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Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping

Daoyin He, Yaqing Sun, Kangqun Zhang

TL;DR

The paper proves sharp global existence for small data in semilinear wave equations with time-dependent, scale-invariant damping by transforming to a semilinear generalized Tricomi equation and establishing new weighted Strichartz estimates. The core strategy combines a time-change and angular-FIO analysis to derive endpoint and inhomogeneous bounds, followed by a Picard iteration to obtain global weak solutions for exponents in the range $p_{\text{crit}}(n,\mu)<p\le p_{\text{conf}}(n,\mu)$ with $n\ge3$ and $\mu\in(0,1)\cup(1,2)$. The work develops a comprehensive toolkit of weighted estimates for the generalized Tricomi operator, including dyadic decompositions, stationary-phase arguments, and complex interpolation to handle delicate endpoint cases. It also discusses the limiting cases $\mu=1$ and the $n=2$ setting, outlining open questions and related results. The results advance understanding of damping-driven global behavior in nonlinear waves and provide a framework for further refinements in low dimensions and borderline damping regimes.

Abstract

In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $n\ge 3$, $t\ge 1$ and $μ\in(0,1)\cup(1,2)$. For critical exponent $p_{crit}(n,μ)$ which is the positive root of $(n+μ-1)p^2-(n+μ+1)p-2=0$ and conformal exponent $p_{conf}(n,μ)=\frac{n+μ+3}{n+μ-1}$, we establish global existence for $n\geq3$ and $p_{crit}(n,μ)<p\leq p_{conf}(n,μ)$. The proof is based on changing the wave equation into the semilinear generalized Tricomi equation $\partial_t^2u-t^mΔu=t^{α(m)}|u|^p$, where $m=m(μ)>0$ and $α(m)\in\Bbb R$ are two suitable constants, then we investigate more general semilinear Tricomi equation $\partial_t^2v-t^mΔv=t^α|v|^p$ and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution $u$ can be obtained.

Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping

TL;DR

The paper proves sharp global existence for small data in semilinear wave equations with time-dependent, scale-invariant damping by transforming to a semilinear generalized Tricomi equation and establishing new weighted Strichartz estimates. The core strategy combines a time-change and angular-FIO analysis to derive endpoint and inhomogeneous bounds, followed by a Picard iteration to obtain global weak solutions for exponents in the range with and . The work develops a comprehensive toolkit of weighted estimates for the generalized Tricomi operator, including dyadic decompositions, stationary-phase arguments, and complex interpolation to handle delicate endpoint cases. It also discusses the limiting cases and the setting, outlining open questions and related results. The results advance understanding of damping-driven global behavior in nonlinear waves and provide a framework for further refinements in low dimensions and borderline damping regimes.

Abstract

In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of , where , and . For critical exponent which is the positive root of and conformal exponent , we establish global existence for and . The proof is based on changing the wave equation into the semilinear generalized Tricomi equation , where and are two suitable constants, then we investigate more general semilinear Tricomi equation and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution can be obtained.
Paper Structure (32 sections, 11 theorems, 248 equations)

This paper contains 32 sections, 11 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem and statement of the main result
  3. Proof of the main theorem
  4. Case I $\mathbf{ 0<\mu<1}$
  5. Case II $\mathbf{ 1<\mu<2}$
  6. Some remarks
  7. Sketch for the proof of Theorem 1.2
  8. Basic Estimates
  9. Homogeneous Strichartz estimates with characteristic weight
  10. Inhomogeneous Strichartz estimates without characteristic weight
  11. Case I $\mathbf { 2s\leq t }$
  12. Case II $\mathbf { 2s>t }$
  13. The proof of Theorem 1.3 at the end point $\mathbf{q=q_0}$
  14. Local estimate at $q=q_0$
  15. Simplifications for the end point estimate
  16. ...and 17 more sections

Key Result

Theorem 1.1

Assume that $n\geq3$, $\mu\in(0,1)\cup(1,2)$ and $p_{crit}(n,\mu)<p\leq p_{conf}(n,\mu)$. Suppose $u_i\in C_c^{\infty}(\Bbb R^n)$ ($i=0, 1$), then for $\varepsilon>0$ small enough problem equ:eff1 admits a global weak solution $u$ with for some $\gamma$ satifying

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.3
  • Theorem 1.4
  • ...and 17 more