2023/04/01 by Razvan Gabriel Iagar, Iagar, Razvan Gabriel, Muñoz, Ana Isabel +2
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2304.00269
openalex publication_date 2023/04/01 · openalex created_date 2023/04/06 · openalex updated_date 2026/08/01
We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation ∂tu=Δum+(1+|x|)σup, posed for (x,t)∈\realN×(0,∞), where m>1, p∈(0,1) and σ>0. Initial data are taken to be bounded, non-negative and compactly supported. In the range when m+p≥2, we prove local existence of solutions together with a finite speed of propagation of their supports for compactly supported initial conditions. We also show in this case that, for a given compactly supported initial condition, there exist infinitely many solutions to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range m+p<2, we establish new Aronson-Bénilan estimates satisfied by solutions to the Cauchy problem, which are of independent interest as a priori bounds for the solutions. We apply these estimates to establish infinite speed of propagation of the supports of solutions if m+p<2, that is, u(x,t)>0 for any x∈\realN, t>0, even in the case when the initial condition u0 was compactly supported.