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where α, β are two positive constants. If (1.1) holds true, the differential operator A(u) is not coercive as u becomes large. This shows that the classical methods (see [22]) can’t be applied to prove the existence of solutions to problem (P ) even if the data f is sufficiently regular. The goal in this paper is to study the problem (P ) under the assumptions of (1.1)–(1.2). The proof is essentially based on the approximate problems (Pn) with some nondegenerate coercivity and a priori estimates on the weak solutions of these problems. Similar problem to elliptic equations has already been studied in [13] (see also [1, 2, 9, 10, 18, 19]). Recently, Porzio and Pozio in [24] have discussed the case of f ≡ 0, u(x, 0) = u0 6= 0. Now we state the main results of this paper.

Existence and regularity results for some parabolic equations with degenerate coercivity