Vinogradova, Polina2024-07-122024-07-122009Vinogradova, P. (2009). On the galerkin method for non-linear evolution equation. Maltepe Üniversitesi. s. 329.9.78605E+12https://hdl.handle.net/20.500.12415/2250Let 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 ? ? 0 such that (A 0 (t)v, v)H ? ?(A(0)v, v)H. 3) The non-linear operator K(·) is subordinate to operator A(0) with order 0 ? ? < 1, i.e. D(K(·)) ? D(A(0)) and for any v ? H1 the inequality kK(v)k ? kA(0)vk ? ?(kvk 2 ) holds, where ?(?) 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 ?1, ?2, . . . , ?n, . . . , so that 0 < ?1 ? ?2 ? . . . ? ?n . . . and ?n ? ? 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.enCC0 1.0 Universalinfo:eu-repo/semantics/openAccessConference Object330329