Forced waves of the Fisher–KPP equation in a shifting environment

This paper concerns the equation

(0.1)$u_t=u_{xx}+f(x?ct,u),x\in \mathbb{R},$

where $c\ge 0$ is a forcing speed and $f:(s,u)\in \mathbb{R}\times {\mathbb{R}}_{+}\to \mathbb{R}$ is asymptotically of KPP type as $s\to ?\infty$. We are interested in the questions of whether such a forced moving KPP nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist. Under a sublinearity condition on $f(s,u)$, we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases. In these cases, we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in $s\in \mathbb{R}$.     

The Stefan problem for the Fisher–KPP equation

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.     

Asymptotic behaviour of travelling waves for the delayed Fisher–KPP equation

In this work we study the behaviour of travelling wave solutions for the diffusive Hutchinson equation with time delay. Using a phase plane analysis we prove the existence of travelling wave solution for each wave speed c?2$c?2$. We show that for each given and admissible wave speed, such travelling wave solutions converge to a unique maximal wavetrain. As a consequence the existence of a nontrivial maximal wavetrain is equivalent to the existence of travelling wave solution non-converging to the stationary state u=1$u=1$.     

Propagation of chaos for the Keller–Segel equation over bounded domains

In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions.     

Tropical discretization: ultradiscrete Fisher–KPP equation and ultradiscrete Allen–Cahn equation

A systematic approach to the construction of ultradiscrete analogues for differential systems is presented. This method is tailored to first-order differential equations and reaction–diffusion systems. The discretizing method is applied to Fisher–KPP equation and Allen–Cahn equation. Stationary solutions, travelling wave solutions and entire solutions of the resulting ultradiscrete systems are constructed.     

On the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation

In this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19 (2014), pp. 1907–1920]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some numerical experiments are conducted in order to assess the validity of the analytical results. We conclude that the methodology under investigation is a fast, nonlinear, explicit, stable, convergent numerical technique that preserves the positivity, the boundedness and the monotonicity of approximations, making it an ideal tool in the study of some travelling-wave solutions of the mathematical model of interest. This note closes proposing new avenues of future research.     

On the Leray–Hopf extension condition for the steady-state Navier–Stokes problem in multiply-connected bounded domains

Employing the approach of Takeshita (Pacific J Math 157:151–158, 1993), we give an elementary proof of the invalidity of the Leray–Hopf Extension Condition for certain multiply connected bounded domains of \({\mathbb {R}}^{n}\) , \(n=2,3\) , whenever the flow through the different components of the boundary is non-zero. Our proof is alternative to and, to an extent, more direct than the recent one proposed by Heywood (J Math Fluid Mech 13:449–457, 2011).     

Oscillatory traveling waves of polytropic filtration equation with generalized Fisher–KPP sources

The polytropic filtration equation with generalized Fisher–KPP sources is considered. We will show that the equation may have finite times oscillatory traveling waves, and try to give a complete classification by virtue of a singular exponent in the source according to the finiteness of the oscillatory times of traveling waves.     

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Given \(\rho >0\), we study the elliptic problem

$$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain and \(p>1\) is Sobolev-subcritical, searching for conditions (about \(\rho \), N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is \(L^2\)-subcritical, i.e. \(1<p<1+4/N\), the problem admits solutions for every \(\rho >0\). In the \(L^2\)-critical and supercritical case, i.e. when \(1+4/N \le p < 2^*-1\), we show that, for any \(k\in {\mathbb {N}}\), the problem admits solutions having Morse index bounded above by k only if \(\rho \) is sufficiently small. Next we provide existence results for certain ranges of \(\rho \), which can be estimated in terms of the Dirichlet eigenvalues of \(-\Delta \) in \(H^1_0(\Omega )\), extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.

     

Fisher–KPP equation on the Heisenberg group

In this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow-up solutions. Moreover, we extend the obtained results to the time-fractional Fisher–KPP equation on the Heisenberg group.     

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

We consider the problem

$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary, the exponent p is greater than 1, \(\epsilon >0\) is a small parameter, V is a uniformly positive, smooth potential on \(\bar{\Omega }\), and \(\nu \) denotes the outward unit normal of \(\partial \Omega \). Let \(\Gamma \) be a curve intersecting orthogonally \(\partial \Omega \) at exactly two points and dividing \(\Omega \) into two parts. Moreover, \(\Gamma \) satisfies stationary and non-degeneracy conditions with respect to the functional \(\int _{\Gamma }V^{\sigma }\), where \(\sigma =\frac{p+1}{p-1}-\frac{1}{2}\). We prove the existence of a solution \(u_\epsilon \) concentrating along the whole of \(\Gamma \), exponentially small in \(\epsilon \) at any fixed distance from it, provided that \(\epsilon \) is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).

     

On the stability of Hosszú’s functional equation

Two stability results are proved. The first one states that Hosszú’s functional equation $$f(x+y-xy)+f(xy)=f(x)-f(y)=0\ \ \ \ \ (x,y \in \rm R)$$ is stable. The second is a local stability theorem for additive functions in a Banach space setting.     

New patterns of travelling waves in the generalized Fisher–Kolmogorov equation

We prove the existence and uniqueness of a family of travelling waves in a degenerate (or singular) quasilinear parabolic problem that may be regarded as a generalization of the semilinear Fisher–Kolmogorov–Petrovski–Piscounov equation for the advance of advantageous genes in biology. Depending on the relation between the nonlinear diffusion and the nonsmooth reaction function, which we quantify precisely, we investigate the shape and asymptotic properties of travelling waves. Our method is based on comparison results for semilinear ODEs.     

On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces

This paper is concerned with the Cauchy problem for the Dullin–Gottwald–Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999