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- W2767656341 abstract "We study the propagation profile of the solution $u(x,t)$ to the nonlinear diffusion problem $u_t-Delta u=f(u); (xin mathbb R^N,;t>0)$, $u(x,0)=u_0(x) ; (xinmathbb R^N)$, where $f(u)$ is of multistable type: $f(0)=f(p)=0$, $f'(0)<0$, $f'(p)<0$, where $p$ is a positive constant, and $f$ may have finitely many nondegenerate zeros in the interval $(0, p)$. The class of initial functions $u_0$ includes in particular those which are nonnegative and decay to 0 at infinity. We show that, if $u(cdot, t)$ converges to $p$ as $ttoinfty$ in $L^infty_{loc}(mathbb R^N)$, then the long-time dynamical behavior of $u$ is determined by the one dimensional propagating terraces introduced by Ducrot, Giletti and Matano [DGM]. For example, we will show that in such a case, in any given direction $nuinmathbb{S}^{N-1}$, $u(xcdot nu, t)$ converges to a pair of one dimensional propagating terraces, one moving in the direction of $xcdot nu>0$, and the other is its reflection moving in the opposite direction $xcdotnu<0$. Our approach relies on the introduction of the notion radial terrace solution, by which we mean a special solution $V(|x|, t)$ of $V_t-Delta V=f(V)$ such that, as $ttoinfty$, $V(r,t)$ converges to the corresponding one dimensional propagating terrace of [DGM]. We show that such radial terrace solutions exist in our setting, and the general solution $u(x,t)$ can be well approximated by a suitablly shifted radial terrace solution $V(|x|, t)$. These will enable us to obtain better convergence result for $u(x,t)$. We stress that $u(x,t)$ is a high dimensional solution without any symmetry. Our results indicate that the one dimensional propagating terrace is a rather fundamental concept; it provides the basic structure and ingredients for the long-time profile of solutions in all space dimensions." @default.
- W2767656341 created "2017-11-17" @default.
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- W2767656341 date "2017-11-02" @default.
- W2767656341 modified "2023-10-03" @default.
- W2767656341 title "Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over $mathbb R^N$" @default.
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- W2767656341 doi "https://doi.org/10.48550/arxiv.1711.00952" @default.
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