The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut (x, t) - Δp 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 RN, f : Ω × (0, T) → R is given and Δp denotes the so-called p-Laplacian defined by Δp u : = ∇ ṡ (| ∇ u |p - 2 ∇ u), with initial data u0 ∈ Lr (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, Lr-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.
- Degenerate parabolic equation
- Local existence
- Reflexive Banach space
ASJC Scopus subject areas
- Applied Mathematics