Approximation of nonlinear wave equations with nonstandard anisotropic growth conditions

Haehnle, Jonas; Prohl, Andreas

doi:10.1090/s0025-5718-09-02231-5

Cited by 17 publications

(7 citation statements)

References 26 publications

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“…The proof of this theorem goes exactly like that of Theorem . More on the local and global existence can be found in Antontsev and Haehnle and Prohl…”

Section: Preliminariesmentioning

confidence: 62%

On the decay of solutions of a damped quasilinear wave equation with variable‐exponent nonlinearities

Messaoudi

2020

Math Methods in App Sciences

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In this work, we consider the following nonlinear wave equation with variable exponents:where Ω is a bounded domain, T > 0, and m(.) and r(.) are continuous functions. We will establish several decay results depending on the range of the variable exponents m and r.KEYWORDS exponential decay, polynomial decay, strong damping, variable exponent, wave MSC CLASSIFICATION 35B35; 35L20; 35L70in a bounded domain Ω, with a smooth boundary, was considered at a large scale. For instance, Nakao 6 looked into Equation (1.2), in the case when (u) = |u| p−2 u, and g(u t ) = |u t | m−2 u t , m, p > 2 and proved a unique global weak solution if 2 ≤ p ≤ 2(n − 1)∕(n − 2), n ≥ 3 and a global unique strong solution if p > 2(n − 1)∕(n − 2), n ≥ 3. Moreover, he addressed Math Meth Appl Sci. 2020;43:5114-5126. wileyonlinelibrary.com/journal/mma

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“…The proof of this theorem goes exactly like that of Theorem . More on the local and global existence can be found in Antontsev and Haehnle and Prohl…”

Section: Preliminariesmentioning

confidence: 62%

On the decay of solutions of a damped quasilinear wave equation with variable‐exponent nonlinearities

Messaoudi

2020

Math Methods in App Sciences

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“…Indeed, it is necessary, for such a numerical study, to approach p(x) by some piecewise constant or piecewise polynomial functions p h (x), h being the discretization parameter (see e.g. Haehnle and Prohl [37] and the forthcoming paper [7] for numerical approximation of problems involving p(x)-laplacian).…”

Section: P(•)mentioning

confidence: 99%

“…In what concerns the passage-to-the-limit techniques, Zhikov's methods include semicontinuity arguments and an ingenious adaptation of the classical Minty-Browder monotonicity argument; see in particular [68,Lemmas 8,9]. Similar approaches were used by Haehnle and Prohl [37] and by Wróblewska (see [62] and references therein). Our argument is longer but more straightforward.…”

Section: Introductionmentioning

confidence: 99%

Structural stability for variable exponent elliptic problems, I: The -Laplacian kind problems

Andreïanov

Bendahmane

Ouaro

2010

Nonlinear Analysis: Theory, Methods & Applications

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We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b(u n) − div a n (x,∇u n) = f n. The equation is set in a bounded domain Ω of R N and supplied with the homogeneous Dirichlet boundary condition on ∂Ω. Here b is a non-decreasing function on R, and a n (x, ξ) n is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent p n (x), 1 < p − ≤ p n (•) ≤ p + < +∞. The need for making vary p(x) arises, for instance, in the numerical analysis of the p(x)−laplacian problem. Uniqueness and existence for these problems are well understood by now. We apply the stability properties to further generalize the existence results. The continuous dependence result we prove is valid for weak and for renormalized solutions. Notice that, besides the interest of its own, the renormalized solutions' framework also permits to deduce optimal convergence results for the weak solutions. Our technique avoids the use of a fixed duality framework (like the W 1,p(x) 0 (Ω)-W −1,p ′ (x) (Ω) duality), and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself, in a local or in a non-local way. The sequel of this paper will be concerned with wellposedness of some p(u)-laplacian kind problems and with existence of solutions to elliptic systems with variable, solution-dependent growth exponent.

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“…[11,13,27,28], and the literature cited therein. Evolutionary equations with nonstandard anisotropic growth conditions have been investigated both, analytically and numerically in [1,2], and [26,17]. In this work, we present an implicit finite element discretization of (1.8)-(1.11), which is referred to as Scheme A, for variable p ∈ C Ω, (1, ∞) .…”

Section: Introductionmentioning

confidence: 99%

“…(1.1)-(1.4)). The numerical analysis combines tools which have been developed in [7] for constant p's, and in [26,17] for the numerical study of related evolutionary problems with nonstandard anisotropic growth conditions. Computational studies, both academic and with applications in computational fluid dynamics, are reported in Section 5.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis for Incompressible Generalized Newtonian Fluid Flows with Nonstandard Anisotropic Growth Conditions

Carelli

Haehnle²,

Prohl³

2010

SIAM J. Numer. Anal.

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We study equations to describe incompressible generalized Newtonian fluids, where the extra stress tensor satisfies a nonstandard anisotropic asymptotic growth condition. An implicit finite element discretization, and a simple, fully practical fixed-point scheme with proper thresholding criterion are proposed, and convergence towards weak solutions of the limiting problem is shown. Computational experiments are included, which motivate nontrivial fluid flow behavior.

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Approximation of nonlinear wave equations with nonstandard anisotropic growth conditions

Cited by 17 publications

References 26 publications

On the decay of solutions of a damped quasilinear wave equation with variable‐exponent nonlinearities

On the decay of solutions of a damped quasilinear wave equation with variable‐exponent nonlinearities

Structural stability for variable exponent elliptic problems, I: The -Laplacian kind problems

Convergence Analysis for Incompressible Generalized Newtonian Fluid Flows with Nonstandard Anisotropic Growth Conditions

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