Parabolic and pseudo-parabolic partial differential equations*

1. Introduction.Let G be a bounded domain in an Euclidean n-space and Ḡ its closure.Let Cᵏ(G), 0 k<∞ , be the class of real-valued functions defined and k-times continuously differentiable in G and C_0ᵏ(G) the subset of Cᵏ(G) consisting of those functions with compact support in G .As usual [1-4], we denote by Ĥᵏ(G) the pre-Hilbert space consisting of functions in Cᵏ(G) with finite k-fold Dirichlet norm, ₖ , and denote by Hᵏ(G) the Hilbert space being the completion of Ĥᵏ(G) under the norm ₖ .In a completely similar way one defines the pre-Hilbert space Ĥ_0ᵏ(G) and the Hilbert space H_0ᵏ(G) .In the following discussions the domain G will be fixed.We shall for simplicity write H_0ᵏ for H_0ᵏ(G) etc..It may be noted that H₀⁰ is the space L₂(G) .Consider the two self-adjoint elliptic partial differential operators, (1.1) L≡∑i,j=1η∂∂ xᵢ(lᵢⱼ(x)∂∂ xⱼ)-l(x) , M≡∑i,j=1ⁿ-∂Xᵢ∂(mᵢⱼ(x)∂∂ xⱼ)-m(x) .It will be assumed that the given real-valued functions lᵢⱼ(x), $l(x),$ mᵢⱼ(x) and $m(x)$ are bounded measurable in G and that l(x) 0, m(x) 0 almost everywhere in G .Further we shall restrict L and M to be elliptic in the sense that there are constants kL, KL, kM and KM such that almost everywhere in G (1.2) kL∑ᵢ₌₁ⁿ(ξᵢ)²∑i,j=1ⁿlᵢⱼ(x)ξᵢξⱼ KL∑ι̇=1ⁿ(ξᵢ)² , kM∑ᵢ₌₁ⁿ(ξᵢ)²∑t,j=1ⁿmᵢⱼ(x)ξᵢξⱼ KM∑i,j=1ⁿ(ξᵢ)² , for all real vectors ξ .