2017/05/26 by Severino H. da Silva, da Silva, Severino H., Antônio Ronaldo Gomes Garcia +4
Computer Science · Engineering · Mathematics · #37B25 #45J05 #45M05 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations #math.DS #msc:37B25 #msc:45J05 #msc:45M05
paper · pdf · doi:10.48550/arxiv.1705.09702
20 pages
arxiv created 2017/05/26 · openalex publication_date 2017/05/26 · arxiv updated 2017/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we consider the non local evolution problem \begincases ∂t u(x,t)=-u(x,t)+g(βK(f∘ u)(x,t)+βh), ~x ∈Ω, ~t∈[0,∞[;
u(x,t)=0, ~x∈ℝN∖Ω, ~t∈[0,∞[;
u(x,0)=u0(x),~x∈ℝN, \endcases where Ω is a smooth bounded domain in ℝN, ~g,f: ℝ→ℝ satisfying certain growing condition and K is an integral operator with symmetric kernel, Kv(x)=∫ℝNJ(x,y)v(y)dy. We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has a gradient property.