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(2) If q = 1 < p, then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data provided that λ is appropriately large.
Therefore, u ( x, t ) is a nonextinction solution of equation (1.1) for any nonnegative initial data provided that λ > 0 is appropriately large.
If q = p < 1, then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data provided that λ > 0 is appropriately large.
Theorem 2.3 (Nonextinction) (1) If q = p < 1, then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data provided that λ > 0 is appropriately large.
Therefore problem (1.1) admits a solution u ( x, t ) satisfying u ( x, t ) ≥ w ( x, t ), which, combined with w ( x, t ) > 0 ( Ω × ( 0, + ∞ ) ) implies that u ( x, t ) is a nonextinction solution of equation (1.1) for any nonnegative initial data provided that λ > 0 is appropriately large.
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