The Cauchy-Dirichlet problem for the heat equation in Besov spaces

Abstract

The solvability in anisotropic spaces $B_{p,q}^{\tfrac{\sigma } {2},\sigma } (\Omega ^T )$ , σ ∈ ℝ₊, p, q ∈ (1, ∞), of the heat equation u_t − Δu = f in Ω^T ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on S^T, u|_t=0 = u₀ in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝⁿ. The existence of a unique solution $B_{p,q}^{\tfrac{\sigma } {2},1,\sigma + 2} (\Omega ^T )$ of the above problem is proved under the assumptions that $S \in C^{\sigma + 2} ,f \in B_{p,q}^{\tfrac{\sigma } {2},\sigma } (\Omega ^T ), g \in B_{p,q}^{\tfrac{\sigma } {2} + 1 - \tfrac{1} {{2P}},\sigma + 2 - \tfrac{1} {P}} (S^T ), u_0 \in B_{p,q}^{\sigma + 2 - \tfrac{2} {P}} (\Omega )$ and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence, the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation. Bibliography: 26 titles.