What does this research mean for the field?

Under an exterior geometric control condition, the local energy of solutions to the damped defocusing energy-critical wave equation outside two strictly convex obstacles decays exponentially at the same rate as the damped linear flow. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish the uniform exponential decay of local energy for solutions to the defocusing energy-critical wave equation with damping in the exterior of two strictly convex obstacles in three dimensions.

August 22, 2026AxiomsOpen Access

Exponential Decay of the Local Energy for Solutions of the Damped Critical Wave Equation Outside the Union of Two Strictly Convex Obstacles

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To establish the uniform exponential decay of local energy for solutions to the defocusing energy-critical wave equation with damping in the exterior of two strictly convex obstacles in three dimensions.
Analyzed the exterior domain under a geometric control condition to address trapped periodic rays bouncing between obstacles.
Combined Lafontaine’s lossless Strichartz estimates with Lax–Phillips and microlocal defect measure analysis of the damped linear flow.
Used a Duhamel–Gronwall scheme to transfer linear decay estimates to the nonlinear energy-critical setting.
Proved that local energy decays exponentially and uniformly for data supported in a fixed ball below any energy threshold.
Demonstrated that the nonlinear decay rate exactly equals that of the damped linear flow, remaining independent of initial data and energy levels.

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Implication

Theoretical analysis demonstrates exponential local energy decay for damped critical wave equations outside two convex obstacles, indicating trapping does not prevent linear-rate stabilization.

Cite This Study

Mehenaoui et al. (2026) studied this question.

synapsesocial.com/papers/6a895fe4ca7ade938187ee46 https://doi.org/10.3390/axioms15080622

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Implication

Theoretical analysis demonstrates exponential local energy decay for damped critical wave equations outside two convex obstacles, indicating trapping does not prevent linear-rate stabilization.

Exponential Decay of the Local Energy for Solutions of the Damped Critical Wave Equation Outside the Union of Two Strictly Convex Obstacles

Key Points

Discussion

Implication

Cite This Study

Exponential Decay of the Local Energy for Solutions of the Damped Critical Wave Equation Outside the Union of Two Strictly Convex Obstacles

Key Points

Discussion

Implication

Cite This Study

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