Universal bounds and blow-up estimates for a reaction-diffusion system.pdf

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Universal bounds and blow-up estimates for a reaction-diffusion system.pdf

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Universal bounds and blow-up estimates for a reaction-diffusion system

Mahmoudi Boundary Value Problems (2015) 2015:228 DOI 10.1186/s13661-015-0491-5 REV I EW Open Access Universal bounds and blow-up estimates for a reaction-diffusion system Nejib Mahmoudi* *Correspondence: mahmoudinejib@yahoo.fr Laboratoire équations aux Dérivées Partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, Tunis, 2092, Tunisia Abstract This paper is concerned with nonnegative solutions of the reaction-diffusion system: ut –u = vp +μ1ur , vt –v = uq +μ2vs . In a suitable range of parameters, we prove (initial and final) blow-up rates, as well as universal bounds for global solutions. This is done in connection with new Liouville-type theorems in a half-space, that we establish. MSC: Primary 35B44; secondary 35K57; 35K58 Keywords: semilinear parabolic systems; reaction-diffusion systems; doubling property; Liouville-type theorem; blow-up rate; universal bound 1 Introduction In this paper, we study (initial and final) blow-up rates, as well as universal bounds for global solutions, for a class of semilinear reaction-diffusion systems, in connection with Liouville-type theorems. Our study is motivated by [?], where Polá?ik et al. developed a general method for obtaining universal initial and final blow-up rates for the scalar equa- tion ut –u = up (p ?), based on rescaling arguments and Liouville-type theorems, com- bined with a key doubling property. In this context, the Liouville-type theoremmeans the nonexistence of nontrivial, nonnegative and bounded solutions defined for all negative and positive times on the whole space Rn, or on a half-space Rn+ = {x ∈Rn;x? ?}. We here consider the system: { ut –u = vp +μ?ur , vt –v = uq +μ?vs, (?) where p,q, r, s ? and μ?,μ? ≥ ?.We use the following notation for the scaling exponents: α = p + ?pq – ? , β = q + ? pq – ? . (?) Let us recall that, even in the scalar case, the optimal exponent for the Liouville-type prop- erty is not presently known (see the