This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation \begin{equation}\label{eq 1} \left\{ \arraycolsep=1pt \begin{array}{lll} (-Δ)_x^α u+(-Δ)_y u+u=f(u)\quad \ \ {\rm in}\ \ \R^N\times\R^M, u>0\ \ {\rm{in}}\ \R^N\times\R^M,\ \ \quad \lim_{|(x,y)|\to+\infty}u(x,y)=0, \end{array} \right. \end{equation} with $N\ge 1$, $M\ge 1$ and $α\in (0,1)$. We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the …
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $λt + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of "mixed operators" for which the required ellipticity comes from a combination of the properties of th…
Motivated by the occurrence of "shattering" mass-loss observed in purely continuous fragmentation models, this work concerns the development and the mathematical analysis of a new class of hybrid discrete--continuous fragmentation models. Once established, the model, which takes the form of an integro-differential equation coupled with a system of ordinary differential equations, is subjected to a rigorous mathematical analysis, using the theory and methods of operator semigroups and their generators. Most notably, by applying the theory relating to the Kato--Voigt perturbation theorem, honest…
We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori estimates, and symmetry of solutions. As an application, we construct a nontrivial bounded saddle solution in the plane.
This is the first of two papers concerning saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only on this set. By the odd symmetry, $L_K$ coincides with a new operator $L_K^{\mathcal{O}}$ which acts on functions defined only on one side of the Simons cone, $\{|x'|>|x''|\}$, and that vanish on it. This …
In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black-Scholes model relies on several restrictive assumptions such as liquidity and compl…
We consider an integro-differential model for evolutionary game theory which describes the evolution of a population adopting mixed strategies. Using a reformulation based on the first moments of the solution, we prove some analytical properties of the model and global estimates. The asymptotic behavior and the stability of solutions in the case of two strategies is analyzed in details. Numerical schemes for two and three strategies which are able to capture the correct equilibrium states are also proposed together with several numerical examples.
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
Paper dedicated to qualitative study of the solution of the Zaremba type problem in Lipschitz domain with respect to the elliptic equation in non-divergent form. Main result is Landis type Growth Lemma in spherical layer for Mixed Boundary Value Problem in the class of "admissible domain". Based on the Growth Lemma Phragmén-Lindelöf theorem is proved at junction point of Dirichlet boundary and boundary over which derivative in non-tangential direction is defined.
In this paper, we study existence results for initial value problems for hybrid fractional integro-differential equations. Our investigation is based on the Dhage hybrid fixed point theorem. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers co…
We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \partial_t^αu - Lu + λu= f \quad \mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^αu$ is the Caputo fractional derivative of order $α\in (0,1]$, $λ\ge 0$, $T\in (0,\infty)$, and $$Lu(t,x) := \int_{ \mathbb{R}^d} \bigg( u(t,x+y)-u(t,x) - y\cdot \nabla_xu(t,x)χ^{(σ)}(y)\bigg)K(t,x,y)\,dy $$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We also derive a weighted mixed-norm …
We study the mixed dispersion fourth order nonlinear Schrödinger equation \begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t ψ-γΔ^2 ψ+βΔψ+|ψ|^{2σ} ψ=0\ \text{in}\ \R \times\R^N, \end{equation*} where $γ,σ>0$ and $β\in \R$. We focus on standing wave solutions, namely solutions of the form $ψ(x,t)=e^{iαt}u(x)$, for some $α\in \R$. This ansatz yields the fourth-order elliptic equation \begin{equation*} %\tag{\protect{*}}\label{4nlsstar} γΔ^2 u -βΔu +αu =|u|^{2σ} u. \end{equation*} We consider two associated constrained minimization problems: one with a constraint on the $L^2$-norm a…
