On the galerkin method for non-linear evolution equation
Küçük Resim Yok
Tarih
2009
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Maltepe Üniversitesi
Erişim Hakkı
CC0 1.0 Universal
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
Özet
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 ? ? 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.
Açıklama
Anahtar Kelimeler
Kaynak
International Conference of Mathematical Sciences
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Vinogradova, P. (2009). On the galerkin method for non-linear evolution equation. Maltepe Üniversitesi. s. 329.
