Abstract
This paper is concerned with qualitative properties of the evolutionary p-Laplacian population model with delay. We first establish the existence of solutions of the model by using the method of parabolic regularization and energy estimate and give the uniqueness by a recursive process. Then, combining the upper and lower solution method and the oscillation theory of functional differential equations, we obtain the oscillation of all positive solutions about the positive equilibrium.
Highlights
1 Introduction This paper is concerned with the following evolutionary p-Laplacian with delay:
Using the upper and lower solution method and theory of functional differential equation, the authors showed that all positive solutions of the model oscillate about the positive equilibrium
As far as we know, few works concerned with the oscillation property were obtained for the quasilinear parabolic equations such as a non-Newtonian filtration equation with delay
Summary
This paper is concerned with the following evolutionary p-Laplacian with delay:. |∇u|p– ∇u · n = , (x, t) ∈ ∂ × R+, u(x, t) = η(x, t), (x, t) ∈ × [–τ , ], where ⊂ RN is a bounded domain with smooth boundary, p ≥ , a, b, c, τ > are all constants, m, n > are integers satisfying m < n, < d(t) ∈ C([ , +∞)), and η ∈ L∞( × [–τ , ]) ∩ Lp(–τ , ; W ,p( )) is a nonnegative function satisfying some suitable compatibility conditions. In the last few decades, there are many works on the existence and uniqueness of solutions for parabolic equations with delay(s) (see [ – ] and the references therein). Pao [ , ] discussed the global existence and uniqueness of coupled system of nonlinear parabolic equations with both continuous and discrete delays. ). Using the upper and lower solution method and theory of functional differential equation, the authors showed that all positive solutions of the model oscillate about the positive equilibrium. ). As far as we know, few works concerned with the oscillation property were obtained for the quasilinear parabolic equations such as a non-Newtonian filtration equation with delay. According to a recursive process, we give the uniqueness of the solution Based on these results, we find that, for the non-Newtonian filtration equation, the oscillation phenomenon may occur. Get d dt uk (t) sk(sk + )dm (sk + p – )p αk sk uk (sk + p – )p sk(sk + )dm aαk p–aαk
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