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<p>In this paper we prove the existence of global attractors for the following semilinear degenerate parabolic equation on <InlineEquation ID="IEq3"> <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>: <Equation ID="Equ46"> <EquationSource Format="TEX">\( \frac{\partial u}{\partial t} - \text {div}(\sigma (x)\nabla u) + \lambda u+ f(x,u) = g(x),\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mi>∂</mi> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mi mathvariant="normal">div</mi> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">x</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>λ</mi> <mi mathvariant="normal">u</mi> <mo>+</mo> <mi mathvariant="normal">f</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">x</mi> <mo>,</mo> <mi mathvariant="normal">u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">g</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>under some new conditions concerning a variable non-negative diffusivity <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sigma (\cdot )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the exponential nonlinearity <i>f</i>. We prove the existence of weak solutions by combining Galerkin’s method and the weak compactness theorem in the Orlicz spaces. To overcome some significant difficulty arising when proving the existence of a global attractor caused by the lack of compactness of the embeddings, we try to combining the tail estimates method and the asymptotic a priori estimate method.</p>

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Global attractors for a class of semilinear degenerate parabolic equations on \({\mathbb {R}}^N\)

  • 期刊论文

  • 发表日期：

    2025年11月24日

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  • Le Thi Thuy,
  • Tran Thi Quynh Chi

摘要

In this paper we prove the existence of global attractors for the following semilinear degenerate parabolic equation on \({\mathbb {R}}^N\) R N : \( \frac{\partial u}{\partial t} - \text {div}(\sigma (x)\nabla u) + \lambda u+ f(x,u) = g(x),\) ∂ u ∂ t - div ( σ ( x ) ∇ u ) + λ u + f ( x , u ) = g ( x ) , under some new conditions concerning a variable non-negative diffusivity \(\sigma (\cdot )\) σ ( · ) and the exponential nonlinearity f. We prove the existence of weak solutions by combining Galerkin’s method and the weak compactness theorem in the Orlicz spaces. To overcome some significant difficulty arising when proving the existence of a global attractor caused by the lack of compactness of the embeddings, we try to combining the tail estimates method and the asymptotic a priori estimate method.