On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks

Abstract

In this paper, we consider a parabolic-elliptic system of partial differential equations in the three dimensional setting that arises in the study of biological transport networks. We establish the local existence of strong solutions and present a blow-up criterion. We also show that the solutions exist globally in time under the some smallness conditions of initial data and of the source.

A Priori Estimates for a Nonlinear System with Some Essential Symmetrical Structures

“…Remark 1. The blow-up criterion (4) in Theorem 1 is similar to the blow-up criterion of the strong solution of the Cauchy problem (1) and (2) on R 3 [17] and to the Serrin-type criterion of Leray-Hopf weak solution of the Navier-Stokes equations [18].…”

“…To our best knowledge, the Cauchy problems (1) and (2) was only studied in [17] when Ω = R 3 , in which local existence as well as blow-up criterion for large initial data and global existence for small initial data were established based onḢ(R 3 ) being a Hilbert space; however, there is no result on the Cauchy problems (1) and (2) on R 2 . Our goal is to fill this gap.…”

“…Hence, in the absence of a global well-posedness theory, establishing a priori estimates is of major importance for both theoretical and practical purposes, which motivates us to investigate an a priori blow-up criterion and a priori upper bounds of strong solutions to system (1) and (2) with Ω = R 2 . We would like remark that, in contrast to system (1) and (2) with Ω = R 3 ,Ḣ(R 2 ) not being a Hilbert space poses obstacles with adopting the same strategies in [17] to study the Cauchy problems (1) and (2) with Ω = R 2 . To overcome this challenge, we will derive the a priori L ∞ -bound for m to rule out the degeneracy and develop some new dissipation mechanisms hidden in the system (1) and (2) by fully utilizing some essential symmetrical structures of system when Ω = R 2 .…”

“…Compared to the initial-value problem (1) and (2) on R 3 in [17], establishing an a priori blow-up criterion (4) and a priori upper bounds (5) are nontrivial in the sense that the benefit emanating fromḢ(R 3 ) being a Hilbert space will be not granted and that our strategy depends on the weight Hardy inequality over R 2 (see (12)) as well as seeking some new estimates.…”

“…The general case can be dealt with by taking an approximation procedure. We proceed along the lines of the proof of ( [17], Lemma 3.1) and have…”