A Liouville-type theorem for a system of parabolic inequalities and its applications

In this work, we study a system of parabolic inequalities. Using the method of test functions, we establish an integral estimate and get a Liouville-type theorem for the solutions of this system. As an application, we observe the well-known Fujita blow-up phenomena for the corresponding parabolic system. Here, the importance of this observation is that we do not impose any regularity assumption on the initial data. For general exponents, we also establish the universal Lp estimates and obtain the nonexistence of solutions bounded below by a positive constant. c © 2007 Elsevier Ltd. All rights reserved.