TY - JOUR

T1 - Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

AU - Akagi, Goro

N1 - Funding Information:
1 Partially supported by Shibaura Institute of Technology grant for Project Research, #211459.

PY - 2007/10/15

Y1 - 2007/10/15

N2 - 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.

AB - 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.

KW - Degenerate parabolic equation

KW - Local existence

KW - p-Laplacian

KW - Reflexive Banach space

KW - Subdifferential

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U2 - 10.1016/j.jde.2007.05.009

DO - 10.1016/j.jde.2007.05.009

M3 - Article

AN - SCOPUS:34548487964

SN - 0022-0396

VL - 241

SP - 359

EP - 385

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -