On the porous medium equations with lower order singular nonlinear terms

Abstract

where rn_~l, 2>0, p>0 , b>0, c > 0 are constants. The function uo(x) is supposed to be bounded, continuous and nonnegative, satisfying uo(x) >0 on land uo(x) =-0 in R \ I for a bounded interval I = (al, a2)c R. Equation (1) is parabolic when u>0 and degenerates when u=0. In the case b = c = 0 (1) is called the equation of nonlinear heat conductivity or filtration equation. The terms cu p and b(uX)~ in (1) correspond to the absorption and to the transport of heat or matter, respectively. The Cauchy problem (1), (2) usually has no classical solution having continuous derivatives which appear in (1) even in the case b = c = 0 (see Zeldovich and Komponeets [7]). Oleinik, Kalastmikov and YuMin [6] proved that the Cauchy problem (1), (2) has a unique generalized solution in the case b = c = 0 .