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to express the solution u ( x, t ).

By Lemma 2.1, the solution u ∈ H.

corresponding to the solution ( u, v ).

The solutions U ± ∗ are stable and satisfy U − ∗ ≤ 0 and U + ∗ > 0. The solution U 0 ∗ is unstable.

The solution u N j has the following property.

Theorem 3.8 provides the existence of the solution u ∈ C ∞.

The solution (u) of (1.3), (1.5) is called a potential.

Then the solution u ( t ) of system (1.1) is stable.

Moreover, the solution u is in E 1 ( X ).

Assume by contradiction that the solution (u, v) is global.

Then the solution u blows up at finite T ∗.