Sentence examples for using the embedding theorem from inspiring English sources

Consequently, we immediately get the desired result by using the embedding theorem again.

By using the embedding theorem and the trace theorem, we obtain the main result.

First, by using the embedding theorem (Theorem 3) and the trace theorem (Theorem 1) in the space W l, p ( R + n ; E ), we obtain the following.

By using the embedding theorem, the higher-order energy estimates and bootstrap arguments, the condition of chemotaxis-driven instability and the nonlinear evolution near an unstable positive constant equilibrium for this chemotaxis model are proved.

Using the embedding theorem [2] a phase space trajectory in dimension m with m > 1 is created from u. Therefore m values from u are used to create a new vector v of dimension m representing the points of the phase space trajectory.

Differentiating (1.2) with respect to y twice, using the embedding theorem and Theorem 1.1, we conclude ∥ u t y y ( t ) ∥ ≤ C 3 ( ∥ v y ( t ) ∥ H 2 + ∥ θ y ( t ) ∥ H 2 + ∥ u y ( t ) ∥ H 3 + ∥ b y ( t ) ∥ H 2 ), (3.5).

Lemma 3.2 F ∩ Λ is a weakly closed subset in H 1. Proof Let { u n } ⊂ F ∩ Λ be a weakly convergent sequence, we use the embedding theorem to find which uniformly converges to u ∈ H 1. Now we claim u ∈ Λ, and then it is obvious that u ∈ F. In fact, if u ∈ ∂ Λ, by V ( q ) → − ∞, q → 0 and the condition (A4)′ we have − V ( u ) ≥ C 1 | u | − β, 0 < | u | < r ′ < r.

Conversely, using the Sobolev embedding theorem (H^{s}hookrightarrow L^{infty}) ((s>frac{1}{2})), we derive that if condition (1.5) in Theorem 1.2 holds, then the corresponding solution blows up in finite time.

Also, using the Sobolev embedding theorem, the compactness of the trace map and the convexity of G, we see that (widehat {varphi }_{v}) is sequentially weakly lower semicontinuous.

By using the Sobolev embedding theorem, it follows that from (4.7) and (4.8) the right of the above inequality is bounded.

Using the Sobolev Embedding Theorem and Hölder's Inequality, we can derive the following results (for more details, see [11, Lemma  3.4] and [13, Lemma  4.2]).