Journal of Prime Research in Mathematics

# Weakened condition for the stability to solutions of parabolic equations with “maxima”

D. Kolev
Department of Mathematics, University of Chemical Technology and Metallurgy, Sofia, Bulgaria.
T. Donchev
Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore, Pakistan.
K. Nakagawa
Department of Information Science, Tohoku Gakuin University, Sendai 981-3193, Japan.

$$^{1}$$Corresponding Author: mkolev@math.uctm.edu

### Abstract

A class of reaction-diffusion equations with nonlinear reaction terms perturbed with a term containing ”maxima” under initial and boundary conditions is studied. The similar problems that have no ”maxima” have been studied during the last decade by many authors. It would be of interest the standard conditions for the reaction function to be weakened in the sense that the partial derivative of the reaction function, w.r.t. the unknown, to be bounded from above by a rational function containing $$(1 + t) ^{−1}$$ where $$t$$ is the time. When we slightly weaken the standard condition imposed on the reaction function then the solution still decays to zero not necessarily in exponential order. Then we have no exponential stability for the solution of the considered problem. We establish a criterion for the nonexponential stability. The asymptotic behavior of the solutions when $$t → +∞$$ is discussed as well. The parabolic problems with ”maxima” arise in many areas as the theory of automation control, mechanics, nuclear physics, biology and ecology.

#### Keywords:

Reaction-diffusion equation, stability, ”maxima”.