A. Suárez

Universidad de Sevilla | US · Differential Equations and Numerical Analysis

In this paper we study the long time behaviour of cooperative $n$- dimensional non-autonomous Lotka-Volterra systems from population dynamics. We provide conditions to obtain extinction of a given subset of species. We also give sufficient conditions to get existence of a globally stable global solution with one species extinct, or with of all spec...

We study the logistic equation with nonlinear advection term −Δu+α→⋅∇up=λu−u2 in Ω with u=0 on ∂Ω, where Ω⊂RN, N≥1, is a bounded domain with smooth boundary, α→=(α1,…,αN) is a flow satisfying suitable condition, p>1 and λ∈R. The classical logistic equation differs from ours by the inclusion of the nonlinear advection term α→⋅∇up, meaning that the v...

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In...

In this paper we consider a diffusive Lotka-Volterra system including nonlocal terms in the reaction functions. We analyze the main types of interactions between species: competition, predator-prey and cooperation. We provide existence and non-existence of positive solutions results. For that, we employ mainly bifurcation method and a priori bounds...

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In...

By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have or...

In this paper, we study a PDE–ODE system as a simplification of a glioblastoma model. Mainly, we prove the existence and uniqueness of global in time classical solution using a fixed point argument. Moreover, we show some stability results of the solution depending on some conditions on the parameters.

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-sy...

Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this...

In this paper we analyse a differential system related to a Glioblastoma growth which will be able to capture different kind of growth. Firstly, we make an adimensional study in order to reduce the number of parameters changing adequately the parameters of the model. Later, using numerical simulations, we detect the main parameters determining diff...

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-sy...

This paper deals with a nonlocal diffusion elliptic eigenvalue problem. Specifically, the diffusion of the unknown variable at a point of the domain depends on its value in a neighborhood of the point. We apply bifurcation arguments and appropriate approximation to obtain our results. Some applications to the population dynamics will be given.

In this paper, we are concerned with the existence of solution to the class of nonlocal quasilinear problem of the type −div(a(|∇u|2)∇u)=f(x,u,B(u))inΩ,u=0on∂Ω,(P) where Ω is a smooth bounded domain on ℝN, a:ℝ+→ℝ+, f:Ω×ℝ×ℝ→ℝ, and B:L∞(Ω)→ℝ are functions whose hypotheses will be detailed later. We use sub‐ and supersolution method to find solutions...

Using the sub-supersolution method, we study the existence of positive solutions for the anisotropic problem $$\begin{aligned} -\sum _{i=1}^N\frac{\partial }{\partial x_i}\left( \left| \frac{\partial u}{\partial x_i}\right| ^{p_i-2}\frac{\partial u}{\partial x_i}\right) =\lambda u^{q-1} \end{aligned}$$ (0.1) where \(\Omega \) is a bounded and regul...

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array} \right. \end{equation*} in a bounded and smooth domain $\Omega$ in ${\rm I}\hskip -0.85mm{\rm R}$. By using c...

We study the existence of positive solutions of a logistic equation in the entire space with a nonlocal reaction term. Mainly, we apply a bifurcation method and singular boundary equations to obtain a priori bounds of the solutions. Our results show a drastic change of behaviour of the set of positive solutions depending on the sign of the nonlocal...

In this paper we study a PDE-ODE system as a simplification of a Glioblastoma model. Mainly, we prove the existence and uniqueness of global in time classical solution using a fixed point argument. Moreover, we show some stability results of the solution depending on some conditions on the parameters.

Using the sub-supersolution method we study the existence of positive solutions for the anisotropic problem \begin{equation} -\sum_{i=1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right)=\lambda u^{q-1} \end{equation} where $\Omega$ is a bounded and regular domain of $\ma...

We study a superlinear elliptic problem with a non-local diffusion coefficient. We show that there exists a drastic change on the structure of the set of positive solutions when the non-local coefficient grows fast enough to infinity. We combine mainly sub-super and bifurcation methods to obtain our results.

This paper deals with nonlinear elliptic problems where the diffusion coefficient is a degenerate non-local term. We show that this degeneration implies the growth of the complexity of the structure of the set of positive solutions of the equation. Specifically, when the reaction term is of logistic type, the continuum of positive solutions breaks...

A logistic equation in the whole space is considered. In this problem, a non-local perturbation is included. We establish a new sub–supersolution method for general nonlocal elliptic equations and, consequently, we obtain the existence of positive solutions of a nonlocal logistic equation.

In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonloc...

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tan...

In this paper we study the existence, uniqueness and multiplicity of positive solutions to a non-linear Schrödinger equation. We describe the set of positive solutions. We use mainly the sub-supersolution method, bifurcation and variational arguments to obtain the results.

A class of generalized Schrodinger problems in bounded domain is studied. A complete overview of the set of solutions is provided, depending on the values assumed by parameters involved in the problem. In order to obtain the results, we combine monotony, bifurcation and variational methods.

In this paper, we prove the existence of coexistence states for a nonlocal singular elliptic system that arises from the interaction between amoeba and bacteria populations. Our study is based on fixed point arguments using a version of the Bolzano's theorem, for which we will first analyze a local system by bifurcation theory. Moreover, we study t...

In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending a bifurcation theorem due to J. L\'opez-G\'omez. To this aim, we use several results, such that Fredholm operator of index $0$ theory, eigenvalues of elliptic operators and the Krein-Rutman theorem. Lastly, we apply this...

In this paper we use the bifurcation method and fixed point arguments to study a logistic equation with nonlocal diffusion coefficient. We prove the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global behaviour of this continuum depends strongly on the value of the nonlocal diffusion coeff...

In this paper, we study the existence and non-existence of coexistence states for a cross-diffusion system arising from a prey–predator model with a predator satiation term. We use mainly bifurcation methods and a priori bounds to obtain our results. This leads us to study the coexistence region and compare our results with the classical linear dif...

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem. © 2018 A...

In this paper we investigate a class of elliptic problems involving a nonlocal Kirchhoff type operator with variable coefficients and data changing its sign. Under appropriated conditions on the coefficients, we have shown existence and uniqueness of solution.

In this work we investigate an elliptic problem with a non-local non-autonomous diffusion coefficient. Mainly, we use bifurcation arguments to obtain existence of positive solutions. The structure of the set of positive solutions depends strongly on the balance between the non-local and the reaction terms.

In this paper we consider a logistic equation with nonlinear diffusion arising in population dynamics. In this model, there exists a refuge where the species grows following a Malthusian law and, in addition, there exists also a non-linear diffusion representing a repulsive dispersion of the species. We prove existence and uniqueness of positive so...

A mathematical system of differential equations for the modelization of mutualistic networks in Ecology has been proposed in Bastolla et al. (2007). Basically, it is studied how the complex structure of cooperation interactions between groups of plants and pollinators or seed dispersals affects to the whole network. In this paper we prove existence...

We study Lotka–Volterra models with fractional Laplacian. To do this we study in detail the logistic problem and show that the sub–supersolution method works for both the scalar problem and for systems. We apply this method to show the existence and non-existence of positive solutions in terms of the system parameters.

In this paper we analyse an elliptic equation that combines linear and nonlinear fast diffusion with a logistic type reaction function. We prove existence and non-existence results of positive solutions using bifurcation theory and sub-supersolution method. Moreover, we apply variational methods to obtain a pair of ordered positive solutions.

Real phenomena from different areas of Life Sciences can be described by complex networks, whose structure is usually determining their intrinsic dynamics. On the other hand, Dynamical Systems Theory is a powerful tool for the study of evolution processes in real situations. The concept of global attractor is the central one in this theory. In the...

We examine a logistic equation with local and non-local reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results d...

In this work we show some multiplicity results for the anisotropic equation where Ω ⊂ℝ N is a bounded smooth domain, 1 < p 1 ≤ p 2 ≤ . . . ≤ p N and λ is a positive parameter. Using genus theory, we study the subcritical case g λ (u) = λ|u| q−2 u with q ∈ (1, p N ) and the critical case g λ (u) = λ|u| q−2 u +|u| p*−2 u with q ∈ (1, p 1 ) and p* =...

In this paper we give a sub-supersolution method for nonlinear elliptic singular systems with quadratic gradient whose model system is the following { -Delta u + v(beta)vertical bar del u vertical bar(2)/u(alpha) = f(1)(x, u, v) in Omega, -Delta v + v(mu)vertical bar del v vertical bar(2)/v(gamma) = f(2)(x, u, v) in Omega, u = v = 0 on partial deri...

In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.

In this paper we prove that the sub-supersolution method works for general Kirchhoff systems. We apply the cited method to prove the existence of positive solutions for some specific models.

We consider a nonlinear eigenvalue problem with indefinite weight under Robin boundary conditions. We prove the existence and multiplicity of positive solutions. To this end, we carry out a detailed study of some linear eigenvalues problems and we use mainly bifurcation and sub–supersolution methods.

In this paper we study an elliptic eigenvalue problem with nonlocal boundary condition. We prove the existence of the principal eigenvalue and its main properties. As consequence, we show the existence and uniqueness of positive solution of a nonlinear problem arising from population dynamics.

In this work, we show existence and non-existence results of coexistence states for a Lotka-Volterra symbiotic model with self and cross-diffusion in one species. We study the behavior of the set of positive solutions when the cross-diffusion or the self-diffusion parameter is large.

In this paper we study a non-homogeneous elliptic Kirchhoff equation with nonlinear reaction term. We analyze the existence and uniqueness of positive solution. The main novelty is the inclusion of non-homogeneous term making the problem without a variational structure. We use mainly bifurcation arguments to get the results.

In this paper, we study the existence of positive solutions for a class of nonlocal problem arising in population dynamic. Basically, we prove our results via bifurcation theory.

In this paper we study a three dimensional mutualistic model of two plants in competition and a pollinator with cooperative relation with plants. We compare the dynamical properties of this system with the associated one under absence of the pollinator. We observe how cooperation is a common fact to increase biodiversity, which it is known that, ge...

We analyze the existence, non-existence and uniqueness of positive solutions of some nonlinear elliptic equations containing singular terms and natural growth in the gradient term. We use an adequate sub-super-solution method to prove the existence of solutions, the characterization of eigenvalues and the integrability of the term |∇u| 2 /u 2 for t...

This paper deals with a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to specific receptors of the endothelial cells. We study the time-dependent problem as well as the stationary problem associated to it.

In this paper, we study the existence, uniqueness, multiplicity, and stability of positive solution of a nonlinear elliptic problem that combines local and nonlocal terms, taking the form of an integral in the space. The proofs are mainly based on fixed point theorems, bifurcation techniques, sub‐supersolutions, and continuation arguments. Copyrigh...

In this paper we study different problems with nonlinear diffusion and non-local terms, some of them arising in population dynamics. We are able to give a complete description of the set of positive solutions and their stability. For that we employ a fixed point argument for the existence and uniqueness or multiplicity results and the study of sing...

In this paper we analyze the existence, uniqueness or multiplicity and stability of positive solutions to some nonlinear heterogeneous problems with nonlocal reaction term and linear diffusion. We employ mainly bifurcation and sub-supersolutions methods.

In this paper we are concerned with the nonlocal elliptic problem where Ω ⊂ℝ N is a bounded smooth domain, f, g : ℝ →ℝ are given functions and p is a fixed real number. We use variational methods to prove multiplicity results.

This paper deals with a nonlinear system of parabolic–elliptic type with a logistic source term and coupled boundary conditions related to pattern formation. We prove the existence of a unique positive global in time classical solution. We also analyze the associated stationary problem. Moreover it is proved, under the assumption of sufficiently st...

In this paper we study in detail the geometrical structure of global pullback and forwards attractors associated to non-autonomous Lotka–Volterra systems in all the three cases of competition, symbiosis or prey–predator. In particular, under some conditions on the parameters, we prove the existence of a unique nondegenerate global solution for thes...

In this paper we study in detail the pullback and forwards attractions to non-autonomous competition Lotka-Volterra system. In particular, under some conditions on the parameters, we prove the existence of a unique non-degenerate global solution for these models, which attracts any other complete bounded trajectory. For that we present the sub-supe...

In this paper we present some theoretical results concerning to a non-local elliptic equation with non-linear diffusion arising from population dynamics. Resumen. En este artículo presentamos algunos resultados teóricos relativos a una ecuación elíptica no local con la difusión no lineal que surge de la dinámica de poblaciones.

We show the existence and non-existence of positive solutions to a system of singular elliptic equations with the Dirichlet boundary condition.

In this paper we consider a parabolic problem as well as its stationary counterpart of a model arising in angiogenesis. The problem includes a chemotaxis type term and a nonlinear boundary condition at the tumor boundary. We show that the parabolic problem admits a unique positive global in time solution. Moreover, by bifurcation methods, we show t...

The main goal of this paper is to study the existence and non-existence of coexistence states for a Lotka–Volterra symbiotic model with cross-diffusion. We use mainly bifurcation methods and a priori bounds to give sufficient conditions in terms of the data of the problem for the existence of positive solutions. We also analyze the profiles of the...

Lotka-Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis, or prey-predator behavior involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extende...

We study a system of equations arising from angiogenesis which contains a nonregular term that vanishes below a certain threshold. We are forced to modify the usual methods of bifurcation theory because of this loss of regularity. Nevertheless, we obtain results on the existence, uniqueness and permanence of a positive solution for the time-depende...

In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation −Δu = λ u−u p in Ω, with the nonlinear boundary condition ∂u/∂ν = u r on ∂Ω. Here Ω is a smooth bounded domain of \mathbbRd{\mathbb{R}}^d with outward unit normal ν, λ is a real pa...

The main goal of this paper is to study a stationary problem arising from angiogenesis, including terms of chemotaxis and flux at the boundary of the tumor. We give sufficient conditions on terms of the data of the problems assuring the existence of positive solutions.

The main goal of this paper is the study of the existence and uniqueness of positive solutions of some nonlinear age-dependent diffusive models, arising from dynamic populations. We use a bifurcation method, for which it has been necessary to study in detail the linear and eigenvalue problems associated to the nonlinear problem in an appropriate sp...

In this work we consider existence and uniqueness of positive solutions to the elliptic equation −∆u = λu in Ω, with the nonlinear boundary conditions ∂u ∂ν = u p on Γ 1 , ∂u ∂ν = −u q on Γ 2 , where Ω is a smooth bounded domain, ∂Ω = Γ 1 ∪ Γ 2 , Γ 1 ∩ Γ 2 = ∅, ν is the outward unit normal, p, q > 0 and λ is a real parameter. We obtain a complete p...

The goal of this work is to study the forward dynamics of positive solutions for the non-autonomous logistic equation ut −∆u = λu−b(t)u p , with p > 1, b(t) > 0, for all t ∈ R, limt→∞ b(t) = 0. While the pullback asymptotic behaviour for this equation is now well understood, several different possibilities are realized in the forward asymptotic reg...

The main goal of this work is to study the existence and uniqueness of a positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.

In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and ar...

In this paper it is shown that the sub-supersolution method works for age-dependent diffusive nonlinear systems with non-local initial conditions. As application, we prove the existence and uniqueness of positive solution for a kind of Lotka-Volterra systems, as well as the blow-up in finite time in a particular case.

We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the non-autonomous problems converge towards the autonomous attractor only in the Hausdorff semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a ‘gradient-like’ st...

In this paper our aim is to present a survey of known results of an optimal control problem with concave non-quadratic cost functional and a state equation arising from population dynamics. First, we study in detail the state equation, and then we show existence and uniqueness of optimal control and also a numerical approximation of the optimal con...

We study a nonlinear predator–prey model in which the prey population is affected by a serious but curable disease which causes mortality and it is assumed an age-structure in the prey population. We suppose that the predator population grows according to a logistic law. Using mainly a decoupling technique and a fixed point theorem, we prove the ex...

In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations.

The main goal of this paper is the study of the existence and uniqueness of a positive solution for a nonlinear age-dependent equation with spatial diffusion. For that, we mainly use the properties of an eigenvalue problem related to the equation and the sub-supersolution method. We justify that this method works for this equation, in which there i...

This work deals with the uniqueness of positive solution for an elliptic equation whose nonlinearity satisfies an specific monotony property. In order to prove the main result, we employ a change of variable used in previous papers and the maximum principle.

This paper concerns with some elliptic equations with nonlinear boundary conditions. Sub-supersolution and bifurcation methods are used in order to obtain existence, uniqueness or multiplicity of positive solutions.

In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the non-autonomous equation ut − ∆u = λu − b(t)u 2 with b(t) ∈ [b0, B0], 0 < b0 < B0 < 2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λ passes through the first eigenvalue of the Laplacian,...

In this paper, we would like compare the spread of an infectious disease in a population without the influence of a predator and under its influence. We show that it is possible to control an epidemic in a population with the help of predators. Copyright © 2004 John Wiley & Sons, Ltd.

In this work, we analyze a nonlinear population-dynamics model with age dependence and spatial diffusion, and where we are assuming the influence of a reaction term. Using a sub-supersolution method we derive existence and uniqueness results. We apply this method to study the existence and uniqueness of a positive solution of a generalized logistic...

This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well...

In this paper we study a generalized porous medium equation where the diffusion rate, say m(x) —spatially heterogeneous—, is assumed to be linear, m = 1, on a piece of the support domain, Ω 1 , and slow nonlinear, m(x) > 1, in its complement, Ω m := Ω \ ¯ Ω 1 . Most precisely, we characterize the existence of positive solutions and construct the co...

Although the pioneering studies of G. I. Barenblatt [8] and A. G. Aronson & L. A. Peletier [7] did result into a huge industry around the porous media equation, none further study analyzed the effect of combining fast, slow, and linear diffusion simultaneously, in a spatially heterogeneous porous medium. Actually, it might be this is the first work...

In this paper we study the eigenvalues associated with a positive eigenfunction of a quasilinear elliptic problem with an operator that is not necessarily bounded. For that, we use the bifurcation theory and obtain the existence of positive solutions for a range of values of the bifurcation parameter. AMS 2000 Mathematics subject classification: Pr...

In this paper we characterize the existence of large solutions for a general class of porous medium logistic equations in the presence of a vanishing carrying capacity. The decay rate of the carrying capacity at the boundary of the underlying domain determines the exact blow-up rate of the large solutions. Its explicit knowledge allows us to obtain...

Lotka–Volterra systems have been extensively studied by many authors, both in the autonomous and non-autonomous cases. In previous papers the time asymptotic behaviour as t→∞ has been considered. In this paper we also consider the `pullback' asymptotic behaviour which roughly corresponds to observing a system `now' that has already been evolving fo...

In this work we consider positive solutions to cooperative elliptic systems of the form -Deltau = lambdau-u(2) +bunu, -Deltanu = munu-nu(2)+cunu in a bounded smooth domain Omega subset of R-N (lambda, mu is an element of R, b, c > 0) which blow up on the boundary an, that is u(x), nu(x) --> +infinity as dist(x, partial derivativeOmega) --> 0. We sh...

The goal of this paper is to study the nonnegative steady-states solutions of the degenerate logistic indefinite sublinear problem. We combine bifurcation method and linking local subsupersolution technique to show the existence and multiplicity of nonnegative solutions. We employ a change of variable already used in a different context and the spe...

An optimal harvesting problem with a concave nonquadratic cost functional and a diffusive degenerate elliptic logistic state equation type is investigated. Under certain assumptions, we prove the existence and uniqueness of an optimal control. A characterization of the optimal control via the optimality system is also derived, which leads to approx...

In this work we study the existence, stability and multiplicity of the positive steady-states solutions of the degenerate logistic indefinite superlinear problem. By an adequate change of variable, the problem is transformed into an elliptic equation with concave and indefinite convex nonlinearities. We use singular spectral theory, the Leray-Schau...

A system of differential equations is permanent if there exists a fixed bounded set of positive states strictly bounded away from zero to which, from a time on, any positive initial data enter and remain. However, this fact does not happen for a differential equation with general non-autonomous terms. In this work we introduce the concept of pullba...

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and d...

In this paper we analyze the existence, the uniqueness and the stability of the pos-itive steady-states of a sublinear logistic porous media equation involving a weight function vanishing on a region of the support domain. Quite surprisingly, the model possesses a unique positive steady state which attracts to all positive solu-tions within the ran...