This paper aims to study the model of Klein–Gordon–Schrödinger equations with a memory term and locally distributed damping : <Equation ID="Equ71"> <EquationSource Format="TEX">$ {\left\{ \begin{array}{ll} i\psi '+ \Delta \psi + i\alpha (|\psi |^{2} +1)\psi =- \phi \psi ~\text{ in }~\Omega \times (0, \infty ), \\ \phi '' - \Delta \phi +\displaystyle \int _{0}^{t} g(t-\tau ) div[a(x) \nabla \phi (\tau )] d \tau +b(x)\phi '= |\psi |^{2\theta }\chi _{\omega } ~\text{ in }~\Omega \times (0, \infty ),\\ \end{array}\right. } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mi>i</mi> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>ψ</mi> <mo>+</mo> <msup> <mrow> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>ψ</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mo>-</mo> <mi>ϕ</mi> <mi>ψ</mi> <mspace width="3.33333pt" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mrow /> <msup> <mi>ϕ</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>t</mi> </msubsup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>-</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>i</mi> <mi>v</mi> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> <mi>τ</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>ψ</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>θ</mi> </mrow> </msup> <msub> <mi>χ</mi> <mi>ω</mi> </msub> <mspace width="3.33333pt" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded domain of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and smooth boundary <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\partial \Omega =\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi mathvariant="normal">Γ</mi> </mrow> </math></EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> are positive constants. In this work, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\chi _{\omega }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>χ</mi> <mi>ω</mi> </msub> </math></EquationSource> </InlineEquation> represents the cutoff function of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>. Assuming that a and b are non-negative functions such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a(x) + b(x) \ge \delta > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, the exponential decay rate is demonstrated for each regular solution of the above system.

Exponential Stability for the Coupled Klein–Gordon–Schrödinger Equations with Competing Viscoelastic and Frictional Dissipative Effects

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2026年7月4日

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L. R. L. Camargo,
M. M. Cavalcanti,
C. M. Webler,
J. P. Zanchetta

摘要

This paper aims to study the model of Klein–Gordon–Schrödinger equations with a memory term and locally distributed damping : $ {\left\{ \begin{array}{ll} i\psi '+ \Delta \psi + i\alpha (|\psi |^{2} +1)\psi =- \phi \psi ~\text{ in }~\Omega \times (0, \infty ), \\ \phi '' - \Delta \phi +\displaystyle \int _{0}^{t} g(t-\tau ) div[a(x) \nabla \phi (\tau )] d \tau +b(x)\phi '= |\psi |^{2\theta }\chi _{\omega } ~\text{ in }~\Omega \times (0, \infty ),\\ \end{array}\right. } $ $\{\begin{matrix} i ψ^{'} + Δ ψ + {i α (| ψ |}^{2} + 1) ψ = - ϕ ψ in Ω \times (0, \infty), \\ ϕ^{''} - Δ ϕ + \int_{0}^{t} g (t - τ) d i v [a (x) \nabla ϕ (τ)] d τ + b (x) ϕ^{'} = {| ψ |}^{2 θ} χ_{ω} in Ω \times (0, \infty), \end{matrix})$ where $\Omega $ $Ω$ is a bounded domain of $\mathbb {R}^n$ $R^{n}$ with $n\le 3$ $n \leq 3$ and smooth boundary $\partial \Omega =\Gamma $ $\partial Ω = Γ$ . Here, $\alpha $ $α$ and $\theta $ $θ$ are positive constants. In this work, $\chi _{\omega }$ $χ_{ω}$ represents the cutoff function of $\omega $ $ω$ . Assuming that a and b are non-negative functions such that $a(x) + b(x) \ge \delta > 0$ $a (x) + b (x) \geq δ > 0$ in $\Omega $ $Ω$ , the exponential decay rate is demonstrated for each regular solution of the above system.