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<p>This paper is concerned with the energy decay rate of the total energy to a wave equation with <i>p</i>(<i>x</i>)-Laplacian damping (nonlinear strong damping) and nonlinear source. With some suitable restrictions on variable growth exponents <i>r</i>(<i>x</i>) and <i>p</i>(<i>x</i>), first, we prove that the local solution can be extended to exist globally. Second, by using suitable and weighted multiplier techniques, it is proved that the total energy decays logarithmically. The key and main difficulty is to give a prior estimate for the wighted integral <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\int_{\Omega}\chi^{p(x)-1}(\tau)\vert\nabla u(\tau)\vert^{p(x)}dx\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mo>∫</mo> <mrow> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <msup> <mi>χ</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">∣</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">∣</mo> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>d</mi> <mi>x</mi> </math></EquationSource> </InlineEquation> by some differential inequality techniques. In the proof of energy decay, the traditional method to eliminate the lower-order terms by exploiting the unique continuation and compactness arguments is not needed in our energy decay estimate.</p>

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Energy Decay for a Nonlinear Wave Equation with p(x)-Laplacian Damping

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Meng-lan Liao,
  • Xiao-lei Li,
  • Zayd Hajjej

摘要

This paper is concerned with the energy decay rate of the total energy to a wave equation with p(x)-Laplacian damping (nonlinear strong damping) and nonlinear source. With some suitable restrictions on variable growth exponents r(x) and p(x), first, we prove that the local solution can be extended to exist globally. Second, by using suitable and weighted multiplier techniques, it is proved that the total energy decays logarithmically. The key and main difficulty is to give a prior estimate for the wighted integral \(\int_{\Omega}\chi^{p(x)-1}(\tau)\vert\nabla u(\tau)\vert^{p(x)}dx\) ∫ Ω χ p ( x ) − 1 ( τ ) ∣ ∇ u ( τ ) ∣ p ( x ) d x by some differential inequality techniques. In the proof of energy decay, the traditional method to eliminate the lower-order terms by exploiting the unique continuation and compactness arguments is not needed in our energy decay estimate.