The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption

We consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term:∂tu−(|ux|p−2ux)x+u−βχ{u>0}=f(u,x,t),inI×(0,T), with the homogeneous zero boundary condition on I=(x1,x2), 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 |ux|. 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 τ0>0 arises such that limt→τ0⁡‖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.