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Tag: Nonlinear Partial Differential Equations

  • Existence, comparison, and uniqueness shown for anisotropic evolution equations

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    What the study found The study found existence of solutions for a Cauchy-Dirichlet problem involving a class of fully nonlinear anisotropic evolution equations. It also proved a comparison principle and concluded that the solutions are unique. Why the authors say this matters The authors present these results as establishing well-posedness properties for this class of…

  • Log-Sobolev inequality holds for some Gibbs measures

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    What the study found The authors show that the Gibbs measures for the focusing Schrödinger equation satisfy a log-Sobolev inequality when 2 ≤ p ≤ 4. For p > 4, they establish a lower bound for the Hessian of the effective potential. Why the authors say this matters The study suggests that the log-Sobolev inequality…

  • Global regularity and blow-up criteria are established for a non-local equation

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    What the study found The study found that higher-order Sobolev solutions for a non-local integrable evolution equation can exist globally under a natural assumption on the initial momentum. It also identified a criterion for the existence of blow-up solutions. Why the authors say this matters The authors say the equation is related to pseudospherical surfaces…

  • Local Lipschitz regularity proved for minimizers of certain functionals

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    What the study found The authors prove local Lipschitz regularity for minimizers of a class of functionals, meaning the minimizers have locally bounded slopes. They also show, as a byproduct, that a locally Lipschitz minimizer exists for a related class of functionals in which the function f may be nonconvex. Why the authors say this…