A semilinear parabolic system with coupling variable exponents

Abstract

This paper deals with semilinear parabolic equations coupled via variable sources, subject to the homogeneous Dirichlet condition in a bounded domain. Since the variable exponents in the sources are just assumed to be positive, the non-linearities may be non-Lipschitz. We establish the existence-uniqueness with comparison principle of local solutions to the regularized problem at first, and then consider the maximal solutions of the original problem as the limits of the solutions of the regularized problem. Some criteria are established for distinguishing global and non-global solutions of the problem, dependent or independent of initial data. Especially, we prove a Fujita type conclusion that the solutions blow up for any non-trivial initial data under certain assumptions on the variable sources and the size of the domain.

Nonnegative solutions to reaction–diffusion system with cross‐diffusion and nonstandard growth conditions

“…The system of semilinear parabolic equations with homogeneous Dirichlet boundary condition and coupled source terms was considered by Bai and Zheng. In addition, they assumed that the sources were positive and non‐Lipschitz and proved that the existence and uniqueness of local solutions to the regularized problem. Moreover, they considered that the limits of the solutions of the regularized problem was the maximal solutions of the original problem.…”

Blowup Properties for Parabolic Equations Coupled via Non-Standard Growth Sources

“…By (19) and 20, we obtain (7). For p + = n + , there is cu q+ (x 0 , t)e n+v(x0,t) ≤ u q+−m+ (x 0 , t)u t (x 0 , t) ≤ Cu q+ (x 0 , t)e n+v(x0,t) .…”

“…have been firstly obtained by Bai and Zheng [19]. Some criteria are established for distinguishing global and non-global solutions of the problem, depending or independent on initial data.…”

A nonlinear Stefan problem with variable exponent and different moving parameters

“…It should be remarked here that paper [16] not only focuses on initial-boundary value problem of (3), but also discusses initial value problem of (3) (Ω = R N ). Since then, many scholars in succession generalize results of [16], and we refer the reader to [5,26] for more references. In contrast to the number of papers available for the case when m(x) = m is a constant and µ 1 = µ 2 and for the case when the domain Ω does not change over time, much less is known when m(x) is a nonnegative function and µ 1 = µ 2 for problem (1).…”