Regularity of solutions to systems governing chemotaxis and spreading of epidemics

We’ve made a study of the regularity of solutions to coupled systems of two nonlinear evolution equations with the coefficients dependent on coordinates and unknowns (but not their derivatives). Boundedness (the estimates of the maximum of modulus) and Holder continuity of solutions to both Dirichlet and Neumann problem to such systems have been established. Our study hinges upon a certain generalization of maximum principle for a system of second order differential equations. The main idea is to work with linearly independent combinations of the components of solution, in general, with some functions of unknowns, rather than the components of solution separately. The resulting estimates for them can be resolved to produce the estimates for the unknowns themselves. The systems on issue exhibit maximum principle in such generalized form. If we had systems with constant coefficients, this approach would be nothing else than eigenvalue problem for the matrix of coefficients. To tackle the dependence of coefficients on unknowns and spatial coordinates a kind of freezing of coefficients must be applied. We illustrate the advantages of this approach on the Neumann problem for triangular systems arising in chemotaxis (the movement of living organisms due to the presence of certain chemicals) and epidemics dynamics in the model “susceptibles–infectives-removed”.