Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.52 (2020), 1st Issue

In this paper we simulate positive solutions, large solutions and metasolutions of the heterogeneous logistic equation in a disk and an annulus. The numerical methods introduced in this paper are extremely innovative because they make unnecessary determining any previous lifting and solving any decoupled system of ordinary differential equations. Moreover, they can be used to solve non-radially symmetric problems. The models are of a huge interest in Spatial Ecology because they enable us to analyse the effects of the spatial heterogeneity on the evolution of the terrestrial ecosystems. The large solutions and the metasolutions have been computed by the first time in this paper.

A spatio-temporal dynamics model is presented to study the effects of mutations on the persistence and extinction of bacteria under the antibiotic inhibition. We construct a mixed type Lyapunov functional to prove the global stability of extinction state and coexistence state for the case of forward mutation and forward-backward mutation respectively.

This paper deals with a free boundary problem for a reaction-diffusion equation with moving boundary, whose dynamics is governed by the Stefan condition. We will mainly discuss the problem for the case of multi-stable nonlinearity, which is a function with a multiple number of positive stable equilibria. The first result is concerned with the classi cation of solutions in accordance with large-time behaviors. As a consequence, one can observe a multiple number of spreading phenomena corresponding for each positive stable equilibrium. Here it is seen that there exists a certain group of spreading solutions whose element accompanies a propagating terrace. We will derive sharp asymptotic estimates of free boundary and profile of every spreading solution including spreading one with propagating terrace.

We consider the Cauchy problem for an attraction-repulsion chemotaxis system in two-dimensional space. The system consists of three partial differential equations; a drift-diffusion equation incorporating terms for both chemoattraction and chemorepulsion, and two elliptic equations. We denote by β1 the coefficient of the attractant and by β2 that of the repellent. The boundedness of nonnegative solutions to the Cauchy problem was shown in the repulsive dominant case β1 < β2 and the balance case β1 = β2. In this paper, we study the boundedness problem to the Cauchy problem in the attractive dominant case β1 > β2.

We give an analytical proof of the presence of complex dynamics for a model of charged particles in a magnetic field. Our method is based on the theory of topological horseshoes and applied to a periodically perturbed Dffing equation. The existence of chaos is proved for sufficiently large, but explicitly computable, periods.

We study an interesting model, with reaction terms of Lotka-Volterra type, where diffusion and reaction have polynomial growth of any order. We establish existence of global attractors as well as exponential attractors. In the sequel we study the long time dynamics of an appropriate semigroup and show that it possesses a global attractor (and exponential attractors) in a certain Banach space.

We consider the nonlinear eigenvalue problem [D(u(t))u(t)']' + λg(u(t)) = 0, u(t) > 0 ; t ∈ I := (0, 1), u(0) = u(1) = 0, which comes from the porous media type equation. Here, D(u) = pu2n+sin u (n ∈ N, p > 0: given constants), g(u) = u or g(u) = u + sin u. λ > 0 is a bifurcation parameter which is a continuous function of α = ||uλ||∞ of the solution uλ corresponding to λ, and is expressed as λ = λ(α). Since our equation contains oscillatory term in diffusion term, it seems significant to study how this oscillatory term gives effect to the structure of bifurcation curves λ(α). We propose a question from a view point of inverse bifurcation problems and show that the simplest case D(u) = u2 + sin u and g(u) = u gives us the most impressible asymptotic formula for global behavior of λ(α).

In this paper we analyze the influence of the spatial heterogeneities in the existence of positive solutions of Logistic problems with heterogeneous sublinear boundary conditions. We will show that the relative positions of the vanishing sets of the potentials in front of the nonlinearities, in the PDE and on the boundary conditions, play a crucial role as for the amplitude of the range of values of the bifurcation parameter for which the problems possess positive solutions. We will compare the cases of the logistic problem with linear and nonlinear boundary conditions. Also, we will show the global bifurcation diagram of positive solutions of the logistic problem with heterogeneous nonlinear boundary conditions, considering the amplitude of the nonlinearity in the boundary conditions as bifurcation-continuation parameter.

Bifurcation is a very useful method to prove the existence of positive solutions for nonlinear elliptic equations. The existence of an unbounded continuum of positive solutions emanating from zero or from infinity can be deduced in many problems. In this paper, we show the applicability of this method in some problems where the classical bifurcation results can not be directly applied.

We consider the problem of existence of a solution u to δ_tu — δ_xxu = 0 in (0, T) x R₊ subject to the boundary condition — u_x(t,0) + g(u(t, 0)) = μ on (0, T) where μ is a measure on (0, T) and g a continuous nondecreasing function. When p > 1 we study the set of self-similar solutions of δ_tu — δ_xxu = 0 in R₊ — R₊ such that —u_x(t,0)+u^p = 0 on (0,∞). At end, we present various extensions to a higher dimensional framework.

It is well known that under appropriate conditions on a double well potential, the associated Hamiltonian system possesses a pair of heteroclinic solutions joining the minima of the potential in addition to in nitely many other homoclinics and heteroclinics that oscillate between these minima. This paper studies the effect on such solutions of replacing the temporal domain, R, by a nite but long time interval.

We consider a superlinear indefinite problem with homogeneous Neumann boundary conditions and a parameter appearing in the domain of the di erential equation. Such a problem is an extension of the one studied in [33], in the sense that also negative values of the parameter are allowed. First, we show how to discretize the problem in a way that is suitable to perform numerical continuation methods and obtain the associated bifurcation diagrams. Then, we analyze the results of the simulations, also studying the stability of the solutions.

We review the inde nite sublinear elliptic equation Δu =a(x)u^q in a smooth bounded domain ΩCR^N, with Dirichlet or Neumann homogeneous boundary conditions. Here 0 < q < 1 and a is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in su_x000E_cient and necessary conditions on a and q for the existence of positive solu- tions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.

A new class of quasilinear reaction-diffusion equations is introduced for which the mass flow never reaches the boundary. It is proved that the initial value problem is well-posed in an appropriate weighted Sobolev space setting.

Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in.

Consider the Oldroyd-B system on exterior domains with nonzero external forces f. It is shown that this system admits under smallness assumptions on f a bounded, global solution (u; τ), which is stable in the sense that any other global solution to this system starting in a sufficiently small neighborhood of (u(0); τ (0)) is tending to (u; τ). In addition, if the outer force is T-periodic and small enough, the Oldroyd-B system admits a T-periodic solution. Note that no smallness condition on the coupling coefficient is assumed.