Related papers: A Computer-Assisted Stability Proof for a Stationa…
In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous…
A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value…
In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of globally exponentially stable…
Recently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on N-balls has been solved by means of the backstepping method. However, the extension of this result to spatially-varying…
In this paper, we present a computer-assisted framework for constructive proofs of existence for stationary solutions to one-dimensional parabolic PDEs and the rigorous determination of their linear stability. By expanding solutions in…
This work focuses on stability analysis of numerical solutions to jump diffusions and jump diffusions with Markovian switching. Due to the use of Poisson processes, using asymptotic expansions as in the usual approach of treating diffusion…
Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological…
In this article we investigate the solution of the steady-state fractional diffusion equation on a bounded domain in $\real^{1}$. From an analysis of the underlying model problem, we postulate that the fractional diffusion operator in the…
Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian…
A general reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady state solutions are proved via studying an equivalent…
We investigate the large-time behavior of solutions to an outflow problem of the full compressible Navier-Stokes equations in the half line. The non-degenerate stationary solution is shown to be asymptotically stable under large initial…
The work deals with establishing the solvability of a system of integro-differential equations in the situation of the double scale anomalous diffusion. Each equation of such system involves the sum of the two negative Laplace operators…
We consider some reaction-diffusion equations describing systems with the nonlocal consumption of resources and the intraspecific competition. Sharp conditions on the coefficients are obtained to ensure the stability and instability of…
We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial…
In this paper we provide detailed information about the instability of equilibrium solutions of a nonlinear family of localized reaction-difussion equations in dimensione one. Beyond we provide explicit formulas to the equilibrium…
This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic…
We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel…
Mass-conserving reaction-diffusion systems with bistable nonlinearity are considered under general assumptions. The existence of stationary solutions with a single internal transition layer in such reaction-diffusion systems is shown using…
We propose a quantitative direct method to prove the local stability of a stationary solution for a rough differential equation and its regular discretization scheme. Using Doss-Sussmann technique and stopping time analysis, we provide…
In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and…