2022/10/11 by Monica Marras, Marras, Monica, Stella Vernier Piro +3 · 1 citation
Mathematics · Medicine · #Mathematical Biology Tumor Growth #MRI in cancer diagnosis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2210.05656
In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begincases ut = Δu - ∇ ⋅ (u f(|∇ v|2 )∇ v) + g(u), amp;
0= Δv -m(t)+ u , ∫Ωv dx=0, amp;
u(x,0)= u0(x), amp; \endcases in Ω× (0,∞), with Ω a ball in ℝN, N≥ 3, under homogeneous Neumann boundary conditions, where g(u)= λu - μuk , λ>0, μ>0, and k >1, f(|∇ v|2 )= kf(1+ |∇ v|2)-α, α>0, which describes gradient-dependent limitation of cross diffusion fluxes. The function m(t) is the time dependent spatial mean of u(x,t) i.e. m(t) := \frac 1 |Ω| ∫Ω u(x,t) dx. Under smallness conditions on α and k, we prove that the solution u(x,t) blows up in L∞-norm at finite time Tmax and for some p>1 it blows up also in Lp-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on α or k, we prove that the solution is global and bounded in time.