Abstract
In this paper we consider the time-periodic Allen-Cahn equation subject to homogeneous boundary value condition and time-periodic condition. For the case of a smooth bounded domain with spatial dimension $N\leq 3$ , we prove the existence of classical nontrivial periodic solutions. For the case of a star shaped domain with $N\geq4$ , we prove the nonexistence of nontrivial periodic solutions. For the case of an annulus domain with $N \geq3 $ , we prove the existence of nontrivial radial periodic solutions. Some numerical simulations are also presented to illustrate our results.
Highlights
∂t and proved the existence of positive periodic classical solutions for the case of < q < (N + )/(N – ) under the assumption m(t + ω) = m(t), m ∈ W ,∞[ , ω], inf t∈[ ,ω]
This paper is concerned with the following time-periodic Allen-Cahn equation:∂u – u = m(t) u – u, x∈, t ∈ R+, ( . )∂t u(x, t) =, x ∈ ∂, t ∈ R+, u(x, t + ω) = u(x, t) >, x ∈, t ∈ R+, where ⊂ RN is a domain, ω is a positive constant, m(t) is a positive ω-periodic function
To describe the population dynamics which is sensitive to time-periodic factors, in the present paper, we investigate the time-periodic Allen-Cahn type problem ( . )-( . )
Summary
∂t and proved the existence of positive periodic classical solutions for the case of < q < (N + )/(N – ) under the assumption m(t + ω) = m(t), m ∈ W ,∞[ , ω], inf t∈[ ,ω] For the case of < q < ( N + )/( N – ), under an additional technical assumption on m(t), the existence of positive periodic classical solutions is true. ∂t and obtained the existence of positive periodic solutions under some structural assumptions on m(t) and f (x, u).
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