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Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

  • *Corresponding author: Grégory Faye

    *Corresponding author: Grégory Faye 

GF acknowledges support from an ANITI (Artificial and Natural Intelligence Toulouse Institute) Research Chair and from Labex CIMI under grant agreement ANR-11-LABX-0040. GF and TG acknowledge support from the ANR project Indyana under grant agreement ANR-21-CE40-0008-01. The research of MH was partially supported by the National Science Foundation (DMS-2007759)

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  • We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at $ x = \pm \infty $. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.

    Mathematics Subject Classification: Primary: 35K57, 35B40, 35K45, 35C07.


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  • Figure 1.  Illustration of $ \chi $ in cases (I) (left) and (II) (right)

    Figure 2.  Numerically computed spreading speed $ c_u^* $ (pink circles) as a function of $ c_{het} $ for Case (I) (left) and Case (II) (right). The purple plain line is the theoretical spreading speed provided by Theorem 2.1 and Theorem 2.3. In both cases parameters are fixed with $ \alpha = 1 $, $ d_+ = 1 $ and $ d_- = 1/4 $, such that the corresponding linear speeds are $ c_+ = 2 $ and $ c_- = 1 $. The function $ \chi $ was set to $ \chi(x) = \frac{d_+ e^{-\lambda x} + d_-}{1+e^{-\lambda x}} $ in Case (I) and to $ \chi(x) = \frac{d_- e^{-\lambda x} + d_+}{1+e^{-\lambda x}} $ in Case (II) with $ \lambda = 2 $. Numerical simulations were performed by discretizing equation (1) via finite differences in space and a semi-implicit scheme in time. Typical discretization step sizes were set to $ \delta t = 0.02 $ in time and $ \delta x = 0.02 $ in  

    Figure 3.  Numerically computed spreading speed $ c_u^* $ (pink circles) as a function of $ c_{het} $ for Case (II) in the degenerate case where $ d_- = 0 $. The purple plain line is the curve $ c_{het}\mapsto \frac{4 d_+ \alpha}{c_{het}} $ obtained by formally taking the limit $ d_- = 0 $ in Theorem 2.3. Other parameters are fixed with $ \alpha = 1 $ and $ d_+ = 1 $, such that the corresponding linear speed is $ c_+ = 2 $. The function $ \chi $ was set to $ \chi(x) = \frac{d_+}{1+e^{-\lambda x}} $ with $ \lambda = 2 $

    Figure 4.  Illustration of the building block of the general sub-solution (10) (before its scaling by $ \epsilon $) which is composed of two parts $ \rho \Psi_+ $ (pink curve) and $ \Psi_- $ (blue curve) in the moving frame $ z = x-ct $. It is of class $ \mathscr{C}^2 $ and compactly supported on $ \left[-\frac{\pi}{2\omega}-z_+^*,\frac{\pi}{2\beta}-z_-^*\right] $

    Figure 5.  Sketch of the super-solution $ u_\tau(t,x) $ given in Lemma 5.1 with $ C = 1 $ which is composed of three parts: it is constant and equal to $ 1 $ for $ x\leq ct-\tau $ (gray curve), and then it is the concatenation of two exponentials (blue and pink curves) for $ x\geq ct-\tau $ which are glued at $ x = c_{het}t-\tau $. Note that the factor $ \rho(t) $ is to ensure continuity between the two exponentials

    Figure 6.  Sketch of the sub-solution given in Proposition 2 which is the concatenation of the sub-solution $ \underline{u}_{1,\tau}(x-ct) $ given in Lemma 5.2 (composed of the difference of two exponentials) and the function $ {\varphi}_{\lambda_\star-\epsilon} $ which solves $ \mathcal{L} \varphi = (\lambda_\star-\epsilon)\varphi $ with prescribed asymptotic behavior at $ -\infty $. Note that the factor $ \rho(t) $ is to ensure continuity at the matching point $ x = c_{het}t-\tau/2 $

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