Abstract quasilinear parabolic equations with variable domains (Q1176214)

The Cauchy problem for the equation \(u'=A(u)u+f(u)\) is studied in a Banach space \(X\). The main supposition is the existence of a constant interpolation space between \(D(A)\) and \(X\), allowing the reduction to the case with constant domain. The result leads to interesting existence and uniqueness theorems (of local or global type) for a quasilinear variational parabolic equation. The author uses the semigroup approach, working in spaces of continuous or Hölder continuous functions; the equations coefficients are nondifferentiable. The proofs and the quotations from literature are fully detailed and commented.