Abstract
Abstract In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.
Highlights
In this paper, we consider the initial boundary value problem of semilinear pseudo-parabolic equation with Neumann boundary condition u t − ∆u ∆ut = |u|p− − Ω |u|p−udx, in Ω × (, T), u(x, ) = u (x) ≢, in Ω, (1.1)−Ω u dx = |Ω| Ω u dx =, in Ω
Xu and Liu [31,32,33,34] studied the initial and boundary value problem of (1.4) for f (u) = up. They proved the existence, asymptotic behavior of the global solutions and global nonexistence of solutions with subcritical and critical initial energy J(u ) ≤ d, and obtained the global nonexistence of solutions with supercritical initial energy J(u ) > d by comparison principle, further estimated the upper bound of the blowup time for supercritical initial energy
(ii) In Section 3, we prove the local existence of solution by the standard Galerkin method. (iii) In Section 4, we prove the existence, uniqueness, decay estimate of the global solution and global nonexistence of solution with J(u ) < d are inspired by [31]. (iv) In Section 5, we extend all the conclusions in Section 4 parallelly to the initial energy J(u ) = d are inspired by [15]. (v) In Section 6, we prove the global nonexistence of solution with J(u ) > by using the two di erent methods in [29, 35, 36], but the global existence and asymptotic behavior of solution at supercritical initial energy level are still open
Summary
We consider the initial boundary value problem of semilinear pseudo-parabolic equation with Neumann boundary condition. Problem (1.1) can be used to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved [1,2,3,4]. The nonlocal term acts to conserve the spatial integral of the unknown function as time evolves. Such equations give insight into biological and chemical problems where conservation properties predominate. The model including this nonlocal term distinguishes the classical heat equation [5,6,7,8,9,10,11,12,13,14,15,16,17], the parabolic systems [18], the fractional Laplacian parabolic
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