Abstract: In this paper, it is considered that the global existence, uniqueness and regularity results for the Cauchy problem of the well-known Kuramoto-Stvashinsky equation
u_t+1/2▽(|u|²)+△u+△²u=0,t>0,x∈R^N, u(0,x)=u₀(x),x∈R^N.
only under the condition u₀(x) ∈L²(R^N, Rⁿ). Where u(t, x)=(u₁ (t, x),…,u_n (t, x))^T is the unknown vector-valued function. Results show that for N < 6,u₀ (x) ∈L² (R^N, Rⁿ), the above Cauchy problem admits a unique global solution u(t, z) which belongs to C^∞,∞ (R^N×(0, ∞)).

Key words: Kuramoto-Sivashinsky equation, Singular initial data, Sobolev imbedding theorem