LAUSR

Abstract We consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term: ∂ t u − ( | u x | p − 2 u x ) x + u − β χ { u > 0 } = f ( u , x , t ) , in I × ( 0 , T ) , with the homogeneous zero boundary condition on I = ( x 1 , x 2 ) , an open bounded interval in R . Through this paper, we assume that p > 2 and β ∈ ( 0 , 1 ) . To show the local existence result, we prove first a sharp pointwise estimate for | u x | . One of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t ∈ ( 0 , ∞ ) , the so called global solutions, or by the contrary a finite time blow-up τ 0 > 0 arises such that lim t → τ 0 ⁡ ‖ u ( t ) ‖ L ∞ ( I ) = + ∞ . Moreover, we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough. Finally, we show that the condition f ( 0 , x , t ) = 0 , ∀ ( x , t ) ∈ I × ( 0 , ∞ ) is a necessary and sufficient condition for the existence of solution of equations of this type.