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

Abstract

In this paper, we are concerned with a nonlinear system containing some essential symmetrical structures (e.g., cross-diffusion) in the two-dimensional setting, which is proposed to model the biological transport networks. We first provide an a priori blow-up criterion of strong solution of the corresponding Cauchy problem. Based on this, we also establish a priori upper bounds to strong solution for all positive times.

Global existence of strong solutions to a biological network formulation model in 2+1 dimensions

“…In particular, we do not impose any additional restrictions on the size of the given data except those already enumerated in (H1)-(H3). Thus our theorem is different from the results in [12,17].…”

“…However, the author specifically mentioned that his method there was not applicable to the case where N = 1 or 2. For N = 2 the same initial value problem was considered in [17]. Here the authors obtained a similar blow-up criterion to that in [12] and the global existence of a strong solution under the additional assumptions that α is sufficiently large and γ ≥ 1.…”

“…Since T can be any positive number, the above theorem asserts that blow-up in |m| [17] does not occur in finite time, the result in [20] can be extended to the time-dependent case, and the singular set in [13] is empty when the space dimension N is 2. It also implies that in this case we can extend the local solution given by Theorem 1.7 in [21] in the time direction as far away as we want.…”

Global existence of strong solutions to a biological network formulation model in $2+1$ dimensions

“…However, the author specifically mentioned that his method there was not applicable to the case where N = 1 or 2. If N = 2, the same initial value problem was considered in [15]. Here the authors obtained a similar blow-up criterion to that in [11] and the global existence of a strong solution under the additional assumptions that α is sufficiently large and γ ≥ 1.…”