We study the Fisher-KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the 'weak solution' for a class of free boundary problems that include the Fisher-KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson-Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here.

We consider positive solutions of quasilinear elliptic problems of the form Δp^u + ƒ(u) = 0 over the quarter-space Q = {x ∈ ℝ^N : x₁ > 0, x₂ > 0}, with u = 0 on ∂Q. For a general class of nonlinearities ƒ ≥ 0 with finitely many positive zeros, we show that, for each z > 0 such that ƒ(z) = 0, there is a bounded positive solution satisfying

lim u(x1, x2, ..., x_N) = V (x2),
x1→∞
lim u(x1, x2, ..., x_N) = V (x1),
x2→∞

where V is the unique solution of the one-dimensional problem

ΔpV + ƒ(V) = 0 in [0,∞), V (0) = 0, V (t) > 0 for t > 0, V (∞) = z.

When p = 2, we show further that such a solution is unique, and there are no other types of bounded positive solutions to the quarter-space problem. Thus in this case the number of bounded positive solutions to the quarter-space problem is exactly the number of positive zeros of ƒ.

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(∇|u|ᵖ⁻²∇u)=b(x)h(u) in Ω∖{0}, where 1∠p≤N and Ω is an open subset of ℝᴺ with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b0(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=|x|θ and some of its perturbations, for instance, b(x)=|x|θ(−log|x|)ͫ for any m ∈ ℝ. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.

In this paper we consider the diffusive competition model consisting of an invasive species with density 'u' and a native species with density 'v', in a radially symmetric setting with free boundary.

In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number R-0 obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if R-0 > 1, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.

We study the behavior of finite Morse index solutions to the weighted elliptic equation...

We study the following nonlinear Stefan problem 'ut−dΔu=g(u)u=0andut=μ|∇xu|2u(0,x)=u0(x)forx∈Ω(t),t>0,forx∈ (t),t>0,forx∈Ω0', where Ω(t)⊂Rn ( n≧2 ) is bounded by the free boundary Γ(t) , with Ω(0)=Ω0 , μ and d are given positive constants. The initial function u 0 is positive in Ω0 and vanishes on ∂Ω0 . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary Γ(t) is smooth outside the closed convex hull of Ω0 , and as t→∞ , either Ω(t) expands to the entire Rn , or it stays bounded. Moreover, in the former case, Γ(t) converges to the unit sphere when normalized, and in the latter case, u→0 uniformly. When g(u)=au−bu2, we further prove that in the case Ω(t) expands to Rn , u→a/b as t→∞ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists μ∗≧0 such that Ω(t) expands to Rn exactly when μ>μ∗.

We show that for large λ>0, the 'generalized' boundary value problem −Δu = λu(u − a(x))(1 − u) in Ω, u|∂Ω = ø, where 1≼ø(x)≼∞, behaves like the special case ø ≡ 1. In particular, we show the existence of solutions with sharp boundary layers and interior spikes, and determine the location of the spikes.

In this paper, we study the boundary behavior of positive solutions of the following equation...

Let u∊ be a single layered radially symmetric unstable solution of the Allen-Cahn equation -∈²Δu=u(u-a(|x|))(1-u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u∊ when ∈ is small, we construct solutions of the form u∊ + v∊, with v∊ non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.