Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity

Abstract

<p style='text-indent:20px;'>In this paper, we consider a partly dissipative reaction-diffusion system with discontinuous nonlinearity in the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} u_t-\Delta u+u+w\in H_0(u-a), \\ w_t-\epsilon(u-\gamma w) = 0, \end{array}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ H_0 $\end{document}</tex-math></inline-formula> is a multi-valued function of Heaviside type. This type of system is used for describing the generation and transmission of electrical signals in neuroscience. We first present an existence result on global solutions. Then, we prove that the system possesses a global attractor having the <inline-formula><tex-math id="M2">\begin{document}$ H^r\times H^r $\end{document}</tex-math></inline-formula>-regularity <inline-formula><tex-math id="M3">\begin{document}$ (0\leq r<2) $\end{document}</tex-math></inline-formula>. Moreover, by showing the Kneser property for the system, the global attractor is proved to be connected. The main characteristic of the system is that the linear part cannot be represented as the subdifferential of a compact-type function.</p>