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Similarly, (u_{infty}(x)) is a super-solution to (3.6).

The solution to problem (1.1) is a super-solution of (4.7).

It is easy to see that (2.10) is valid for sufficiently small ρ > 0 since q < r ≤ m < 1. Next, we turn our attention to construct a super-solution of (1.1).

A vector-valued function ( u ¯, v ¯ ) is said to be a T-periodic supersolution of problem (1.1 - 1.3 1.1 - 1.3s a super-solutifn it [ 0, T ] such that u ¯ ( ⋅, 0 ) ≥ u ¯ ( ⋅, T ), v ¯ ( ⋅, 0 ) ≥ v ¯ ( ⋅, T ) is Ω.

So z is a super-solution of problem (4.1).

Therefore, u ¯ is a super-solution of problem (1.1).

Then, ((overline{u}(t),overline{v}(t))) is a super-solution of (1.6).

Then it can be verified that v ( x, t ) is a super-solution of (1.1).

Since u ̲ is a super-solution of (4.7), ϕ cannot exist globally.

It can be verified that, for the case m = q < r < 1, a sufficiently large constant L is a super-solution of (1.1).

Moreover, as in the proof of Theorem 1.2, it can be verified that g ( t ) is a super-solution of (1.1).