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Recently, it was claimed that the two-phase local/nonlocal constitutive models give well-posed nonlocal field problems and eliminates the ill-posedness of the fully nonlocal constitutive models. In this study, it is demonstrated that, both,…
In this contribution we investigate the application of phase-field fracture models on non-linear multiscale computational homogenization schemes. In particular, we introduce different phase-fields on a two-scale problem and develop a…
We establish global-in-time decay estimates for the multi-phase Muskat problem in the case where the density takes exactly n+1 distinct constant values. We first linearize the system around a flat stable configuration, followed by the study…
We consider a weighted nonlocal area functional in which the coefficients do not satisfy the triangle inequality. In the context of three phase transitions, this means that one of the weights is larger than the sum of the other two, say…
I argue that ``good'' mathematical models of spatio-temporal dynamics in two-dimensions require non-local operators in the nonlinear terms. Consequently, the often used Swift-Hohenberg equation requires modification as it is purely local.…
We construct, from first principles, a covariant local model for scalar fractonic matter coupled to a symmetric tensor gauge field. The free gauge field action is just the one of the Blasi-Maggiore model. The scalar sector, describing…
We study the higher H\"older regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained…
Multipolaron solutions were studied in the framework of the Holstein one-dimensional molecular crystal model. The study was performed in the continuous limit where the crystal model maps into the nonlinear Schr\"odinger equation for which a…
We develop a unified framework for a broad class of nonlocal elliptic problems, encompassing a wide spectrum of nonlocal terms, including the classical Kirchhoff and Carrier-type equations as particular cases, and nonlinearities having…
This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the $n$-dimensional Euclidean space or a…
In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized strain, the rate…
In this paper we study the homogenization of a stochastic process and its associated evolution equations in which we mix a local part (given by a Brownian motion with a reflection on the boundary) and a nonlocal part (given by a jump…
In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a…
We establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line. The class of equations includes most importantly nonlinear…
We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $\alpha$-entropy…
We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to…
Phase-field (PF) simulation provides a powerful framework for predicting microstructural evolution but suffers from prohibitive computational costs that severely limit accessible spatiotemporal scales in practical applications. While…
We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By ex- tending the least-squares stabilization to the overlap…
Local polynomial regression of order one or higher often performs poorly in areas with sparse data. In contrast, local constant regression tends to be more robust in these regions, although it is generally the least accurate approach,…
In this work we study a nonlinear Volterra equation with non-symmetric feedback that arises as a particular case of the Gurtin-MacCamy model in population dynamics. We are particularly interested in the existence of slowly oscillating…