• Title/Summary/Keyword: Semilinear degenerate elliptic equations

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LONG TIME BEHAVIOR OF SOLUTIONS TO SEMILINEAR HYPERBOLIC EQUATIONS INVOLVING STRONGLY DEGENERATE ELLIPTIC DIFFERENTIAL OPERATORS

  • Luyen, Duong Trong;Yen, Phung Thi Kim
    • Journal of the Korean Mathematical Society
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    • v.58 no.5
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    • pp.1279-1298
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    • 2021
  • The aim of this paper is to prove the existence of the global attractor of the Cauchy problem for a semilinear degenerate hyperbolic equation involving strongly degenerate elliptic differential operators. The attractor is characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.

EXISTENCE THEOREMS FOR CRITICAL DEGENERATE EQUATIONS INVOLVING THE GRUSHIN OPERATORS

  • Huong Thi Thu Nguyen;Tri Minh Nguyen
    • Communications of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.137-151
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    • 2023
  • In this paper we prove the existence of nontrivial weak solutions to the boundary value problem -G1u = u3 + f(x, y, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain with smooth boundary in ℝ3, G1 is a Grushin type operator, and f(x, y, u) is a lower order perturbation of u3 with f(x, y, 0) = 0. The nonlinearity involved is of critical exponent, which differs from the existing results in [11, 12].