EXISTENCE OF THE CLASSICAL SOLUTION FOR DEGENERATE QUASILINEAR PARABOLIC PROBLEMS WITH SLOW DIFFUSIONS

Let T ≤ ∞, b be a positive number, m be a positive number such that m > 1, and q be a nonnegative number. Existence and uniqueness of a classical solution are studied for the following degenerate quasilinear parabolic problem,

                                  x^qu_t = ( u^m )_xx + bf( u )  in (0, 1) × (0, T),

                                 u( x, 0 )= u₀( x ) in [0, 1], u (0, t) = 0 = u ( 1, t ) for t ∈ (0, T),

where u₀( x ) is a positive function for 0 < x < 1,  u₀^m( x ) ∈ C^2+α ( [0, 1] ) for some α ∈ ( 0, 1 ), u₀( 0 ) = u₀( 1 ) = 0,  and f (u) is a given function such that f (0) ≥ 0 and f′ (u) ≥ 0 for u ≥ 0. Furthermore, a criterion for u to blow up in a finite time is given.