Global solutions of reaction-diffusion systems.

*(English)*Zbl 0546.35003
Lecture Notes in Mathematics. 1072. Berlin etc.: Springer-Verlag. V, 216 p. DM 31.50; $ 11.10 (1984).

Reaction-diffusion systems occupy by now an important place without nonlinear parabolic equations, both because of their relevance as models in applied sciences, and of the mathematical challenge they often give rise to. Indeed, in contrast to the case of a single parabolic equation, not even a complete classification of such systems is available, so that it may be not clear, at first look, whether a given system enjoys basic properties such as the global existence of the solution. Of course, one can confine his attention to systems whose structure is strongly reminiscent of that of a single equation - but the resulting classes of treatable cases are too small for most practical applications (cf. the quasi-monotone systems, and the systems admitting invariant rectangles).

This monograph takes a different, more realistic way, in that it considers systems for which a ”weak” invariance property holds, expressed by the boundedness of a certain functional of the solution (this is the case for most systems modelling physical and chemical processes). Then the following question arises: ”Can such a weak invariance information be exploited to deduce a bound in a stronger (hopefully uniform) norm, suitable to prove global existence and asymptotic convergence of the solution?”. The work under review contains a detailed answer to such question, accompanied by a clear-cut analysis of equations of the form \[ (1)\quad U_ t=\Delta U+F(x,t,U), \] which includes existence, maximality, local and global \(L^ p\) bounds, under standard and weakened assumptions on the initial data and on the source terms.

The answer to the above question consists of the following steps: (a) select one component, say \(u_ 1\) - then the other components in the corresponding equation are regarded as coefficients, on which only a little a-priori information is assumed: thus \(u_ 1\) satisfies (1) with \[ (2)\quad| F(x,t,U)|\leq c(x,t)(| U|^{\gamma}+1), \] where \(c\in L^{q_ 1,q_ 2}\) (in some cases the estimate (2) can be replaced by a one-sided bound); b) prove that an a-priori bound for \(u_ 1\) in \(L^ r(\Omega)\) (with r ”small”) implies a uniform estimate; c) deduce a uniform bound for all remaining components, so to imply global existence. In the same way even stronger bounds can be derived, which permit to investigate the solution’s asymptotics.

The first part of the book is devoted to eq. (1), and deals with step b). Here two basic approaches are developed (both use in a crucial way Duhamel’s formula and the linearity of the differential part of (1), thus an extension to quasilinear equations is not straightforward). Special mention deserves the ”bootstrap-feedback” method: it allows, for instance, to deal with \[ (3)\quad u_ t-\Delta u\leq u^{\gamma};\quad 0\leq u \] and to prove that if \(\sup\{\| u(.,t)\|_ r\); \(t>0\}\), with \((\gamma -1)N/2r<1\), then \(\sup (\| u(.,t)\|_{\infty}\); \(t>0\}\) is finite, too, and an explicit estimate for the latter quantity is provided; notice that the ”r” in the primary estimate might be less than 1. If \(\gamma =1\) (sublinear case) the above result can be further improved.

Steps a), c) (as well as the specific form of the bound a-priori available for the solution) depend on the system’s structure. They are carried out in the second part of the book, dealing with important and well-known models in chemistry, biology, engineering, for which - in spite of their popularity, no complete results on global existence and asymptotics had been proved before. These are the equations of the ”Brusselator” and other chemical reactions, the diffusive Volterra-Lotka predator-prey model, the Gierer-Meinhardt equations for the onset of patterns in developmental biology, the FitzHugh-Nagumo system in arbitrary dimension, and a supercritical nuclear reactor model with negative feedback. Combining the various methods constructed in the first part and developing a number of original devices, the author proves global existence of the solutions and establishes their asymptotic behaviour.

The proofs, carried out in great detail, reveal a mastery of differential equation techniques, and display quite an amount of ingenious ideas. It should be added that the book is sometimes not easy to read: to find his way in the first part (more than one hundred pages, almost 400 formulae numbered consecutively, essentially no partition in sections) the reader should constantly keep an eye on the Introduction and the table of contents - in this respect, a subdivision in chapters corresponding to the seven theorems proved in this part would have been of great help. The general introduction and the introductory pages of both sections contain a critical presentation of the material developed in the sequel (some of the ideas are sketched in such a concise way that a grain of salt is needed to seize them correctly: cf. the scheme on page 6).

Other authors had previously put forward the same underlying idea of obtaining uniform bounds starting from weaker estimates: cf., after the pioneering work of A. Friedman [Proc. Symp. Appl. Math. 17, 3-23 (1965; Zbl 0192.196)], the more recent contributions of N. D. Alikakos [Commun. Partial Differ. Equations 4, 827-868 (1979; Zbl 0421.35009)], and P. Massatt [Obtaining \(L^{\infty}\)-bounds from \(L^ p\)-bounds for solutions of various parabolic partial differential equations, Manuscr. Symp. Dynam. Systems, Univ. of Florida, Gainesville (1981)]. As compared with these works, the present monograph marks a significant progress: indeed, not only it fills substantial gaps in the existing literaure on reaction-diffusion models, but as well it provides the attentive reader with a wealth of skillful methods and hints that can be of great value for research in related fields.

This monograph takes a different, more realistic way, in that it considers systems for which a ”weak” invariance property holds, expressed by the boundedness of a certain functional of the solution (this is the case for most systems modelling physical and chemical processes). Then the following question arises: ”Can such a weak invariance information be exploited to deduce a bound in a stronger (hopefully uniform) norm, suitable to prove global existence and asymptotic convergence of the solution?”. The work under review contains a detailed answer to such question, accompanied by a clear-cut analysis of equations of the form \[ (1)\quad U_ t=\Delta U+F(x,t,U), \] which includes existence, maximality, local and global \(L^ p\) bounds, under standard and weakened assumptions on the initial data and on the source terms.

The answer to the above question consists of the following steps: (a) select one component, say \(u_ 1\) - then the other components in the corresponding equation are regarded as coefficients, on which only a little a-priori information is assumed: thus \(u_ 1\) satisfies (1) with \[ (2)\quad| F(x,t,U)|\leq c(x,t)(| U|^{\gamma}+1), \] where \(c\in L^{q_ 1,q_ 2}\) (in some cases the estimate (2) can be replaced by a one-sided bound); b) prove that an a-priori bound for \(u_ 1\) in \(L^ r(\Omega)\) (with r ”small”) implies a uniform estimate; c) deduce a uniform bound for all remaining components, so to imply global existence. In the same way even stronger bounds can be derived, which permit to investigate the solution’s asymptotics.

The first part of the book is devoted to eq. (1), and deals with step b). Here two basic approaches are developed (both use in a crucial way Duhamel’s formula and the linearity of the differential part of (1), thus an extension to quasilinear equations is not straightforward). Special mention deserves the ”bootstrap-feedback” method: it allows, for instance, to deal with \[ (3)\quad u_ t-\Delta u\leq u^{\gamma};\quad 0\leq u \] and to prove that if \(\sup\{\| u(.,t)\|_ r\); \(t>0\}\), with \((\gamma -1)N/2r<1\), then \(\sup (\| u(.,t)\|_{\infty}\); \(t>0\}\) is finite, too, and an explicit estimate for the latter quantity is provided; notice that the ”r” in the primary estimate might be less than 1. If \(\gamma =1\) (sublinear case) the above result can be further improved.

Steps a), c) (as well as the specific form of the bound a-priori available for the solution) depend on the system’s structure. They are carried out in the second part of the book, dealing with important and well-known models in chemistry, biology, engineering, for which - in spite of their popularity, no complete results on global existence and asymptotics had been proved before. These are the equations of the ”Brusselator” and other chemical reactions, the diffusive Volterra-Lotka predator-prey model, the Gierer-Meinhardt equations for the onset of patterns in developmental biology, the FitzHugh-Nagumo system in arbitrary dimension, and a supercritical nuclear reactor model with negative feedback. Combining the various methods constructed in the first part and developing a number of original devices, the author proves global existence of the solutions and establishes their asymptotic behaviour.

The proofs, carried out in great detail, reveal a mastery of differential equation techniques, and display quite an amount of ingenious ideas. It should be added that the book is sometimes not easy to read: to find his way in the first part (more than one hundred pages, almost 400 formulae numbered consecutively, essentially no partition in sections) the reader should constantly keep an eye on the Introduction and the table of contents - in this respect, a subdivision in chapters corresponding to the seven theorems proved in this part would have been of great help. The general introduction and the introductory pages of both sections contain a critical presentation of the material developed in the sequel (some of the ideas are sketched in such a concise way that a grain of salt is needed to seize them correctly: cf. the scheme on page 6).

Other authors had previously put forward the same underlying idea of obtaining uniform bounds starting from weaker estimates: cf., after the pioneering work of A. Friedman [Proc. Symp. Appl. Math. 17, 3-23 (1965; Zbl 0192.196)], the more recent contributions of N. D. Alikakos [Commun. Partial Differ. Equations 4, 827-868 (1979; Zbl 0421.35009)], and P. Massatt [Obtaining \(L^{\infty}\)-bounds from \(L^ p\)-bounds for solutions of various parabolic partial differential equations, Manuscr. Symp. Dynam. Systems, Univ. of Florida, Gainesville (1981)]. As compared with these works, the present monograph marks a significant progress: indeed, not only it fills substantial gaps in the existing literaure on reaction-diffusion models, but as well it provides the attentive reader with a wealth of skillful methods and hints that can be of great value for research in related fields.

Reviewer: P.de Mottoni

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B35 | Stability in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35K55 | Nonlinear parabolic equations |

35B45 | A priori estimates in context of PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

92D40 | Ecology |

92Exx | Chemistry |