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A compactness result for solutions to an equation with boundary singularity

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 Publication date 2018
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and research's language is English




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We give a blow-up behavior for solutions to a problem with singularity and with Dirichlet condition. An application, we have a compactness of the solutions to this Problem with singularity and Lipschitz conditions.

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We consider the nonlinear damped Klein-Gordon equation [ partial_{tt}u+2alphapartial_{t}u-Delta u+u-|u|^{p-1}u=0 quad text{on} [0,infty)times mathbb{R}^N ] with $alpha>0$, $2 le Nle 5$ and energy subcritical exponents $p>2$. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate $t^{-1}$ to the excited state.
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