Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over RN\mathbb R^N

We study the propagation profile of the solution $u(x,t)$ to the nonlinear diffusion problem $u_t-\Delta u=f(u)\; (x\in \mathbb R^N,\;t>0)$, $u(x,0)=u_0(x) \; (x\in\mathbb R^N)$, 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 $t\to\infty$ 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 $\nu\in\mathbb{S}^{N-1}$, $u(x\cdot \nu, t)$ converges to a pair of one dimensional propagating terraces, one moving in the direction of $x\cdot \nu>0$, and the other is its reflection moving in the opposite direction $x\cdot\nu<0$. 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 $t\to\infty$, $V(r,t)$ 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

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component

Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type

International audienc

Anisotropic singularities of solutions of nonlinear elliptic equations in R2

AbstractWe study the solutions of Δu = u ¦u¦q − 1, q > 1, that are singular at 0. We prove that ¦x¦2(q − 1) u(x) = r2(q − 1)u(r, θ) converges to some limit ω(θ) when r = ¦x¦ tends to 0 and that ω is a 2π-periodic solution of −d2ωdθ2 + ω ¦ω¦q − 1 = (2(q − 1))2ω. If ω = 0, then either ¦x¦k u(x) = rku(r, θ) converges as r → 0 to some element of Ker(d2dθ2 + k2I) for some integer k in [1, 2(q − 1)) oru(x)Log (1¦x¦) converges to a constant. Global solutions in R2β{0}—their existence and their behavior near x = ∞ as well as near x = 0—are also studied. We use an infinite-dimensional dynamical systems theory to prove the existence of various different types of global singular solutions