Solutions to a model for compressible immiscible two phase flow in porous media.
Stability analysis of a ratio-dependent predator-prey system with diffusion and stage structure.
Stability for a diffusive delayed predator-prey model with modified Leslie-Gower and Holling-type II schemes
A diffusive delayed predator-prey model with modified Leslie-Gower and Holling-type II schemes is considered. Local stability for each constant steady state is studied by analyzing the eigenvalues. Some simple and easily verifiable sufficient conditions for global stability are obtained by virtue of the stability of the related FDE and some monotonous iterative sequences. Numerical simulations and reasonable biological explanations are carried out to illustrate the main results and the justification...
Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system
A one-dimensional version of a gradient system, known as “Kobayashi-Warren-Carter system”, is considered. In view of the difficulty of the uniqueness, we here set our goal to ensure a “stability” which comes out in the approximation approaches to the solutions. Based on this, the Main Theorem concludes that there is an admissible range of approximation differences, and in the scope of this range, any approximation method leads to a uniform type of solutions having a certain common features. Further,...
Stochastic FitzHugh-Nagumo equations on networks with impulsive noise.
Systems of quasilinear parabolic equations with discontinuous coefficients and continuous delays.
The analysis of blow-up solutions to a semilinear parabolic system with weighted localized terms
This paper deals with blow-up properties of solutions to a semilinear parabolic system with weighted localized terms, subject to the homogeneous Dirichlet boundary conditions. We investigate the influence of the three factors: localized sources , vⁿ(x₀,t), local sources , , and weight functions a(x),b(x), on the asymptotic behavior of solutions. We obtain the uniform blow-up profiles not only for the cases m,q ≤ 1 or m,q > 1, but also for m > 1 q < 1 or m < 1 q > 1.
Time-optimal boundary control of an infinite order parabolic system with time lags
In this paper the time-optimal boundary control problem is presented for a distributed infinite order parabolic system in which time lags appear in the integral form both in the state equation and in the boundary condition. Some specific properties of the optimal control are discussed.
Travelling Waves in Near-Degenerate Bistable Competition Models
We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species...
Unique continuation principle for systems of parabolic equations
In this paper we prove a unique continuation result for a cascade system of parabolic equations, in which the solution of the first equation is (partially) used as a forcing term for the second equation. As a consequence we prove the existence of ε-insensitizing controls for some parabolic equations when the control region and the observability region do not intersect.
Volume Filling Effect in Modelling Chemotaxis
The oriented movement of biological cells or organisms in response to a chemical gradient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized models have been proposed to describe the aggregation phase of this process. Most of...
Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems
Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.