On some parabolic equations involving superlinear singular gradient terms

In this paper we prove existence of nonnegative solutions to parabolic Cauchy–Dirichlet problems with (eventually) singular superlinear gradient terms. The model equation is ut-Δpu=g(u)|∇u|q+h(u)f(t,x)in(0,T)×Ω,where Ω is an open bounded subset of RN with N> 2 , 0 < T< + ∞, 1 < p< N, and q< p is superlinear. The functions g,h are continuous and possibly satisfying g(0) = + ∞ and/or h(0) = + ∞, with different rates. Finally, f is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of q, the regularity of the initial datum and the forcing term, and the decay rates of g,h at infinity.