Abstract
In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.
Highlights
Introduction and Preliminaries e Fisher equation arises in abundance in many fields, including chemistry, biology, and the environment [1–15]
This equation is closely related to biology, applied mathematics, parasites, bacteria, and genes
We refer the reader to the following research papers, see, for example, [17–21]
Summary
We state and prove the local existence result of our problem. Theorem 1. En, ∃t1 t1(t0, b0) ∈ ]t0, T[, and there exists a weak solution Ψ ∈ W(t0, t1) of problem (5), satisfying (6) and (7). (10) have a solution Ψm(t) defined on a maximal interval (see, e.g., [22]). By using Young’s inequality, we obtain ddt Ψm(t) 2L2(Ω) + ∇Ψm(t) 2L2(Ω) ≤ μ Ψm(t) 2L2(Ω) + μ Ψm(t) 3L2(Ω). From H1/2(Ω) ⟶ L3(Ω), using the interpolation between L2(Ω) and H10(Ω), Young’s and Poincare’s inequalities, we obtain μ Ψm(t) 3L3(Ω). By adding up (19) and applying the resulting estimate, we find ddt Ψm(t) 2L2(Ω) + ∇Ψm(t) 2L2(Ω) ≤ μ Ψm(t) 2L2(Ω), ≤ C1 Ψm(t) 2L2(Ω) + C2 Ψm(t) 6L2(Ω) + ∇Ψm (t) 2L2 (Ω) , (20). By (22), (11), and (28), we obtain Ψm∞ m 1 belongs to a bounded set of L∞ t0, t1; H10(Ω). It rests to show that Ψ verifies the initial condition Ψ(t0) b0
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