Strong solutions and the initial data space for some non-uniformly parabolic equations

Abstract

The solution for this problem can be considered in different meanings, for example, a classical solution, strong solution, weak (or mild) solution, entropy solution, etc. This paper is devoted to a strong solution for this problem. If A is an unbounded linear operator in Hilbert space H, then it is proved that there does not exist a strong solution for all u0 ∈ H (see [2, 11]). It is proved in [13] that there exists a strong solution if and only if u0 ∈ [D(A), H]1/2, where [D(A), H]1/2 is an interpolation space (see [9,15]). If A is a nonlinear operator, the weak solutions of these problems is often investigated. We consider the case where the operator A has the form: