Global Existence and Blow-Up for the Fractional p -Laplacian with Logarithmic Nonlinearity
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Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity Tahir Boudjeriou Abstract. In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity ⎧ s p−2 u = |u|p−2 u log(|u|) in Ω, t > 0, ⎨ ut + (−Δ)p u + |u| u=0 in RN \Ω, t > 0, ⎩ in Ω, u(x, 0) = u0 (x), where Ω ⊂ RN (N ≥ 1) is a bounded domain with Lipschitz boundary and 2 ≤ p < ∞. The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian. Mathematics Subject Classification. 35K59, 35K55, 35B40. Keywords. Fractional p-Laplacian, global existence, blow-up, potential well.
1. Introduction and the Main Results This paper is concerned with the following parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity ⎧ ⎨ ut + (−Δ)sp u + |u|p−2 u = |u|p−2 u log(|u|) in Ω, t > 0, (1.1) u=0 in RN \Ω, t > 0, ⎩ in Ω, u(x, 0) = u0 (x), where Ω ⊂ RN (N ≥ 1) is a bounded domain with Lipschitz boundary ∂Ω, u0 = 0 is the initial function on Ω, and (−Δ)sp is the fractional p-Laplacian which is nonlinear nonlocal operator defined on smooth functions by 0123456789().: V,-vol
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(−Δ)sp ϕ(x) = 2 lim ↓0
RN \B (x)
MJOM
|ϕ(x) − ϕ(y)|p−2 (ϕ(x) − ϕ(y)) dy. (1.2) |x − y|N +sp
This definition is consistent, up to a normalization constant depending on N and s. We refer the reader to [1,2] and the references therein for further details on the fractional Laplacian and on the fractional Sobolev spaces. Throughout the paper, without further mentioning, we always assume that s ∈ (0, 1), N > sp and 2 ≤ p < ∞. The interest in studying problems like (1.1) relies not only on mathematical purposes but also on their significance in real models, as explained by Caffarelli [3] and Gilboa et al. [4]. Applebaum et al. [5] stated that the fractional 2-Laplacian operator of the form (−Δ)s , 0 < s < 1, is an infinitesimal generator of stable L´evy processes. In recent years, the global existence and blow-up of solutions to the following model: ⎧ ⎨ ut − Δu = f (u), x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω, t > 0, (M1 ) ⎩ u(x, 0) = u0 (x), x ∈ Ω have been studied extensively by many authors, see, for example, [6–10] and the references therein where the authors have assumed the following conditions on the nonlinearity f (u): (1) f ∈ C 1 (R) and f (0) = f (0) = 0. (2) f is monotone and convex for u > 0, concave for u < 0. (3) (p + 1)F (s) < s sf (s), and |sf (s)| ≤ μF (s), where 2 < p < μ < 2N , F (s) = f (r) dr. N −2 0 Recently, the logarithmic heat equation given by vt = Δv + |v|p−2 log |v|, v
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