Mathematics

We consider here a problem of population dynamics modeled on a logistic equation with both classical and nonlocal diffusion, possibly in combination with a pollination term. The environment considered is a niche with zero-flux, according to a new type of Neumann condition. We discuss the situations that are more favorable for the survival of the species, in terms of the first positive eigenvalue. Quite surprisingly, the eigenvalue analysis for the one dimensional case is structurally different than the higher dimensional setting, and it sensibly depends on the nonlocal character of the dispersal. The mathematical framework of this problem takes into consideration the equation ?�αΔu+β(?��?) s u=(m?�μu)u+?J?�uin Ω, where m can change sign. This equation is endowed with a set of Neumann condition that combines the classical normal derivative prescription and the nonlocal condition introduced in [S. Dipierro, X. Ros-Oton, E. Valdinoci, Rev. Mat. Iberoam. (2017)]. We will establish the existence of a minimal solution for this problem and provide a throughout discussion on whether it is possible to obtain non-trivial solutions (corresponding to the survival of the population). The investigation will rely on a quantitative analysis of the first eigenvalue of the associated problem and on precise asymptotics for large lower and upper bounds of the resource. In this, we also analyze the role played by the optimization strategy in the distribution of the resources, showing concrete examples that are unfavorable for survival, in spite of the large resources that are available in the environment.

We consider the sharp interface limit of a convective Allen-Cahn equation, which can be part of a Navier-Stokes/Allen-Cahn system, for different scalings of the mobility m ε = m 0 ε θ as ε?? . In the case θ>2 we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case θ=0 . Moreover, we show that an associated mean curvature functional does not converge the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case θ=0,1 by the method of formally matched asymptotics.

We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property does hold true for the parabolic Anderson model.

We study the limit behaviour of singularly-perturbed elliptic functionals of the form F k (u,v)= ??A v 2 f k (x,?�u)dx+ 1 ε k ??A g k (x,v, ε k ?�v)dx, where u is a vector-valued Sobolev function, v?�[0,1] a phase-field variable, and ε k >0 a singular-perturbation parameter, i.e., ε k ?? , as k????. Under mild assumptions on the integrands f k and g k , we show that if f k grows superlinearly in the gradient-variable, then the functionals F k ? -converge (up to subsequences) to a brittle energy-functional, i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of u . This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in F k decouple in the limit. The abstract ? -convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients, we are interested in a Carleman-type inequality for these solutions satisfying an additional growth condition in elliptic periodic homogenization, which implies a three-ball inequality without an error term at a macroscopic scale. Moreover, if we replace the additional growth condition by the doubling condition at a macroscopic scale, then the three-ball inequality without an error term holds at any scale. The proof relies on the convergence of H 1 -norm for the solution and the compactness argument.

Let u(t,x) be a solution of the heat equation in R n . Then, each k??th derivative also solves the heat equation and satisfies a maximum principle, the largest k??th derivative of u(t,x) cannot be larger than the largest k??th derivative of u(0,x) . We prove an analogous statement for the solution of the heat equation on bounded domains Ω??R n with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction ?��?? k = λ k ? k with Dirichlet conditions on smooth domains Ω??R n .

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail. In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.

We prove existence and nonexistence results concerning elliptic problems whose basic model is ???????????????????�Δu+μ(x) |?�u | 2 (u+δ ) γ =λ u p , u>0, u=0, x?��? x?��? x?��?Ω, where Ω??R N (N??) is a bounded smooth domain, λ>0 , p>1 , δ?? , γ>0 and μ??L ??(Ω) . The main achievement resides in handling a possibly singular ( δ=0 ) first order term having a nonconstant coefficient μ in the presence of a superlinear zero order term. Our approach for the existence results is based on fixed point theory. With the aim of applying it, a previous analysis on a related non-homogeneous problem is carried out. The required a priori estimates are proven via a blow-up method.

We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighborhood of its minimum. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential.

In this article, we prove that the least energy nodal solutions to Lane-Emden equation ?�Δu=|u | p?? u with zero Dirichlet boundary conditions on a square are odd with respect to one diagonal and even with respect to the other one when p is close to 2. We also show that this symmetry breaks on rectangles close to squares.