Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation

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.

$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

“…by a formal approach based on spectral analysis. Similar consideration can been found in [12,21]. In this paper, we derive the L ∞ decay estimate like (1.7) for the solutions of problems (1.1) and (1.2).…”

Semilinear Caputo time-fractional pseudo-parabolic equations

“…The existing literature on this topic is so separated that the Problems (P 1 ) and (P 2 ) generalize different diffusion-parabolic equations according to the values of the parameters α, m, and σ. If α = 1, m > 0, σ = 1, Problem (P 1 ) becomes classical pseudo-parabolic problem as follows [49,24,9,55]. In case of N (u) = u p with p ≥ 1, Cao et al [11] consider the following semilinear pseudo-parabolic equation u t − m∆u t − ∆u = u p .…”

“…For the source terms in form of polynomial nonlinearities (without the logarithmic functions), i.e. N p (u) = φ p (u), the model has been considered in [1,4,59,55,47,23].…”

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

“…For the non-local source case, we refer to some very recent related references, e.g. [24] for the threshold results of the global existence and non-existence for the signchanging weak solutions of thin film equation; [14] and [8] for the Kirchhoff type problem with non-local source 1 |Ω| Ω |u| q−1 udx; [15] for the finite time blow-up of solutions with non-positive initial energy J(u 0 ) and non-local source 1 |Ω| Ω |u| q dx with q > 1; [18] for the well-posedness of pseudo-parabolic equation with singular potential term at three initial energy levels, the logarithmic nonlinearity in [7] and the non-local source 1 |Ω| Ω |u| q−1 udx in [30]. As far as we know, there are few research works concerned with the sign-changing solutions for the mixed pseudoparabolic Kirchhoff equation.…”