Related papers: Strong competition versus fractional diffusion: th…
In this paper, we investigate the uniform regularity and asymptotic behavior of solutions to the following Lotka-Volterra type system of strong competition with Dirichlet boundary conditions: \begin{align*} \left\{ \begin{array}{ll} -\Delta…
For a class of systems of semi-linear elliptic equations, including \[ -\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k, \] for $p=2$ (variational-type interaction) or $p = 1$ (symmetric-type interaction), we…
We study regularity issues for systems of elliptic equations of the type \[ -\Delta u_i=f_{i,\beta}(x)-\beta \sum_{j\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \] set in domains $\Omega \subset \mathbb{R}^N$, for $N \geq 1$. The paper is…
We investigate the long term behavior for a class of competition-diffusion systems of Lotka-Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system…
We investigate the controllability of the competition-diffusion Lotka-Volterra system. Our primary focus is on the one-dimensional setting with Dirichlet boundary controls, interpreted as ecological management policies regulating the…
Motivated by the traditional Lotka-Volterra competitive models, this paper proposes and analyzes a class of stochastic reaction-diffusion partial differential equations. In contrast to the models in the literature, the new formulation…
We consider a family of positive solutions to the system of $k$ components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in $\Omega$}, \] where $\Omega \subset \mathbb{R}^N$…
This paper describes the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction-diffusion systems with strongly competitive interactions of Lotka-Volterra type, when the matrix of…
We consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $\alpha$, $\beta$ and $\gamma$. By the method of energy inequalities, for the…
We study the regularity of segregated profiles arising from competition - diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power $s \in (0,1)$. Among others, our results apply to…
This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =…
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed by balls connected with thin corridors, we show the occurrence of pattern…
Spatial segregation occurs in population dynamics when $k$ species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competition-diffusion system of $k$ differential equations \[ -\Delta…
In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*} \left\lbrace \begin{array}{l} (-\Delta)^{s}u + |\nabla u|^{p} =f \quad\text{ in } \Omega \qquad \qquad \qquad…
We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-\Delta)^{-1/2}u).$ For definiteness, the problem is posed…
For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies H\"older boundedness for…
Super- and sub- diffusions are two typical types of anomalous diffusions in the natural world. In this work, we discuss the numerical scheme for the model describing the competition between super- and sub- diffusions driven by fractional…
In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 &…
In this article we study optimization problems ruled by $\alpha$-fractional diffusion operators with volume constraints. By means of penalization techniques we prove existence of solutions. We also show that every solution is locally of…
In this work we study a strongly coupled system between the equation of plates with fractional rotational inertial force $\kappa(-\Delta)^\beta u_{tt}$ where the parameter $0 <\beta\leq 1$ and the equation of an electrical network…