where T is the maximum existence time of solutions u 1 and u 2 and c depends on ∥ u 1 ( 0, x ) ∥ H 1 ( R ) and ∥ u 2 ( 0, x ) ∥ H 1 ( R ).
Here D ⊂ R N ( N ≥ 1 ) is a bounded domain with a smooth boundary ∂D, D ¯ is the closure of D, q = | ∇ u | 2, n is the outer normal vector and T is the maximum existence time of u ( x, t ). a ( u ) b ( x ) c ( t ), f ( x, u, q, t ) and h ( x, t ) r ( u ) are nonlinear diffusion coefficient, reaction term and boundary flux, respectively.
Let u 0 ( x ) ∈ H s ( R n ) with s > n 2. Then the Cauchy problem (3) has a unique solution u ( t, x ) ∈ C ( [ 0, T ) ; H s ( R n ) ) ∩ C 1 ( [ 0, T ) ; H s − 1 ( R n ) ) where T is the maximum existence time. Moreover, lim t → T ∥ u ( t, ⋅ ) ∥ H s ( R n ) = ∞. if and only if lim t → T ∥ u ( t, ⋅ ) ∥ L ∞ ( R n ) = ∞. For the case of space dimension n = 1, we have the result.
which implies that T is not the maximum existence time and thus the solution ( u, w, b ) can be extended beyond T by the standard arguments of continuation of local solutions.
That is to say, the solution (g z)) can be extended to the interval ([alpha-h_{0},1]), which contradicts the fact that ((alpha,1]) is the maximum interval of the existence of solutions.
Ten minutes was the maximum".
Size 10 was the maximum".
Furthermore, Weinstein's theory predicts the existence of more than 150 new subatomic particles, most of them with exotic properties (such as electric charges that are greater than one, which is the maximum seen in nature at present).
The punishment was the maximum allowed.
