On the galerkin method for non-linear evolution equation
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CitationVinogradova, P. (2009). On the galerkin method for non-linear evolution equation. Maltepe Üniversitesi. s. 329.
Let H1 be a Hilbert space densely and compactly embedded in a Hilbert space H. In the space H we consider the Cauchy problem u 0 (t) + A(t)u(t) + K(u(t)) = h(t), u(0) = 0. (0.1) We assume that the operators A(t) and K(·) have the following properties. 1) A(t) is self-adjoint operator in H with domain D(A(t)) = H1. A(t) is positive definite operator. 2) The operator A(t) is strongly continuously differentiable on [0, T]. There is a constant β ≥ 0 such that (A 0 (t)v, v)H ≤ β(A(0)v, v)H. 3) The non-linear operator K(·) is subordinate to operator A(0) with order 0 ≤ τ < 1, i.e. D(K(·)) ⊃ D(A(0)) and for any v ∈ H1 the inequality kK(v)k ≤ kA(0)vk τ ϕ(kvk 2 ) holds, where ϕ(ξ) is a continuous positive function on [0, ∞). The operator K(t) is compact. 4)There is given a positively definite self-adjoint operator B which is similar to A(0), i.e., D(B) = D(A(0)). 5) The operators A(t) and B satisfy the inequality (A(t)v, Bv)H ≥ mkA(0)vkkBvk, where a constant m > 0 is independent of the choice v ∈ H1 and t. By e1, e2, . . . , en, . . . we denote a complete orthonormalized system of eigenvectors of B with the corresponding eigenvalues λ1, λ2, . . . , λn, . . . , so that 0 < λ1 ≤ λ2 ≤ . . . ≤ λn . . . and λn → ∞ as n → ∞. Let Pn be the orthogonal projection in H onto the linear span Hn of the elements e1, e2, . . . , en. In Hn we consider the problem: u 0 n(t) + PnA(t)un(t) + PnK(un(t)) = Pnh(t), un(0) = 0. (0.2) Let h(t) ∈ L2(0, T; H). It was proved, that problems (1) and (2) have at least one solution at each n and that from the sequence un(t) it is possible to select the subsequence, which converges to the solution of problem (1) in strong norm.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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