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<p>The multiplier method is a vital implement for investigating the decay and long-time behavior of solutions to hyperbolic equations with nonlinear damping. This paper examines the role of the strong damping term on the decay estimate of solutions, which extends our previous work (Li and Li in Evol. Equ. Control Theory 13(1):116–127, <CitationRef CitationID="CR12">2024</CitationRef>). For the subcritical cases, energy estimates combined with Komornik inequality yield exponential decay under strong damping – a sharp contrast to the algebraic decay induced by single nonlinear damping. For the supercritical case, to overcome the failure of the embedding <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mrow> <mn>0</mn> </mrow> <mn>1</mn> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>↪</mo> <msup> <mi>L</mi> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$H_{0}^{1}(\Omega )\hookrightarrow L^{m(\cdot )}(\Omega )$</EquationSource> </InlineEquation>, we establish a priori estimate and utilize a weighted multiplier method to demonstrate energy decay estimates, taking into account the effects of both single nonlinear damping and mixed damping. The main results are compiled in Table&#xa0;<InternalRef RefID="Tab1">1</InternalRef>, and a comparative summary of the decay laws is provided in the final section.</p>

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Long-Time Behavior of Solutions to the Fourth-Order Hyperbolic Equations with Variable-Exponent Nonlinearities

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Xiaolei Li,
  • Shuke Xiao,
  • Jingjing Zhang

摘要

The multiplier method is a vital implement for investigating the decay and long-time behavior of solutions to hyperbolic equations with nonlinear damping. This paper examines the role of the strong damping term on the decay estimate of solutions, which extends our previous work (Li and Li in Evol. Equ. Control Theory 13(1):116–127, 2024). For the subcritical cases, energy estimates combined with Komornik inequality yield exponential decay under strong damping – a sharp contrast to the algebraic decay induced by single nonlinear damping. For the supercritical case, to overcome the failure of the embedding H 0 1 ( Ω ) ↪ L m ( ⋅ ) ( Ω ) $H_{0}^{1}(\Omega )\hookrightarrow L^{m(\cdot )}(\Omega )$ , we establish a priori estimate and utilize a weighted multiplier method to demonstrate energy decay estimates, taking into account the effects of both single nonlinear damping and mixed damping. The main results are compiled in Table 1, and a comparative summary of the decay laws is provided in the final section.