We study non-autonomous parabolic equations with critical exponents in a scale of Banach spaces Eσ, σ ∈ [0,1 + μ). We consider a suitable E1+ε-solution and describe continuation properties of the solution. This concerns both a situation when the solution can be continued as an E1+ε-solution and a situation when the E1+ε-norm of the solution blows up, in which case a piecewise E1+ε-solution is constructed.

In the early 70s A. Kaneko studied the problem of continuation of regular solutions of systems of linear partial differential equations with constant coefficients to compact convex sets. We show here that the conditions be obtained for real analytic solutions also hold in the quasi-analytic case. In particular we show that every quasi-analytic solution of the system p(D)u = 0 defined outside a compact convex subset K or Rn can be continued as a quasi-analytic solution to K if and only if the system is determined and the -module Ext1(Coker p′, ) has no elliptic component; here is the ring of polynomials in n variables, p is a matrix with elements from and p′ is the transposed matrix. In the scalar case, i.e. when p is a single polynomial, these conditions mean that p has no elliptic factor.