Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: u_t (x, t) - Δ_p u (x, t) - | u |^{q - 2} u (x, t) = f (x, t), (x, t) ∈ Ω × (0, T), where 2 ≤ p < q < + ∞, Ω is a bounded domain in R^N, f : Ω × (0, T) → R is given and Δ_p denotes the so-called p-Laplacian defined by Δ_p u : = ∇ ṡ (| ∇ u |^{p - 2} ∇ u), with initial data u₀ ∈ L^r (Ω) is proved under r > N (q - p) / p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, L^r-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0, T₀] in which the problem admits a solution. More precisely, T₀ depends only on | u₀ |_Lr and f.