In this paper we discuss a c1ass of partial differential equations with singular coefficients，the initial problem for which can be solved uniquely. We have proved the fo11owing theorem: The problem: Lu≡[(t^a/2?_t-λ₁(x,t)?_x)(t^a/2?_t-λ₂(x,t)?_x)+a(x,t)?_t+b(x,t)?_x+c(x,t)]u(x,t)=f(x,t) (x,t)∈R×(0,T] u∣_t=0=φ(x),limt^a/2u_t=ψ(x) can be solved uniquely by u∈C([0,T],H₁(R))∩C¹((0,T],L₂(R)) if the following conditions are satisfied. （A） a is a constant and 0j(x,t)(j=1,2)∈C’([0,T], C²(R)) and all the derivatives of the above functions are bounded by the constant; （C） φ(x),ψ(x)∈C₀⁴(R) （D） f(x,t)∈C((0,T],C₀²(R)),and sup{t^a/2(∣f∣+∣f_x∣+∣f_xx∣}＜+∞ (E) There exists a constantδ＞0,such that ∣λ₁(x,t)-λ₂(x,t)∣≥δwhere (x,t)?R×[0,T]. In the proof,we use the particular sets of energy inequalities.A generalizatitn of the theorem is obtained.