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Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the…
Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of…
We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration…
We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the…
This article is devoted to the proof of the well-posedness of a model describing waves propagating in shallow water in horizontal dimension $d=2$ and in the presence of a fixed partially immersed object. We first show that this…
The well-posedness of a generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory was recently established by the author with C.G. Gal. In this article we report advances concerning the asymptotic behavior…
We present our deep learning framework to solve and accelerate the Time-Dependent partial differential equation's solution of one and two spatial dimensions. We demonstrate DiffusionNet solver by solving the 2D transient heat conduction…
We consider elliptic operators with operator-valued coefficients and discuss the associated parabolic problems. The unknowns are functions with values in a Hilbert space $W$. The system is equipped with a general class of coupled boundary…
We study the initial boundary value problem for a heat equation in a domain containing a thin layer. The thermal conductivity of the layer is drastically different from that of the bulk of the domain; moreover, the layer is anisotropic and…
We present a new formulation and generalization of the classical theory of heat conduction with or without fading memory which includes the usual heat equation subject to a dynamic boundary condition as a special case. We investigate the…
We investigate linear boundary value problems for first-order one-dimensional hyperbolic systems in a strip. We establish conditions for existence and uniqueness of bounded continuous solutions. For that we suppose that the non-diagonal…
We establish two-sided Gaussian bounds on the heat kernel of divergence-form parabolic equation with singular time-inhomogeneous vector field satisfying some minimal assumptions.
We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable…
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 introduce the most general class of linear boundary-value problems for systems of first-order ordinary differential equations whose solutions belong to the complex H\"older space $C^{n+1,\alpha}$, with $0\leq n\in\mathbb{Z}$ and…
We propose methods that augment existing numerical schemes for the simulation of hyperbolic balance laws with Dirichlet boundary conditions to allow for the simulation of a broad class of differential algebraic conditions. Our approach is…
In this work, we consider not only a discontinuous boundary-value problem with retarded argument and four supplementary transmission conditions at the two points of discontinuities but also, eigenparameter-dependent boundary conditions and…
Based on a combination of insights afforded by Rainer Picard and Serge Nicaise, we extend a set of abstract piezo-electromagnetic impedance boundary conditions. We achieve this by accommodating for the influence of heat with the inclusion…
Like any fluid heated from below, the atmosphere is subject to vertical instability which triggers convection. Convection occurs on small time and space scales, which makes it a challenging feature to include in climate models. Usually…
We study well posedness of time--dependent Hamilton--Jacobi equations on a network, coupled with a continuous initial datum and a flux limiter. We show existence and uniqueness of solutions as well as stability properties. The novelty of…