Аннотация:This paper deals with the global existence of the weak solutions to the abstract Cauchy problem for three kinds of nonlinear operator-differential equations, such as, e.g.,
ddt(A0u+∑j=1NAj(u))=F(u),ddt(A0u+∑j=1NAj(u))+Lu=F(u),ddt(A0u+∑j=1NAj(u))+D(P(u))=F(u),
with the Cauchy datum u(0)=u0. The authors state that by applying Galerkin's method combined with monotonicity and compactness methods, they obtain several results on the global existence and the blow-up of the solutions for these problems (no proofs are given in the paper). As applications of the results, the authors give some examples, such as, e.g.,
∂∂t(Δu+∑j=1ndiv(|∇u|pj−2∇u))+|u|qu=0,(x,t)∈Ω×[0,T],
u=0on ∂Ω,u(x,0)=u0(x)for x∈Ω.