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This paper deals with a doubly degenerate parabolic equation with absorption (u n ) t  = (|u x | m-1 u x ) x  - ηu p  in (0, 1) × (0, T) subject to the boundary source (|u x | m-1 u x )(1, t) = u q (1, t). The critical boundary source exponent q c  is determined under different dominations to identify global and non-global solutions. A complete classification for all of the four nonlinear exponents m, n, p, q with the coefficient η draws a very clear picture of interactions among the multi-nonlinearities. It is shown that q c  depends on the absorption exponent p if and only if p &gt; n. It is found as well that q c  is related to small m only: q c  relies on m if and only if m &lt; p when p &gt; n, also q c  relies on m if and only if m &lt; n when p ≤ n. The behavior of solutions in the critical case of q = q c  is interesting. The absorption coefficient η affects the critical property of solutions with q = q c  if and only if p &gt; n: the case q = q c  belongs to the global existence situation (regardless of the absorption coefficient η) if p ≤ n; while for q = q c  with p &gt; n, the solutions may be both global and non-global, depending on the size of the absorption coefficient.

Critical Boundary Source Exponent in a Doubly Degenerate Parabolic Equation