Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

We consider the quasilinear diffusion problem of 𝒖 = (𝑢¹ , … , 𝑢^𝑚) (Formula presented.) for an open set Ω ⊂ ℝ^𝑛, 𝒖₀ ∈ 𝐇^𝑠₀ (Ω) ∶= [𝐻^𝑠₀ (Ω)]^𝑚, for 0 < 𝑠 ≤ 1, and any 𝑇 ∈]0, ∞[. Here, Σ denotes an operator which may involve the distributional Riesz fractional gradient D^𝜎 of order 𝜎, with 0 < 𝜎 < 2s, the classical gradient D¹ = 𝜕 or/and nonlocal derivatives D^𝜎, with 0 < 𝜎 < min{2s, 1}. We show global existence results for various quasilinear diffusion systems in nondivergence form for linear elliptic operators 𝔸, including classical elliptic systems, anisotropic fractional equations and systems, and anisotropic local and nonlocal operators of the following type: (Formula presented.) for coercive, invertible matrices Π and suitable vectorial functions 𝒇. © 2024 Wiley-VCH GmbH.