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This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping. We study the case where the damping is effective on the complement of visible umbilical sets.
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This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping. We study the case where the damping is effective in a well-chosen subset of arbitrarily small measure.
We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven by additive space-time white-noise, and with initial data distributed according to the Gibbs measure. By introducing a suitable space-dependent renormalization, we prove local well-posedness of the renormalized equation. Bourgains invariant measure argument then allows us to establish almost sure global well-posedness and invariance of the Gibbs measure for the renormalized stochastic damped NLW. (ii) Similarly, we study the random data defocusing NLW (without stochastic forcing), and establish the same results as in the previous setting. (iii) Lastly, we study the stochastic NLW without damping. By introducing a space-time dependent renormalization, we prove its local well-posedness with deterministic initial data in all subcritical spaces. These results extend the corresponding recent results on the two-dimensional torus obtained by (i) Gubinelli-Koch-Oh-Tolomeo (2018), (ii) Oh-Thomann (2017), and (iii) Gubinelli-Koch-Oh (2018), to a general class of compact manifolds. The main ingredient is the Greens function estimate for the Laplace-Beltrami operator in this setting to study regularity properties of stochastic terms appearing in each of the problems.
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
In this paper, we first establish a criterion based on contractive function for the existence of polynomial attractors. This criterion only involves some rather weak compactness associated with the repeated limit inferior and requires no compactness, which makes it suitable for critical cases. Then by this abstract theorem, we verify the existence of a polynomial attractor and estimate its attractive speed for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity.
We concern the blow up problem to the scale invariant damping wave equations with sub-Strauss exponent. This problem has been studied by Lai, Takamura and Wakasa (cite{Lai17}) and Ikeda and Sobajima cite{Ikedapre} recently. In present paper, we extend the blowup exponent from $p_F(n)leq p<p_S(n+2mu)$ to $1<p<p_S(n+mu)$ without small restriction on $mu$. Moreover, the upper bound of lifespan is derived with uniform estimate $T(varepsilon)leq Cvarepsilon^{-2p(p-1)/gamma(p,n+2mu)}$. This result extends the blowup result of semilinear wave equation and shows the wave-like behavior of scale invariant damping wave equations solution even with large $mu>1$.
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