Existence for reaction diffusion systems. A compactness method approach
The paper deals with the reaction diffusion systems of the form \[ u_ t - \Delta \varphi (u) \in F(u,v), \quad v_ t - \Delta \psi (v) \in G(u,v) \quad \text{in} \quad (0,T) \times \Omega \tag{1} \] \[ \varphi (u) = \psi (v) = 0 \quad \text{on} \quad (0,T) \times \partial \Omega, \quad u(0,x) = u_ 0(x),\quad v(0,x) = v_ 0(x) \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^ n\), \(n \geq 1\), with smooth boundary \(\partial \Omega\); \(\varphi, \psi : \mathbb{R} \to \mathbb{R}\) are continuous and nondecreasing with \(\varphi (0) = \psi (0) = 0\); \(F,G : \mathbb{R}^ 2 \to 2^ \mathbb{R}\) and \(u_ 0\), \(v_ 0 \in L^ \infty (\Omega)\). Results on local and global existence of weak solutions of the system (1) are proved.
local and global existence of weak solutions
scientific article; zbMATH DE number 710338