Inhomogeneous wave equation with t-dependent singular coefficients

Discacciati, Marco; Garetto, Claudia; Loizou, Costas

doi:10.1016/j.jde.2022.02.039

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…The final section of this paper is devoted to a wave equation toy model with discontinuous space-dependent coefficient. This complements the study initiated in [DGL22] where the equation's coefficient was depending on time only. In detail, we consider the Cauchy problem…”

Section: Toy Models and Numerical Experimentssupporting

confidence: 69%

“…Note that we do not expect that a change in the mollifier will impact our results as we have shown similarly in [DGL22].…”

Section: Toy Models and Numerical Experimentsmentioning

confidence: 49%

“…However, very weak solutions of higher Sobolev order can also be obtained with a suitable choice of nets involved in the regularisation process by following the techniques developed in Section 2. In analogy to Section 7 in [DGL22] it is immediate to see from the estimate (31) that perturbing the equation's coefficients, right-hand side and initial data with negligible needs leads to a negligible change in the solution, where with negligible we intend negligible net of Sobolev order 0. So, we can conclude that our Cauchy problem is very weakly well-posed.…”

Section: Case 2 We Assume That Our Coefficientsmentioning

confidence: 95%

“…It is of physical interest to understand the wellposedness of this Cauchy problem when the coefficients are less than continuous. This kind on investigation has been initiated in [GR14] for second order hyperbolic equations with t-dependent coefficients and recently extended to inhomogeneous equations in [DGL22]. Here we want to work with space-dependent coefficients with minimal assumptions of regularity, namely distributions with compact support.…”

Section: Introductionmentioning

confidence: 99%

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On the wave equation with space dependent coefficients: singularities and lower order terms

Discacciati¹,

Garetto²,

Loizou³

2022

Preprint

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This paper complements the study of the wave equation with discontinuous coefficients initiated in [DGL22] in the case of time-dependent coefficients.Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in [GR14]. As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.(a + b 1 ) 2 ≺ a.Note that a is bounded by √ a as a direct consequence of Glaeser's inequality: If a ∈ C 2 (R), a(x) ≥ 0 for all x ∈ R and a L ∞ ≤ M 1 , then |a (x)| 2 ≤ 2M 1 a(x),

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Section: Toy Models and Numerical Experimentssupporting

confidence: 69%

“…Note that we do not expect that a change in the mollifier will impact our results as we have shown similarly in [DGL22].…”

Section: Toy Models and Numerical Experimentsmentioning

confidence: 49%

Section: Case 2 We Assume That Our Coefficientsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the wave equation with space dependent coefficients: singularities and lower order terms

Discacciati¹,

Garetto²,

Loizou³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms

Discacciati,

Garetto,

Loizou

2023

Acta Appl Math

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This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ.319 (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in Arch. Ration. Mech. Anal.217 (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.

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