On some parabolic equations involving superlinear singular gradient terms

Abstract

In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with superlinear gradient terms which are possibly singular. The model equation iswhere Ω is an open bounded subset of R N 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.