## On the galerkin method for non-linear evolution equation

#### Citation

Vinogradova, P. (2009). On the galerkin method for non-linear evolution equation. Maltepe Üniversitesi. s. 329.#### Abstract

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.

#### Source

International Conference of Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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