It follows from [26, Theorem 5.2.1] that (U_{2}(t)geq0) in its maximal interval of existence.
To show that the maximal interval of existence of (2.1) is indeed all of (mathbb {R}) we assume that there is a maximal interval of existence and then show that we can extend the solution beyond that interval.
Then, on a common interval of existence of u 0 ( t ) and v ( t ) v ( t ) ≤ u 0 ( t ).
In addition, the interval of existence for the solutions types II, III varies with the expansion ratio.
By the maximum principle, it follows that u ( x, t ) ≥ 0 in the time interval of existence.
Because of (C3), the last solution cannot meet the surface (Gamma_{i}) again, then ([theta_{i},theta_{i+1}]) is the maximal right interval of existence of (x t)) here.
In this work, we recall the method of existence of a solution by the upper and lower solution methods, which guarantees the interval of existence.
Suppose that [0, T*] is the maximal interval of existence of the solution θ ∈ C ( [ 0, T * ], Ḣ 1 . Now, we show the global existence for (1.3).
Also, let [ t 0, η be the maximal right-interval of existence of N t ( t 0, φ ). The remaining part of this paper is structured as follows.
{ d ϕ d t = − ϕ + T n, λ , ϕ = u ∈ W 0 1, p ∖ K. Denote τ ( u ) to be the maximal interval of existence of ϕ u ( t ).
