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We study the global (in time) existence of nonnegative solutions of the Gierer-Meinhardt system with mixed boundary conditions. In the research, the Robin boundary and Neumann boundary conditions were used on the activator and the inhibitor conditions respectively. Based on the priori estimates of solutions, the considerable results were obtained.

Classical language dynamics explain language shift as a process in which speakers adopt a higher status language in lieu of a lower status language. This is well documented with English having out-competed languages such as Scottish Gaelic, Welsh, and Mandarin. The 1961–1991 Indian censuses report a sharper increase in Hindi/English Bilinguals compared to Monolingual Hindi speakers, suggesting that English is on the rise in India — and is out-competing Hindi. However, the 1991–2011 data show that Bilingual numbers have saturated, while Monolingual Hindi speakers continue to rise exponentially. To capture this counter-intuitive dynamic, we propose a novel language dynamics model of interaction between Monolingual Hindi speakers and Hindi/English Bilinguals, which captures the Indian census data of the last 50 years with greater accuracy, than other well-known language dynamics models. We thus provide a first example of a lower status language having out competed English, a higher status language.