Stability of asymptotic waves in the Fisher-Stefan equation

Abstract

We establish spectral, linear, and nonlinear stability of the vanishing and slow-moving travelling waves that arise as time asymptotic solutions to the Fisher-Stefan equation. Nonlinear stability is in terms of the limiting equations that the asymptotic waves satisfy.

Figures (5)

• Figure 1: Left: a plot of the limiting solution $\bar{u}_0(x)$\ref{['eq:u0']} on $(-\infty,0]$ (solid line - blue online). The black dashed line illustrates how it compares to a standing wave solution of the classical Fisher-KPP equation on the line. Right: the corresponding phase portrait of \ref{['eq:nonlinsys']} with $c=0$. The solution (solid blue line) to the Fisher-Stefan equation is realised as the part of the unstable manifold of the saddle point $(1,0)$ of \ref{['eq:nonlinsys']} in the fourth quadrant.
• Figure 2: A sketch of the essential spectrum, $\mathop{\mathrm{\sigma_\textrm{ess}}}\nolimits{(\mathcal{L}_\infty)},$ (shaded region, blue online) of the operator $\mathcal{L}_\infty$ and consequently of $\mathcal{L}$. The operator is not Fredholm on the boundary of the essential spectrum, and has non-zero index in the interior. It is invertible in the region to the right of the essential spectrum, $\rho(\mathcal{L}_\infty)$.
• Figure 3: The angular variable $\theta(z;\lambda)$ for a general $\lambda >0$ is bounded for each $z \in (-\infty, 0]$ by the solutions when $\lambda = 0$ and for a large $\lambda_\infty \gg 0$. As neither of these solutions are ever an integer multiple of $\pi$, neither is the solution for any $\lambda$ in the interval $[0, \lambda_\infty]$. This leads to the conclusion that there is no point spectrum of \ref{['eq:qeq']}, and subsequently \ref{['eq:bvp']} with $\lambda \geq0$.
• Figure 4: Plots of the unstable manifold of the saddle at $(1,0)$ in the phase plane of \ref{['eq:nonlinsys']}. The curve (red online) intersects the vertical axis (black dashed) at the (blue online) dot. The curve is computed by numerically solving the ODE \ref{['eq:nonlinsys']}, while the blue dot is computed using the series expression for $J(-1)$. Left is $c=0.5$. Right is $c=1.5$
• Figure 5: Left: a plot of $\mu$ vs $c$ for $0<c<2$. Right: The inversion, showing $c$ vs $\mu$.

Theorems & Definitions (8)

• Theorem 3.1: Weyl's essential spectrum theorem (Theorem 2.26 from KP13)
• Lemma 3.2
• proof
• Remark 3.3
• Theorem 3.4: Sturm Oscillation theorem
• proof
• Theorem 4.1: Theorem 1.3.2 from Hen80
• Theorem 4.2: Theorem 5.1.1 from Hen80