Uniqueness and continuous dependence results for solutions of singular differential equations

AbstractSolutions of Cauchy problems for the singular equations utt + (Ψ(t)t) ut = Mu (in a Hilbert space setting) and ut + Δu + ∑mi=1 ((kixi)(∂i∂i)) + g(t)u=0 in ω × |0,T), ω={(x1,…,xM)εRm: 0 < xi < ci for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0⩽t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced