-}(t)$ and $J*\Gamma =\int _{\mathbb {R}}J(x-y)\Gamma (y-ct-\xi ^{-}(t))dy$. We consider three cases.

Case 1. $x-ct-\xi ^{-}(t)\le -L_{1}-1$ In this case, $\Gamma =1$, $\Gamma '=0$ and hence $J*\Gamma -\Gamma \le 1-1=0$. Moreover, $\phi \ge \frac{1+\tilde{\theta }_{I}}{2}$ by the monotonicity of $\phi$ and the choice of $L_{1}$, which implies that $u_{-}\ge \phi -\epsilon _{I}\ge \tilde{\theta }_{I}$ if we choose

$$\begin{aligned} \epsilon _{I}\le \frac{1-\tilde{\theta }_{I}}{2}. \end{aligned}$$

(7.4)

It then follows that $f_{I}(\phi )-f_{I}(u^{-})\le -\epsilon \tilde{\beta }_{I}e^{-\omega t}\Gamma$. Hence, we obtain

$$\begin{aligned} u^{-}_{t}-[J*u^{-}-u^{-}]-f_{I}(u^{-})\le \omega \epsilon e^{-\omega t}\Gamma -\epsilon \tilde{\beta }_{I}e^{-\omega t}\Gamma \le 0 \end{aligned}$$

if we choose

$$\begin{aligned} \omega \le \tilde{\beta }_{I}. \end{aligned}$$

(7.5)

Case 2. $x-ct-\xi ^{-}(t)\in [-L_{1}-1,L_{2}]$ In this case,

$$\begin{aligned}&A\epsilon e^{-\omega t}\phi '\le A\epsilon e^{-\omega t}\sup _{x\in [-L_{1}-1,L_{2}]}\phi '(x)<0,\\&\epsilon \omega e^{-\omega t}\Gamma -\epsilon e^{-\omega t}(A\epsilon e^{-\omega t}-c)\Gamma '+\epsilon e^{-\omega t}[J*\Gamma -\Gamma ]\le \epsilon e^{-\omega t}(\omega +1) \end{aligned}$$

if we choose

$$\begin{aligned} \epsilon _{I}\le \frac{c}{A}, \end{aligned}$$

(7.6)

and $f_{I}(\phi )-f_{I}(u^{-})\le (\sup _{u\in [0,2]}|f'_{I}(u)|)\epsilon e^{-\omega t}$ (note that it’s safe to extend $f_{I}$ to (1, 2] so that $\sup _{u\in [0,2]}|f'_{I}(u)|<\infty$). It then follows that

$$\begin{aligned} u^{-}_{t}-[J*u^{-}-u^{-}]-f_{I}(u^{-})\le \epsilon e^{-\omega t}\bigg [A\sup _{x\in [-L_{1}-1,L_{2}]}\phi '(x)+\omega +1+\sup _{u\in [0,2]}|f'_{I}(u)|\bigg ]\le 0 \end{aligned}$$

if we choose

$$\begin{aligned} A\ge -\bigg [\sup _{x\in [-L_{1}-1,L_{2}]}\phi '(x)\bigg ]^{-1}\bigg [1+2\sup _{u\in [0,2]}|f'_{I}(u)|\bigg ], \end{aligned}$$

(7.7)

since $\omega \le \tilde{\beta }_{I}\le \sup _{u\in [0,2]}|f'_{I}(u)|$ due to (7.5).

Case 3. $x-ct-\xi ^{-}(t)\ge L_{2}$ In this case, $\Gamma =e^{-\alpha (x-ct-\xi ^{-}(t)-L_{1})}$, $\Gamma '=-\alpha \Gamma$ and hence,

$$\begin{aligned} \epsilon \omega e^{-\omega t}\Gamma -\epsilon e^{-\omega t}(A\epsilon e^{-\omega t}-c)\Gamma '=\epsilon e^{-\omega t}[\omega +A\alpha \epsilon e^{-\omega t}-c\alpha ]\Gamma . \end{aligned}$$

By Lemma 7.2, we have $\epsilon e^{-\omega t}[J*\Gamma -\Gamma ]\le \epsilon e^{-\omega t}\frac{\alpha c}{4}\Gamma$. Since $f_{I}(\phi )=0=f_{I}(u^{-})$ (note it’s safe to do zero extension of f on $(-\infty ,0)$), we obtain

$$\begin{aligned} u^{-}_{t}-[J*u^{-}-u^{-}]-f_{I}(u^{-})\le \epsilon e^{-\omega t}\bigg [\omega +A\alpha \epsilon e^{-\omega t}-c\alpha +\frac{\alpha c}{4}\bigg ]\Gamma \le 0 \end{aligned}$$

if we choose

$$\begin{aligned} \omega \le \frac{\alpha c}{4}\quad \text {and}\quad \epsilon _{I}\le \frac{c}{4A} \end{aligned}$$

(7.8)

Consequently, if we choose A as in (7.7), $\omega$ as in (7.5) and (7.8), and $\epsilon _{I}$ as in (7.4) and (7.8), then we have $u^{-}_{t}-[J*u^{-}-u^{-}]-f_{I}(u^{-})\le 0$ for $t\ge 0$. This completes the proof. $\square$