[Paper Review] Planar Traveling Waves For Nonlocal Dispersion Equation With Monostable Nonlinearity

Rui Huang, Ming Mei+1 more|arXiv (Cornell University)|Mar 13, 2011

Advanced Mathematical Physics Problems28 references3 citations

TL;DR

This paper establishes the existence and global stability of planar traveling waves for a nonlocal dispersion equation with monostable nonlinearity and time delay in n-dimensional space. Using Fourier analysis and a weighted energy method with an optimal weight function, it proves that noncritical waves converge exponentially at rate $ t^{-n/eta}e^{-\mu t} $, while critical waves converge algebraically as $ t^{-n/\alpha} $, providing optimal convergence rates and extending stability theory for nonlocal equations with fractional diffusion-like kernels.

ABSTRACT

In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space u_t - J\ast u +u+d(u(t,x))= \int_{\mathbb{R}^n} f_β(y) b(u(t-τ,x-y)) dy, u(s,x)=u_0(s,x), s\in[-τ,0], \ x\in \mathbb{R}^n} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_β(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(ξ)=1-\mathcal{K}|ξ|^α+o(|ξ|^α)$ for $00$, and the critical wavefronts $ϕ(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/α}$. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function. These rates are optimal and the stability results significantly develop the existing studies for nonlocal dispersion equations.

Motivation & Objective

• To establish the existence of planar traveling wave solutions for a nonlocal dispersion equation with monostable nonlinearity and time delay in $ \mathbb{R}^n $.
• To analyze the global stability of these traveling waves under general initial data with spatial decay.
• To derive sharp convergence rates—exponential for noncritical waves and algebraic for critical waves—under the nonlocal dispersion governed by $ \hat{J}(\xi) = 1 - \mathcal{K}|\xi|^\alpha + o(|\xi|^\alpha) $ with $ 0 < \alpha \leq 2 $.
• To generalize existing results on Fisher-KPP type equations by incorporating nonlocal dispersal and time delays.

Proposed method

• Employing the Fourier transform to analyze the linearized operator around the traveling wave profile.
• Applying a weighted energy method with a carefully constructed weight function to control the decay and regularity of perturbations.
• Using comparison principles to bound solutions between upper and lower solutions, ensuring monotonicity and convergence.
• Deriving the critical wave speed $ c_* $ via the equality of two characteristic functions $ \mathcal{H}_{c_*}(\lambda_*) = \mathcal{G}_{c_*}(\lambda_*) $ and their derivatives.
• Proving global existence of solutions to the Cauchy problem using energy estimates and Sobolev embedding in weighted spaces.
• Extending the framework to a general class of nonlocal equations with a convolution-type birth rate term involving the heat kernel $ f_\beta(y) $.

Experimental results

Research questions

• RQ1 What is the minimal wave speed $ c_* $ for which planar traveling wave solutions exist in the nonlocal dispersion equation with monostable nonlinearity and time delay?
• RQ2 How do the convergence rates of traveling waves depend on the nonlocal dispersion parameter $ \alpha $, and are they optimal?
• RQ3 Can the stability of both noncritical and critical traveling waves be established under general initial data with spatial decay?
• RQ4 What is the role of time delay $ \tau $ in modifying the stability and convergence rate of traveling wave solutions?
• RQ5 How does the nonlocal dispersal kernel $ J(x) $, with $ \hat{J}(\xi) \sim 1 - \mathcal{K}|\xi|^\alpha $, affect the wave propagation and stability compared to classical diffusion?

Key findings

• Noncritical planar wavefronts $ \phi(x \cdot \mathbf{e} + ct) $ with $ c > c_* $ are globally exponentially stable with convergence rate $ t^{-n/\alpha} e^{-\mu t} $ for some $ \mu > 0 $, and this rate is optimal.
• Critical wavefronts $ \phi(x \cdot \mathbf{e} + c_* t) $ are globally stable with algebraic convergence rate $ t^{-n/\alpha} $, which is also optimal.
• The critical wave speed $ c_* $ is characterized by the condition $ \mathcal{H}_{c_*}(\lambda_*) = \mathcal{G}_{c_*}(\lambda_*) $ and $ \mathcal{H}_{c_*}'(\lambda_*) = \mathcal{G}_{c_*}'(\lambda_*) $, where $ \lambda_* > 0 $.
• The stability results hold for a broad class of monostable nonlinearities $ d(u) $, $ b(u) $, and a general birth rate kernel involving the heat kernel $ f_\beta(y) $, with $ \beta > 0 $.
• For $ \tau > 0 $, the exponential decay rate depends on a factor $ \varepsilon_1 < 1 $, while for $ \tau = 0 $, the full exponential rate is recovered.
• The framework applies to generalized equations with nonlocal birth rates, including Nicholson's blowflies and Fisher-KPP equations with nonlocal dispersal, extending stability results beyond classical PDEs.