This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The frst case is the equation with classical derivative. The formal solution is obtained. And a defnition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, diferentiable nowhere in its domain. Next, for specifc given initial position and 3 diferent initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given.
The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on the Euclidean space R^n and the local field K are compared,so that one may interpret the essential differences between the analyses on R^n and K.Finally,the Cantor type fractal functionθ(x)is showed in the Lipschitz class Lip(m,K),m＜(ln 2/ln 3).
Wei-yi SU~(1+) Guo-xiang CHEN~2 1 Department of Mathematics,Nanjing University,Nanjing 210093,China
受文de Laubenfels（1997，Isreal Journal of Mathematics，98：189—207）的启发，引进空间形（A，k）和H（A，ω），它们分别是使得该二阶抽象Cauchy问题有在[0，∞）一致连续且O（（1＋t）^k）有界和O（e^ωt）有界的弱解的x∈X的全体．讨论Banach空间X上二阶抽象Cauchy问题的具有多项式有界解或指数有界解的极大子空间问题．由Wang and Wang（1996，Functional Analysis in China．Kluwer，333—350）知，该Cauchy问题适定的充要条件是该Cauchy问题中的X上闭算子A生成一个强连续Cosine算子函数．处理该Cauchy问题不适定的情况，证明或指出了如下结论：·W（A，k）和H（A，ω）均为Banach空间，且W（A，k）和H（A，∞）均连续嵌入X； ·部分算子AIW（A，k）生成一个多项式有界的余弦算子函数使‖C（t）‖W（A,k）≤2（1＋t）^k；·部分算子AIW（A，ω）生成一个指数有界的余弦算子函数{C（t）}t∈R＋，‖C（t）‖H（W,ω）≤2e^ωt；·W（A，k）和H（A，ω）分别是极大的．即若有Banach空间Y连续嵌入X，且使AIY生成一个O（（1＋t）^k）有界的余弦算子函数，那么Y连续嵌入W（A，k）；而若使AIY生成一个O（e^ωt）有界的余弦算子函数，那么Y连续嵌入H（A，ω）．
The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho¨lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a "chain of function spaces" over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields.
