Large diffusivity stability of attractors in the \(C(\Omega)\)-topology for a semilinear reaction and diffusion system of equations (Q851464)

The following weakly coupled system of reaction and diffusion equations \[ \begin{aligned} u_t-\text{div}\bigl(d^\varepsilon(x)\nabla u\bigr) + V^\varepsilon(x)u+ \lambda u & =f(u)\text{ in }\Omega,\\ d^\varepsilon (x)\frac{\partial u}{\partial n}+b^\varepsilon(x)u & =g(u) \text{ on } \Gamma,\\ u(0) & = u^\varepsilon_0\text{ in }\Omega \end{aligned} \] with nonlinear boundary conditions is considered, when \(u= (u_1,\dots,u_m)^T\) is a vector function, \(\Omega\) is an open bounded regular domain in \(\mathbb{R}^N\) with boundary \(\Gamma, \varepsilon>0\) is a given parameter and \(\lambda>0\) is a real number, \(\vec n=(n_1, \dots,n_N)^T\) denotes the unit external vector at the boundary \(\Gamma\), \[ d^\varepsilon\in L^\infty(\Omega;\mathbb{M}_{m \times m}) \quad V^\varepsilon\in L^{p_0}(\Omega;\mathbb{M}_{m \times m})\quad b^\varepsilon\in L^{q_0}(\Omega;\mathbb{M}_{m\times m})\tag{2} \] are diagonal matrices with the first being positive definite; \(p_0\) and \(q_0\) depend on \(N\) respectively. In the last two matrices of (2) the functions are assumed to have at most their integrated values converging as \(\varepsilon\to 0\). First, the existence of at most one mild solution in the space \(C((0,T);H^1(\Omega))\) is proved and the Hölder regularity of solutions is studied. Later the existence of a global compact attractor in the space \(H^1(\Omega)\cap C (\overline\Omega)\) for the nonlinear evolution problem (1), and the stability in the \(C(\overline\Omega)\)-topology of attractors is shown.