In this paper, we study the general decay result of the viscoelastic wave equation of Kirchhoff-type with Balakrishnan-Taylor damping, weakly nonlinear time-dependent damping, and a logarithmic source term under acoustic boundary conditions. Our analysis is conducted under minimal conditions on the relaxation function g in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2048_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$L^{1} (0, \infty )$</EquationSource> </InlineEquation>, specifically, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2048_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="151" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$g'(t) \leq -\mu (t) G(g(t))$</EquationSource> </InlineEquation>, where G is an increasing and convex function near the origin and μ is a nonincreasing function. Moreover, we derive the energy decay rates that depend on the functions μ, ξ, and G, as well as the function K defined by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2048_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$k_{0}$</EquationSource> </InlineEquation>, which characterizes the growth of k at the origin. This result is novel and extends earlier results in the literature.

General decay for the viscoelastic wave equation for Kirchhoff-type containing Balakrishnan-Taylor damping, nonlinear damping and logarithmic source term under acoustic boundary conditions

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2025年4月29日

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Mi Jin Lee,
Jum-Ran Kang

摘要

In this paper, we study the general decay result of the viscoelastic wave equation of Kirchhoff-type with Balakrishnan-Taylor damping, weakly nonlinear time-dependent damping, and a logarithmic source term under acoustic boundary conditions. Our analysis is conducted under minimal conditions on the relaxation function g in $L^{1} (0, \infty)$ $L^{1} (0, \infty )$ , specifically, $g^{'} (t) \leq - μ (t) G (g (t))$ $g'(t) \leq -\mu (t) G(g(t))$ , where G is an increasing and convex function near the origin and μ is a nonincreasing function. Moreover, we derive the energy decay rates that depend on the functions μ, ξ, and G, as well as the function K defined by $k_{0}$ $k_{0}$ , which characterizes the growth of k at the origin. This result is novel and extends earlier results in the literature.