In this paper, we study the initial-boundary value problem involving the p(x)-Laplacian operator with the Robin boundary condition <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u_t(x,t) -\Delta _{p(x)}u(x,t) = \vert u \vert ^{q(x)-2}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. The local existence of solutions is established using the Galerkin method. By combining the potential well method with the Nehari manifold, we derive decay estimates for global weak solutions under suitable conditions for the initial energy to be less than or equal to d. Additionally, we apply the concavity method to obtain blow-up solutions to the problem. Finally, we use an extended form of the concavity method to prove that the weak solutions blow up in finite time for arbitrarily large initial energy.

Global existence, asymptotic bihavior and blow-up of solutions for parabolic equations involving the $p(x)-$Laplacian under Robin boundary conditions

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发表日期：

2025年9月15日

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Abdelkader El minsari,
Anass Ourraoui

摘要

In this paper, we study the initial-boundary value problem involving the p(x)-Laplacian operator with the Robin boundary condition $u_t(x,t) -\Delta _{p(x)}u(x,t) = \vert u \vert ^{q(x)-2}u$ $u_{t} (x, t) - Δ_{p (x)} u (x, t) = {| u |}^{q (x) - 2} u$ . The local existence of solutions is established using the Galerkin method. By combining the potential well method with the Nehari manifold, we derive decay estimates for global weak solutions under suitable conditions for the initial energy to be less than or equal to d. Additionally, we apply the concavity method to obtain blow-up solutions to the problem. Finally, we use an extended form of the concavity method to prove that the weak solutions blow up in finite time for arbitrarily large initial energy.

Global existence, asymptotic bihavior and blow-up of solutions for parabolic equations involving the \(p(x)-\)Laplacian under Robin boundary conditions

摘要