Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m
Scattering for 1D cubic NLS and singular vortex dynamics
We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions form a family of evolving regular curves in that develop a singularity in finite time, indexed by a parameter . We consider curves that are small regular perturbations of for a fixed time . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...
Second order quasilinear functional evolution equations
We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in (boundedness and stabilization as ) are shown.
Self-similarity in chemotaxis systems
We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.
Semiclassical states of nonlinear Schrödinger equations with bounded potentials
Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential .
Semigroup Analysis of Structured Parasite Populations
Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral...
Semilinear elliptic problems in unbounded domains
We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interval only.
Sharp stability estimates for hyperbolic conservation laws.
Simplicity and stability of the first eigenvalue of a nonlinear elliptic system.
Singular integral inequalities and stability of semilinear parabolic equations
Using a method developed by the author for an analysis of singular integral inequalities a stability theorem for semilinear parabolic PDEs is proved.
Sistemi iperbolici di leggi di conservazione
This survey paper provides a brief introduction to the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After reviewing some basic concepts, we describe the fundamental theorem of Glimm on the global existence of BV solutions. We then outline the more recent results on uniqueness and stability of entropy weak solutions. Finally, some major open problems and research directions are discussed in the last section.
Slowly oscillating solutions of a parabolic inverse problem: boundary value problems.
Solitary waves in massive nonlinear -sigma models.
Solution of a linear model of a single-piston pump by means of methods for differential equations in Hilbert spaces
A mathematical model of a fluid flow in a single-piston pump is formulated and solved. Variation of pressure and rate of flow in suction and delivery piping respectively is described by linearized Euler equations for barotropic fluid. A new phenomenon is introduced by a boundary condition with discontinuous coefficient describing function of a valve. The system of Euler equations is converted to a second order equation in the space where is length of the pipe. The existence, unicity and stability...
Solution semigroup and invariant manifolds for functional equations with infinite delay
It is proved that parabolic equations with infinite delay generate -semigroup on the space of initial conditions, such that local stable and unstable manifolds can be constructed for a fully nonlinear problems with help of usual methods of the theory of parabolic equations.
Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation
We study solutions of the 2D Ginzburg–Landau equation subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, , and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small . For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy...
Some decay properties for the damped wave equation on the torus
This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that...
Some exact estimates of weak solutions to a mixed boundary value problem for nonlinear equations in domains with nonsmooth boundary.
Some geometric properties for a class of non-Lipschitz domains.
Spatiotemporal Dynamics in a Spatial Plankton System
In this paper, we investigate the complex dynamics of a spatial plankton-fish system with Holling type III functional responses. We have carried out the analytical study for both one and two dimensional system in details and found out a condition for diffusive instability of a locally stable equilibrium. Furthermore, we present a theoretical analysis of processes of pattern formation that involves organism distribution and their interaction of spatially...