Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

Abstract

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space. II

“…As the spaces H from Sections 1-3 in [1] we take H 2,0 . Then (−Δ) is a nonnegative selfadjoint operator on H, and [E + grad(−Δ) −1 div](u, ∇)u is a bilinear operator on H. Denote them by A and B(·, ·).…”

“…The conditions (2) and (3) in [1] are satisfied because of Lemma 3. Aside from the conditions of Sections 1-3 in [1], another condition is satisfied:…”

“…Note first that the a priori estimate (11) implies (see [1]) that if (w 1 , w 2 ) ∈ D a and w + Aw + B(w, w) ∈ H 2 [0, a] with w(0) = 0 then B w w 1 ∈ H 2 [0, a].…”

“…Therefore, in Definition 3d in [1] instead of saying "if (w 1 , w 2 ) ∈ D a and B w w 1 ∈ H 2 [0, a] then . .…”

“…Proof. By Definition 1, (12) is equivalent to the system (4) from Theorem A in [1] that corresponds to (10 ); i.e., (4) in [1] is an abstract form of (12). Therefore, Theorem 2 is a corollary to Theorem A in [1].…”