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.”

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.

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.

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.

Numerical solution of parabolic equations in high dimensions

Tobias Von Petersdorff, Christoph Schwab (2004)

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

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We consider the numerical solution of diffusion problems in $(0, T) \times Ω$ for $Ω \subset ℝ^{d}$ and for $T > 0$ in dimension $d \geq 1$ . We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p \geq 1$ , and $h p$ discontinuous Galerkin time-discretization of order $r = O (|log h|)$ on a geometric sequence of $O (|log h|)$ many time steps. The linear systems in each time step are solved iteratively by $O (|log h|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an $L^{2} (Ω)$ -error of $O (N^{- p})$ for $u (x, T)$ where $N$ is the total number of...

Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan, Masahiro Yamamoto (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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Let $y (h) (t, x)$ be one solution to $\partial_{t} y (t, x) - \sum_{i, j = 1}^{n} \partial_{j} (a_{i j} (x) \partial_{i} y (t, x)) = h (t, x), 0 < t < T, x \in Ω$ with a non-homogeneous term $h$ , and ${y |}_{(0, T) \times \partial Ω} = 0$ , where $Ω \subset ℝ^{n}$ is a bounded domain. We discuss an inverse problem of determining $n (n + 1) / 2$ unknown functions $a_{i j}$ by ${\partial_{ν} y (h_{ℓ}) |_{(0, T) \times Γ_{0}}$ , $y (h_{ℓ}) (θ, \cdot)}_{1 \leq ℓ \leq ℓ_{0}}$ after selecting input sources $h_{1}, . . ., h_{ℓ_{0}}$ suitably, where $Γ_{0}$ is an arbitrary subboundary, $\partial_{ν}$ denotes the normal derivative, $0 < θ < T$ and $ℓ_{0} \in ℕ$ . In the case of $ℓ_{0} = {(n + 1)}^{2} n / 2$ , we prove the Lipschitz stability in the inverse problem if we choose $(h_{1}, . . ., h_{ℓ_{0}})$ from a set $ℋ \subset {C_{0}^{\infty} ((0, T) \times ω)}^{ℓ_{0}}$ with an arbitrarily fixed subdomain $ω \subset Ω$ . Moreover we can take $ℓ_{0} = (n + 3) n / 2$ by making...

Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity

Jerzy August Gawinecki

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CONTENTS1. Introduction..................................................................................................................................... 5 1.1. Main Theorem 1.1................................................................................................................. 8 1.2. Main Theorem 1.2................................................................................................................. 92. Radon transform.......................................................................................................................................

$C^{\infty}$ -well posedness of the Cauchy problem for quasi-linear hyperbolic equations with coefficients non-Lipschitz in time and smooth in space

Akisato Kubo, Michael Reissig (2003)

Banach Center Publications

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In this paper we prove the $C^{\infty}$ -well posedness of the Cauchy problem for quasi-linear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0,T] and smooth in x ∈ ℝⁿ.

Boundary estimates for certain degenerate and singular parabolic equations

Benny Avelin, Ugo Gianazza, Sandro Salsa (2016)

Journal of the European Mathematical Society

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We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$ -Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_{T}$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of...

Asymptotics for quasilinear elliptic non-positone problems

Zuodong Yang, Qishao Lu (2002)

Annales Polonici Mathematici

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In the recent years, many results have been established on positive solutions for boundary value problems of the form $- d i v (| \nabla u (x) |^{p - 2} \nabla u (x)) = λ f (u (x))$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).