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For a singularly perturbed system of reaction--diffusion equations, assuming that the 0th order solutions in regular and singular regions are all stable, we construct matched asymptotic expansions for formal solutions to any desired order…
In this paper, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method we find explict periodic waves and we also present a characterization for all…
We consider a lattice regularization for an ill-posed diffusion equation with trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of…
It is well known that the seven-step backward difference formula (BDF) is unstable for the parabolic equations, since it is not even zero-stable. However, a linear combination of two non zero-stable schemes, namely the seven-step BDF and…
This paper concerns the stability on the inverse source scattering problem for the one-dimensional Helmholtz equation in a two-layered medium. We show that the increasing stability can be achieved by using multi-frequency wave field at the…
We devote this paper to the issue of existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In…
The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. This is because the phase-field…
We consider the stability of a system of equations which are a singular perturbation of the incompressible rigid-plastic flow equations used to model granular flow. A linear stability analysis shows that solutions of these equations are…
There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such waves are the connecting orbits of an autonomous second-order Lagrangian system and the corresponding kinetic energy is an indefinite…
The possibility that disorder may stabilize a superfluid phase of para-hydrogen in two dimensions is investigated theoretically by means of Quantum Monte Carlo simulations. We model disorder using a random distribution of scatterers, and…
We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in $\Omega\times (0,T)$ -…
We discuss the diffusion phenomenon in the parabolic and hyperbolic regimes. New effects related to the finite velocity of the diffusion process are predicted, that can partially explain the strange behavior associated to adsorption…
We propose a system of two coupled parabolic equations that have degenerate diffusion constants depending on the energy-like variable. The dissipation of the velocity-like variable is fed as a source term into the energy equation leading to…
Using the short-wavelength instability method, we investigate the linear instability of an exact solution describing upward-propagating mountain waves, derived in A. Constantin, \emph{J. Phys. A: Math. Theor.} (2023), under the assumption…
We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify…
One key issue in the probability density function (PDF) approach for disperse two-phase turbulent flows is to close the diffusion term in the phase space. This study aimed to derive a kinetic equation for particle dispersion in turbulent…
This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new…
Explicit solutions for extended objects of a Q-ball type were found analytically in a model describing complex scalar field with piecewise parabolic potential in (3+1)- and (1+1)-dimensional space-times. Such a potential provides a variety…
A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is also…
In complex power systems, nonlinear load flow equations have multiple solutions. Under typical load conditions only one solution is stable and corresponds to a normal operating point, whereas the second solution is not stable and is never…