Degenerate Kirchhoff-type wave problem involving the fractional Laplacian with nonlinear damping and source terms
Degenerate Kirchhoff-type wave problem involving the fractional Laplacian with nonlinear damping and source terms
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涉及具有非线性阻尼和源项的分数拉普拉斯算子的简并基尔霍夫型波问题
DOI：
10.1007/s00028-019-00489-6
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
张彬林
中科院分区：
文献类型：
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作者：
Ning Pan;Patrizia Pucci;Runzhang Xu;张彬林
本文考虑具有非线性阻尼项和源项的分数拉普拉斯方程{u_tt+u^2γ-2_S(-Δ)^su+|u_t|^a-2u_t+u=|u|^b-2u，inΩ*R^+，\u(⋅，0)=u_0，\\u_t(⋅，0)=u_1，&inΩ，\u=0，&in(R^N∖Ω)*R^+_0，.U tt+u S 2γ-2(-Δ)su+|ut|a-2ut+u=|u|b-2 u，inΩ×R+，u(·，0)=u0，ut(·，0)=u1，inΩ，u=0，in(RN\Ω)×R0+，其中(-Δ)S(-Δ)S是分数拉普拉斯算子，u_S u S是u的Gagliardo半范数，S∈(0，1)S∈(0，1)，2<；a<；2γ<；b<；2_S^*=2N/(N-2S)2<；a<；2γ<；b<；2 S∗=2 N/(N-2 S)，Ω⊂R^NΩ⊂RN是一个具有Lipschitz边界∂Ω∂Ω的有界域。在一定的自然假设下，将Galerkin方法与势井理论相结合，得到了上述问题解的整体存在性、真空隔离性、渐近性和爆破。该问题的显著特点和难点是(-Δ)S(-Δ)S的系数可以为零。
In this paper, we consider the following Kirchhoff-type wave problems, with nonlinear damping and source terms involving the fractional Laplacian,{u_ tt+ u^ 2 γ-2 _ s (-Δ)^ su+| u_t|^ a-2 u_t+ u=| u|^ b-2 u, in Ω * R^+,\u (⋅, 0)= u_0,\\u_t (⋅, 0)= u_1, & in Ω,\u= 0, & in (R^ N ∖ Ω) * R^+ _0,. u tt+ u s 2 γ-2 (-Δ) su+| ut| a-2 ut+ u=| u| b-2 u, in Ω× R+, u (·, 0)= u 0, ut (·, 0)= u 1, in Ω, u= 0, in (RN\Ω)× R 0+, where (-Δ)^ s (-Δ) s is the fractional Laplacian, u _ s u s is the Gagliardo semi-norm of u, s ∈ (0, 1) s∈(0, 1), 2< a< 2 γ< b< 2_s^*= 2N/(N-2s) 2< a< 2 γ< b< 2 s∗= 2 N/(N-2 s), Ω ⊂ R^ N Ω⊂ RN is a bounded domain with Lipschitz boundary ∂ Ω∂ Ω. Under some natural assumptions, we obtain the global existence, vacuum isolating, asymptotic behavior and blowup of solutions for the problem above by combining the Galerkin method with potential wells theory. The significant feature and difficulty of the problem are that the coefficient of (-Δ)^ s (-Δ) s can vanish at zero.