On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space
On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space
复制标题
双曲空间中半线性抛物线和椭圆问题的适定性
DOI：
10.1016/j.jde.2011.05.033
复制
发表时间：
2011
影响因子：
2.4
通讯作者：
F. Punzo
中科院分区：
文献类型：
--
作者：
F. Punzo
我们研究了n维双曲空间（n大于或等于3）的有界域中半线性抛物线和椭圆方程解的存在性和唯一性。得到了由Laplace-Beltrami算子生成的半群的Lp→lq估计，并用它证明了抛物型问题的存在唯一性结果。此外，在适当的非线性函数假设下，我们建立了椭圆型问题正经典解的唯一性和奇异解的非唯一性；进一步，对相应的半线性抛物型问题，给出了弱解的非唯一性。
We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n⩾3). Lp→Lqestimates for the semigroup generated by the Laplace–Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.