Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

Catharine W. K. Lo and José Francisco Rodrigues

Mathematische Nachrichten, 2024, vol. 297, issue 6, 2122-2147

Abstract: We consider the quasilinear diffusion problem of u=(u1,…,um)$\bm {u}=(u^1,\ldots ,u^m)$ u′+Π(t,x,u,Σu)Au=f(t,x,u,Σu)in]0,T[×Ω,u=0in]0,T[×Ωc,u(0,·)=u0(·)inΩ$$\begin{equation*} \hspace*{52pt}\begin{cases} \bm{u}'+\Pi(t,x,\bm{u},\Sigma \bm{u})\mathbb{A}\bm{u}=\bm{f}(t,x,\bm{u},\Sigma \bm{u})&\text{ in }]0,T[\times\Omega, \\ \bm{u}=\bm{0}&\text{ in }]0,T[\times\Omega^c, \\ \bm{u}(0,\cdot)=\bm{u}_0(\cdot)&\text{ in }\Omega \end{cases} \end{equation*}$$for an open set Ω⊂Rn$\Omega \subset \mathbb {R}^n$, u0∈H0s(Ω):=[H0s(Ω)]m$\bm {u}_0\in \mathbf {H}^s_0(\Omega):=[H^s_0(\Omega)]^m$, for 0

Date: 2024
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