A remark on \(L^{\infty}\) bounds for solutions to quasilinear reaction- diffusion equations (Q1801934)

The objective of this paper is to obtain an a priori \(L^ \infty\) bound for solutions to certain rection-diffusion equations. This is accomplished by transforming the system into a single second order parabolic equation with a nonlocal nonlinearity to which may be applied earlier results of the author [J. Math. Anal. Appl. 138, No. 1, 1-20 (1989; Zbl 0685.35021), and ibid. 160, No. 1, 203-222 (1991; Zbl 0768.35040)]. This approach makes it possible to consider systems which involve nonlocal nonlinearities. The most obvious application of our estimates is to obtain lower bounds for the blow-up time for a system of quasilinear reaction-diffusion equations, irrespective of whether or not the system contains nonlocal terms.