Well-Posedness of a Mixed Problem in a Cylindrical Domain for One Class of Multidimensional Hyperbolic-Parabolic Equations

2020 ◽  
Vol 72 (2) ◽  
pp. 317-326
Author(s):  
S. A. Aldashev
Keyword(s):  
Mixed Problem ◽  
Parabolic Equations ◽  
Well Posedness ◽  
Cylindrical Domain

scholarly journals A Boundary Estimate for Singular Sub-Critical Parabolic Equations

2020 ◽  
Author(s):  
Ugo Gianazza ◽  
Naian Liao
Keyword(s):  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Boundary Point ◽  
Cylindrical Domain ◽  
Boundary Estimate ◽  
Critical Range ◽  
Type Integral ◽  
Wiener Type ◽  
Singular Parabolic Equations

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

scholarly journals Further results on a space-time FOSLS formulation of parabolic PDEs

2020 ◽  
Author(s):  
Gregor Gantner ◽  
Rob Stevenson
Keyword(s):  
Finite Element Method ◽  
Least Squares ◽  
Parabolic Equations ◽  
Space Time ◽  
Adaptive Finite Element Method ◽  
Order System ◽  
Well Posedness ◽  
Adaptive Finite Element ◽  
Parabolic Pdes ◽  
First Order

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer&Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven.  In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated.  The proof of the latter easily extends to a large class of least-squares formulations.

scholarly journals A semidiscrete Galerkin scheme for a coupled two-scale elliptic–parabolic system: well-posedness and convergence approximation rates

2020 ◽  
Vol 60 (4) ◽  
pp. 999-1031
Author(s):  
Martin Lind ◽  
Adrian Muntean ◽  
Omar Richardson
Keyword(s):  
Parabolic Equations ◽  
Spatial Scales ◽  
A Priori ◽  
Coupled System ◽  
Scale Model ◽  
Reactive Flow ◽  
Well Posedness ◽  
Approximation Rates ◽  
Priori Error Estimates ◽  
Model Equations

AbstractIn this paper, we study the numerical approximation of a coupled system of elliptic–parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the well-posedness of our two-scale model, we design two-scale convergent numerical approximations and prove a priori error estimates for the semidiscrete case. We complement our analysis with simulation results illustrating the expected behaviour of the system.

scholarly journals Space-Time Estimates of Mild Solutions of a Class of Higher-Order Semilinear Parabolic Equations in Lp

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Albert N. Sandjo ◽  
Célestin Wafo Soh
Keyword(s):  
Thin Film ◽  
Epitaxial Growth ◽  
Parabolic Equations ◽  
Mild Solutions ◽  
Film Theory ◽  
Well Posedness ◽  
Semilinear Parabolic Equations ◽  
Unified Framework ◽  
Time Estimates ◽  
Semilinear Parabolic

AbstractWe establish the well-posedness of boundary value problems for a family of nonlinear higherorder parabolic equations which comprises some models of epitaxial growth and thin film theory. In order to achieve this result, we provide a unified framework for constructing local mild solutions in C

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