We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,where 
\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}
$\phi *u$ is a spatial convolution with the top hat kernel, 
$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$. After observing that the problem is globally well-posed, we demonstrate that positive, spatially periodic solutions bifurcate from the spatially uniform steady state solution 
$u=1$ as the diffusivity, 
$D$, decreases through 
$\Delta _1 \approx 0.00297$ (the exact value is determined in Section 3). We explicitly construct these spatially periodic solutions as uniformly valid asymptotic approximations for 
$D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly spaced, compactly supported regions with width of 
$O(1)$ where 
$u=O(1)$, separated by regions where 
$u$ is exponentially small at leading order as 
$D \to 0^+$. From numerical solutions, we find that for 
$D \geq \Delta _1$, permanent form travelling waves, with minimum wavespeed, 
$2 \sqrt{D}$, are generated, whilst for 
$0 \lt D \lt \Delta _1$, the wavefronts generated separate the regions where 
$u=0$ from a region where a steady periodic solution is created via a distinct periodic shedding mechanism acting immediately to the rear of the advancing front, with this mechanism becoming more pronounced with decreasing 
$D$. The structure of these transitional travelling wave forms is examined in some detail.
We propose and investigate a stage-structured SLIRM epidemic model with latent period in a spatially continuous habitat. We first show the existence of semi-travelling waves that connect the unstable disease-free equilibrium as the wave coordinate goes to − ∞, provided that the basic reproduction number 
$\mathcal {R}_0 > 1$ and 
$c > c_*$ for some positive number 
$c_*$. We then use a combination of asymptotic estimates, Laplace transform and Cauchy's integral theorem to show the persistence of semi-travelling waves. Based on the persistent property, we construct a Lyapunov functional to prove the convergence of the semi-travelling wave to an endemic (positive) equilibrium as the wave coordinate goes to + ∞. In addition, by the Laplace transform technique, the non-existence of bounded semi-travelling wave is also proved when 
$\mathcal {R}_0 > 1$ and 
$0 < c < c_*$. This indicates that 
$c_*$ is indeed the minimum wave speed. Finally simulations are given to illustrate the evolution of profiles.
We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension 
$d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order 
$2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in 
$C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.
