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# Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

• In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in $n$–dimensional space. More precisely, we prove that all planar traveling waves with speed $c>c^*$ are exponentially stable in $L^{\infty}(\mathbb{R}^n )$ in the form of $t^{-\frac{n}{2\alpha }}{\rm{e}}^{-\varepsilon_{\tau} \sigma t}$ for some constants $\sigma >0$ and $\varepsilon_{\tau} \in (0,1)$, where $\varepsilon_{\tau} = \varepsilon(\tau)$ is a decreasing function refer to the time delay $\tau>0$. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed $c = c^*$, we show that they are algebraically stable in the form of $t^{-\frac{n}{2\alpha}}$. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.

Mathematics Subject Classification: 35C07, 92D25, 35B35.

 Citation: • • Figure 1.  Exact planar traveling wave $(\phi_1, \phi_2)$ of the system (5.1) with $b_1 = \frac{1}{2}, b_2 = \frac{19}{2}, r_1 = r_2 = 16, d_1 = 1, d_2 = 7$

Figure 2.  The left picture denotes the solution $u_1$ of system (1.4) with Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution $u_1(t, x)$ plots at times $t = 0, 1, 2, 10, 30, 50$ and behaves as a stable monotone increasing traveling wave (no change of the waves's shape after a large time in the sense of stability) and travels from right to left.

Figure 3.  The left picture denotes the solution $u_2$ of system (1.4) with Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution $u_2(t, x)$ plots at times $t = 0, 1, 2, 10, 30, 50$ and behaves as a stable monotone increasing traveling wave (no change of the waves's shape after a large time in the sense of stability) and travels from right to left.

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