An energy analysis of degenerate hyperbolic partial differential equations.

William J. Layton

Displaying similar documents to “An energy analysis of degenerate hyperbolic partial differential equations.”

Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

Papri Majumder (2021)

Applications of Mathematics

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We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in $ℝ^{d}$ $(d = 2, 3)$ . For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete...

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

$L^{p} - L^{q}$ time decay estimates for the solution of the linear partial differential equations of thermodiffusion

Arkadiusz Szymaniec (2010)

Applicationes Mathematicae

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We consider the initial-value problem for a linear hyperbolic parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove $L^{p} - L^{q}$ time decay estimates for the solution of the associated linear Cauchy problem.

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Peter Oswald (2017)

Applications of Mathematics

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Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^{1}$ norm best approximation error estimates for $H^{2}$ functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation....

Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Peter Poláčik (2002)

Mathematica Bohemica

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We consider three types of semilinear second order PDEs on a cylindrical domain $Ω \times (0, \infty)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 2$ . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $Ω \times (0, \infty)$ is reserved for time $t$ , the third type is an elliptic equation with a singled out unbounded variable $t$ . We discuss the asymptotic behavior, as $t \to \infty$ , of solutions which are defined and bounded on $Ω \times (0, \infty)$ .

Approximation of a semilinear elliptic problem in an unbounded domain

Messaoud Kolli, Michelle Schatzman (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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Let $f$ be an odd function of a class $C^{2}$ such that $f (1) = 0, f^{'} (0) < 0, f^{'} (1) > 0$ and $x \mapsto f (x) / x$ increases on $[0, 1]$ . We approximate the positive solution of $- Δ u + f (u) = 0,$ on $ℝ_{+}^{2}$ with homogeneous Dirichlet boundary conditions by the solution of $- Δ u_{L} + f (u_{L}) = 0,$ on ${] 0, L [}^{2}$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_{L} - u$ tends to zero exponentially fast, in the uniform norm.

Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition

Yong-Hyok Jo, Myong-Hwan Ri (2022)

Applications of Mathematics

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We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value $u_{0} \in H^{1} (Ω)$ is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when $u_{0} \in L^{2} (Ω)$ and the integral kernel in the nonlocal boundary condition is symmetric. ...

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

Alessandra Lunardi (1998)

Studia Mathematica

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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.