Global existence for quadratic systems of reaction-diffusion. (English) Zbl 1330.35211
Summary: We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapunov structure of \(L\log L\)-entropy type holds. The approach relies on an a priori dimension-independent \(L^2\)-estimate, valid for a wider class of systems including also some classical Lotka-Volterra systems, and which provides an \(L^1\)-bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of a renormalized solution with defect measure, arising in the theory of kinetic equations.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
reaction-diffusion system; Lotka-Volterra system; weak solutions; renormalized solutions; global existence; entropy methodsReferences:
| [1] | DiPerna, On the Cauchy problem for Boltzmann equations : global existence and weak stability of no, Math pp 130– (1989) · Zbl 0698.45010 |
| [2] | Bisi, From Reactive Boltzmann Equations to Reaction - Diffusion Systems To appear in, Stat Phys · Zbl 1134.82323 |
| [3] | Fitzgibbon, Stability and Lyapunov Functions for Reaction - Diffusion Systems SIAM no, Math Anal pp 28– (1997) |
| [4] | Pierre, Weak solutions and supersolutions in for reaction - diffusion systems no, Evol Equ pp 1– (2003) |
| [5] | Desvillettes, Exponential Decay toward Equilibrium via Entropy Methods for Reaction - Diffusion Equations no, Math Anal Appl pp 319– (2006) · Zbl 1096.35018 |
| [6] | Ladyzenskaya, Linear and Quasi - linear Equations of Parabolic Type Monographs Providence, Trans Math Am Math Soc pp 23– (1968) |
| [7] | Amann, Global existence for semilinear parabolic problems, Reine Angew Math pp 360– (1985) · Zbl 0564.35060 |
| [8] | Baras, Proble mes paraboliques semi - line aires avec donne es mesures, Applicable Analysis 18 pp 111– (1984) · Zbl 0582.35060 · doi:10.1080/00036818408839514 |
| [9] | Pierre, Blowup in reaction - diffusion systems with dissipation of mass electronic, SIAM Rev pp 42– (2000) |
| [10] | Leung, Systems of Nonlinear Partial Differential Equations Kluwer Academic, Publ (1989) |
| [11] | Alexandre, On the Landau approximation in plasma physics Anal Non Line aire no, Inst pp 61– (2004) |
| [12] | Alexandre, On the Boltzmann equation for long - range interactions Pure no, Comm Appl Math pp 55– (2002) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
