Partly dissipative systems in uniformly local spaces
We study the existence of attractors for partly dissipative systems in ℝⁿ. For these systems we prove the existence of global attractors with attraction properties and compactness in a slightly weaker topology than the topology of the phase space. We obtain abstract results extending the usual theory to encompass such two-topologies attractors. These results are applied to the FitzHugh-Nagumo equations in ℝⁿ and to Field-Noyes equations in ℝ. Some embeddings between uniformly local spaces are also...
Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources
We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources in population dynamics. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. They can be related to the emergence of biological species due to the intra-specific competition and random mutations. Various types of travelling waves are observed.
Pattern Formation Induced by Time-Dependent Advection
We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that...
PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic
Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on...
PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages
This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a...
PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages
This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a...
Peano type theorem for abstract parabolic equations
Periodic parabolic problems with nonlinearities indefinite in sign.
Let Ω ⊂ RN be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) up - b (x, t) uq in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the existence of solutions uλ in the interior of the positive cone and that uλ can...
Periodic problems and problems with discontinuities for nonlinear parabolic equations
In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that...
Periodic solutions for an evaporation problem with a Signorini type boundary condition
Periodic solutions for Sellers type diffusive energy balance model in climatology
Periodic solutions in superlinear parabolic problems.
Periodic solutions of a three-species periodic reaction-diffusion system
We study a periodic reaction-diffusion system of a competitive model with Dirichlet boundary conditions. By the method of upper and lower solutions and an argument similar to that of Ahmad and Lazer, we establish the existence of periodic solutions and also investigate the stability and global attractivity of positive periodic solutions under certain conditions.
Periodic solutions of abstract differential equations: the Fourier method
Periodic solutions of degenerate differential equations in vector-valued function spaces
Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multiplier theorems, we obtain necessary and sufficient conditions for the existence and uniqueness of periodic solutions to the equation d/dt(Mu(t)) = Au(t) + f(t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our results are obtained in the scales of periodic Besov and periodic Lebesgue vector-valued spaces.
Periodic solutions of evolution -Laplacian equations with a nonlinear convection term.
Periodic solutions of evolution problem associated with moving convex sets
This paper is concerned with periodic solutions for perturbations of the sweeping process introduced by J.J. Moreau in 1971. The perturbed equation has the form where C is a T-periodic multifunction from [0,T] into the set of nonempty convex weakly compact subsets of a separable Hilbert space H, is the normal cone of C(t) at u(t), f:[0,T] × H∪H is a Carathéodory function and Du is the differential measure of the periodic BV solution u. Several existence results of periodic solutions for this...
Periodic solutions of nonlinear boundary value problems of the second kind
Periodic solutions of -Laplacian systems with a nonlinear convection term.