Sentence examples for by constructing a subsolution from inspiring English sources

We begin the proof by constructing a subsolution.

We first prove (6) has a positive solution for every a > 0. We begin by constructing a subsolution.

We shall establish Theorem 2.1 by constructing a positive subsolution (Φ1, Φ2) and supersolution (z1, z2) of (P), such that Φ1 ≤ z1 and Φ2 ≤ z2.

Case I: In the case q = p < 1 with λ ∫ Ω ψ p ( x ) d x − M m p − 1 > 0, we shall prove that problem (1.1) admits at least one nonextinction solution for any nonnegative initial data by constructing a suitable subsolution of equation (1.1).

In Section 2 (Section 2.1), we consider a spatial domain (Omega subsetmathbb{R}^{N}) and derive an upper bound of the blow-up time by constructing a blowing up subsolution of our problem, which implies that our solution also blows up.

We first construct a subsolution.

We construct a subsolution of (P).

We will construct a subsolution of.

The main idea is to construct a subsolution of the equation.

We construct a subsolution (widetilde{w}) to problem (5.5), which satisfies begin{cases} partial _{tau} widetilde{w}-mathcal{L}widetilde{w} =-frac{c 1-gamma)}{P}, &(y,tau)in Rtimes(0,T], widetilde{w}(y,0)=-frac{c 1-gamma R. end{cases} (5.6) It is clear that (wgeq widetilde{w}) by the comparison principle.

Similarly to the proof of Proposition 4.1, we construct a subsolution as w x,t)=-lnbigl(V x,t -tbigr),t -tbigrV(x,t)=U(x,varepsilon t)), and (U(x,tau)) satisfies (U_{tau}=Delta U).