Related papers: Evans function, parity and nonautonomous bifurcati…
We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality…
Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate…
Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet…
In this paper we investigate spectral stability of traveling wave solutions to 1-$D$ quantum hydrodynamics system with nonlinear viscosity in the $(\rho,u)$, that is, density and velocity, variables. We derive a sufficient condition for the…
Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution…
We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric…
We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the…
We reduce the problem of proving decay estimates for viscosity solutions of fully nonlinear PDEs to proving analogous estimates for solutions of one-dimensional ordinary differential inequalities. Our machinery allow the ellipticity to…
The existence of periodic solutions is proven for some neuroscience models with a small parameter. Moreover, the stability of such solutions is investigated, as well. The results are based on a theoretical research dealing with the…
Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for…
We consider a continuum active polar fluid model for the spreading of epithelial monolayers introduced by R. Alert, C. Blanch-Mercader, and J. Casademunt, 2019. The corresponding free boundary problem possesses flat front traveling wave…
In this paper we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of second kind in a Hilbert space. The challenge of the present work is to take into account a perturbation on all the data of the…
This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on…
This paper studies the linear stability problem for solitary wave solutions of Hamiltonian PDEs. The linear stability problem is formulated in terms of the Evans function, a complex analytic function denoted by $D(\lambda)$, where $\lambda$…
Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators…
We consider a sequence of finite irreducible Markov chains with exponentially small transition rates: the transition graph is a fixed, finite, strongly connected directed graph; the transition rates decay exponentially on a paramenter N…
Systems of non-autonomous parabolic partial differential equations over a bounded domain with nonlinear term of Carath\'eodory type are considered. Appropriate topologies on sets of Lipschitz Carath\'eodory maps are defined in order to have…
The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the…
We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that…
A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations -- such as coefficient identification in partial differential…