[Paper Review] Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

Masahiro Ikeda, Yuta Wakasugi|arXiv (Cornell University)|Aug 26, 2017

Advanced Mathematical Physics ProblemsMathematics43 references19 citations

TL;DR

This paper establishes global well-posedness for the semilinear wave equation with time-dependent damping in the overdamping regime, where the inverse damping integral is finite. By transforming the equation and leveraging the overdamping condition to uniformly bound the L²-norm of solutions, the authors prove small data global existence in the energy-subcritical case (1 ≤ p < p₁), contrasting sharply with the small data blow-up observed in non-overdamping regimes.

ABSTRACT

We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the whole energy-subcritical case. This result implies that small data blow-up does not occur in the overdamping case, different from the other cases, i.e. effective or non-effective damping.

Motivation & Objective

• To resolve the open problem of global existence for semilinear wave equations with time-dependent damping in the overdamping case, where b(t)⁻¹ ∈ L¹(0, ∞).
• To establish global well-posedness in the energy space H¹(Rᵈ) × L²(Rᵈ) for small initial data in the energy-subcritical regime (1 ≤ p < p₁).
• To demonstrate that small data blow-up does not occur in the overdamping regime, unlike in the non-overdamping case (b(t)⁻¹ ∉ L¹(0, ∞)).
• To prove non-existence of local weak solutions for large data or supercritical nonlinearities (p > p₁) under suitable singularity assumptions.
• To show that the small data assumption can be removed under a defocusing condition (e.g., N(z) = −|z|ᵖ⁻¹z).

Proposed method

• Transform the original damped wave equation into a new semilinear wave equation without first-order time derivatives via the change of variables: v(t,x) = exp(½∫₀ᵗ b(s)ds) u(t,x).
• Use the overdamping condition (b(t)⁻¹ ∈ L¹(0,∞)) to derive a uniform L²-norm bound for local solutions, which is key to extending existence globally.
• Apply local well-posedness theory for energy-subcritical semilinear wave equations in H¹ × L², relying on Strichartz estimates and interpolation theory.
• Employ test functions with compact support and homogeneity to derive lower bounds on the L²-norm of solutions, leading to blow-up criteria.
• Use contradiction arguments with estimates on integrals involving |x|⁻ᵏφₗ(x) to prove non-existence of local solutions for p > p₁ or large data.
• Establish the blow-up criterion by showing that the L²-norm of the solution must diverge in finite time if the lifespan is finite.

Experimental results

Research questions

• RQ1 Does global existence hold for the semilinear wave equation with time-dependent damping in the overdamping case (b(t)⁻¹ ∈ L¹(0,∞)) for small initial data in the energy-subcritical regime?
• RQ2 How does the overdamping condition (b(t)⁻¹ ∈ L¹(0,∞)) prevent small data blow-up, unlike in the non-overdamping case?
• RQ3 Can the small data assumption be removed under a defocusing nonlinearity condition (e.g., N(z) = −|z|ᵖ⁻¹z)?
• RQ4 What happens to the solution when the nonlinearity is energy-supercritical (p > p₁) or when initial data are large and singular?
• RQ5 Is it possible to prove non-existence of local weak solutions for certain classes of initial data and nonlinearities in the supercritical regime?

Key findings

• Global well-posedness holds for small initial data (u₀, u₁) ∈ H¹(Rᵈ) × L²(Rᵈ) in the energy-subcritical case (1 ≤ p < p₁) under the overdamping condition b(t)⁻¹ ∈ L¹(0,∞).
• A uniform L²-norm bound for local solutions is derived using the overdamping condition, enabling global extension of solutions.
• The small data blow-up observed in the non-overdamping case (b(t)⁻¹ ∉ L¹(0,∞)) does not occur in the overdamping regime.
• For the defocusing nonlinearity N(z) = −|z|ᵖ⁻¹z, the small data assumption can be removed, and global existence holds for all initial data in H¹ × L².
• For focusing nonlinearities N(z) = ±|z|ᵖ with p ∈ (1, p₁], large initial data with a singularity at x = 0 lead to finite-time blow-up, and the lifespan T₊ satisfies T₊ ≤ C∗₂ λ⁻¹/(p+1)/(p−1)−k for λ > λ₀.
• No local weak solutions exist for nonlinearities with p > p₁ when initial data are singular at the origin, under suitable conditions on b(t).