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We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
We study reaction-diffusion equations of various types in the half-space. For bistable reactions with Dirichlet boundary conditions, we prove conditional uniqueness: there is a unique nonzero bounded steady state which exceeds the bistable…
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of…
In quantum theory particles are represented as wave packets. Shock wave analysis of quantum equations of motion shows that wave function representation in general and wave packet description in particular contains discontinuities due to a…
A new asymptotic method is presented for the analysis of the traveling waves in the one-dimensional reaction-diffusion system with the diffusion with a finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics. The analysis makes use of…
Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a non-monotonic dispersion relation. The effect of a parametric driving on such waves is studied within suitable coupled Ginzburg-Landau equations. Due to the…
A wave equation, that governs finite amplitude acoustic disturbances in a thermoviscous Newtonian fluid, and includes nonlinear terms up to second order, is proposed. In contrast to the model known as the Kuznetsov equation, the proposed…
This paper is concerned with reaction-diffusion-advection equations in spatially periodic media. Under an assumption of weak stability of the constant states 0 and 1, and of existence of pulsating traveling fronts connecting them, we show…
We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities (including multistable ones) and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis…
A multiscale analysis of 1D stochastic bistable reaction-diffusion equations with additive noise is carried out w.r.t. travelling waves within the variational approach to stochastic partial differential equations. It is shown with explicit…
Using pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential…
We consider a bistable ($0\textless{}\theta\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This…
This paper studies the stability and large-time behavior of the three-dimensional (3-D) Boltzmann equation near shock profiles. We prove the nonlinear stability of the composite wave consisting of two shock profiles under general…
This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in…
We investigate a two-component reaction-diffusion system with a slow-fast structure and spatially varying coefficients $f_1$ and $f_2$ appearing in the slow equation. Under mild boundedness and regularity conditions on $f_1$ and $f_2$ the…
This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the…
The dispersing equation was derived from system of the hydrodynamic equations that take into account the gravity, and from boundary conditions of shock front. The dispersing equation made it possible to study unstable stability of front not…
We consider a diffusion model with limit cycle reaction functions, in the presence of convection. We select a set of functions derived from a realistic reaction model: the Schnakenberg equations. This resultant form is unsymmetrical. We…
We study a non-linear convective-diffusive equation, local in space and time, which has its background in the dynamics of the thickness of a wetting film. The presence of a non-linear diffusion predicts the existence of fronts as well as…
We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $\epsilon^{-1}$ ($\epsilon\textgreater{}0$), and proliferate according to a reaction…