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We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

In this article, we consider the predator–prey system with Dirichlet boundary conditions which is used in the modelling of ecology. Under the assumptions of no growth conditions and integrable data, we prove the existence of weak-renormalized solutions to the predator–prey system. [ABSTRACT FROM PUBLISHER]