On the well-posedness of a nonlinear pseudo-parabolic equation
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Journal of Fixed Point Theory and Applications
On the well-posedness of a nonlinear pseudo-parabolic equation Nguyen Huy Tuan, Vo Van Au, Vo Viet Tri
and Donal O’Regan
Abstract. In this paper we consider the Cauchy problem for the pseudoparabolic equation: ∂ (u + μ(−Δ)s1 u) + (−Δ)s2 u = f (u), x ∈ Ω, t > 0. ∂t Here, the orders s1 , s2 satisfy 0 < s1 = s2 < 1 (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source f is globally Lipschitz. In the case when the source term f satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given. Mathematics Subject Classification. 35A01, 35B40, 35B44, 35K70. Keywords. Pseudo-parabolic equation, existence, regularity, asymptotic behavior, blow-up.
1. Introduction We consider the initial value problem of the pseudo-parabolic equation (PPE for short): ⎫ ∂ ⎬ (u + μ(−Δ)s1 u) + (−Δ)s2 u = f (u), x ∈ Ω, t > 0, ⎪ ∂t (P) u(x, t) = 0, x ∈ ∂Ω, t > 0, ⎪ ⎭ x ∈ Ω, u(x, 0) = u0 (x), where μ > 0, and Ω ⊂ Rd (d ≥ 1) is a bounded domain with smooth boundary ∂Ω and the initial data u0 ∈ L2 (Ω). The operator (−Δ)s with 0 < s < 1 is the fractional Laplacian defined by (see e.g. [7]) u(x, t) − u(z, t) s dz, s ∈ (0, 1), (−Δ) u(x, t) = Cd,s |x − z|d+2s d R with Cd,s =
4s Γ (d/2 + s) , Γ is the Gamma function. π d/2 |Γ (−s)| 0123456789().: V,-vol
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PPEs describe many physical processes, for example, the leakage of liquid through cracks in rocks or materials [3] (here μ is a characteristic of the fissured rock, a decrease of μ corresponds to a reduction in block dimension and an increase in the degree of fissuring), the unidirectional propagation of nonlinear, dispersive, long waves [4,26,27] (where u is typically the amplitude or velocity) and in biology in the aggregation of populations [20] (where u represents the population density). The classical form of the PPE is as follows: ut − μΔut − Δu = 0,
x ∈ Ω, t > 0.
(1)
If μ = 0, (1) is just the homogeneous heat equation, see e.g. [13,22,26] where the existence and uniqueness of solutions and asymptotic behavior and regularities are investigated. In the inhomogeneous case of (1), when f = up , p ≥ 1, in [5], the authors investigate large time behavior of solutions and in [24] the authors prove the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with the initial energy J(u0 ) ≤ d and finite time blow-up with high initial energy J(u0 ) > d. When f = |u|p−2 u we refer the reader to [9,29,31] and the references therein, when f = |u|p u log |u| see [12] which focuses on the initial condition which ensure the solutions exist globally, blow up in finite time or blow up in infinite time and the asymptotic behavior for solutions was considered in [7,10,11,14,15] and the references g(x, ξ)|u|p+1 (ξ)dξ, ξ ∈ Ω we
therein. For the nonlocal source f = |u|p Ω
refer the reader to [16]. Results on the asymptotic
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