Let be the dimension of . Let us consider a real number satisfying the following condition:
Let and be the following real numbers:
Then the authors prove:
Theorem 1. Let be a regular bounded domain of and a real number satisfying the condition (1) if or if . Let
where . Then there exists functions u,v: ]0,T[ such that:
satisfying the nonlinear systems:
and the initial conditions:
Theorem 2. Let u,v: ]0,T[ be functions in the classes (7), (8) and (9) satisfying from (10) to (13). Then, provided that in case or 2; if in case .